import Erdos1135.Tao.Renewal.Prop78Case3ActiveTriangle import Erdos1135.Tao.Renewal.Prop78Case3CutoffAlignment /-! # Section 7 Case 3 Q-Relative First-Stopping Support This module records the q-relative first-hit/first-stopping support for Proposition 7.8 Case 3. It minimizes only after the current pivot index `q`. That shape is closer to Tao's recursive stopping times `t_i`, which are first hits after the previous triangle top has been exceeded, rather than first hits from absolute time zero. It is support-only. It preserves the later-window facts around the produced black-pivot candidate and makes source rebase provenance at the minimized first hit explicit. It does not produce source-path data, source `(7.11)`, the source `r`/`t_R` bridge, Lemma 7.9, Lemma 7.10, Tao's many-whites inequality, `(7.41)`, Proposition 7.8, Proposition 7.3/7.1, or Tao's theorem. -/ namespace Erdos1135 namespace Tao /-- A post-exit triangle hit after the old triangle. -/ def TaoSection7Case3AfterTriangleHit (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (old : TaoSection7Triangle) (t : ℕ) : Prop := old.cornerL < (pointAt t).l ∧ ∃ Γ : TaoSection7Triangle, Γ ∈ family ∧ Γ.Mem (pointAt t) /-- The first post-exit triangle hit after the pivot index `q`. This is deliberately relative to `q`; it avoids treating unrelated earlier hits in an absolute `pointAt` path as the next stop of Tao's local iteration. -/ def TaoSection7Case3FirstAfterTriangleHitFrom (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (old : TaoSection7Triangle) (q t : ℕ) : Prop := q < t ∧ TaoSection7Case3AfterTriangleHit pointAt family old t ∧ ∀ s : ℕ, q < s → s < t → ¬ TaoSection7Case3AfterTriangleHit pointAt family old s theorem taoSection7_exists_first_nat_between_of_candidate (Pred : ℕ → Prop) [DecidablePred Pred] {start candidate : ℕ} (hstart : start < candidate) (hcandidate : Pred candidate) : ∃ first : ℕ, start < first ∧ first ≤ candidate ∧ Pred first ∧ ∀ k : ℕ, start < k → k < first → ¬ Pred k := by let S : Finset ℕ := Finset.filter Pred (Finset.Icc (start + 1) candidate) have hnonempty : S.Nonempty := by refine ⟨candidate, ?_⟩ simp [S, hcandidate, Nat.succ_le_iff.mpr hstart] let first : ℕ := S.min' hnonempty have hfirst_mem : first ∈ S := Finset.min'_mem S hnonempty have hfirst_mem_Icc : first ∈ Finset.Icc (start + 1) candidate := (Finset.mem_filter.1 hfirst_mem).1 have hfirst_bounds := Finset.mem_Icc.mp hfirst_mem_Icc have hstart_lt_first : start < first := Nat.lt_of_succ_le hfirst_bounds.1 have hfirst_le_candidate : first ≤ candidate := hfirst_bounds.2 have hfirst_pred : Pred first := (Finset.mem_filter.1 hfirst_mem).2 refine ⟨first, hstart_lt_first, hfirst_le_candidate, hfirst_pred, ?_⟩ intro k hk_start hk_first hpred have hk_mem : k ∈ S := by have hk_lower : start + 1 ≤ k := Nat.succ_le_iff.mpr hk_start have hk_upper : k ≤ candidate := (Nat.le_of_lt hk_first).trans hfirst_le_candidate simp [S, hk_lower, hk_upper, hpred] have hmin_le : first ≤ k := Finset.min'_le S k hk_mem omega theorem taoSection7Case3_afterTriangleHit_of_blackPivotStep {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hstep : taoSection7Case3BlackPivotStep black family pointAt old Γ candidate) : TaoSection7Case3AfterTriangleHit pointAt family old candidate := by exact ⟨hstep.1, Γ, hstep.2.2.1, hstep.2.2.2.1⟩ theorem taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_candidate {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {q candidate : ℕ} (hq_candidate : q < candidate) (hcandidate : TaoSection7Case3AfterTriangleHit pointAt family old candidate) : ∃ first : ℕ, first ≤ candidate ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first := by classical rcases taoSection7_exists_first_nat_between_of_candidate (TaoSection7Case3AfterTriangleHit pointAt family old) hq_candidate hcandidate with ⟨first, hq_first, hfirst_le, hfirst, hminimal⟩ exact ⟨first, hfirst_le, hq_first, hfirst, hminimal⟩ theorem taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_blackPivotStep {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {old Γ : TaoSection7Triangle} {q candidate : ℕ} (hq_candidate : q < candidate) (hstep : taoSection7Case3BlackPivotStep black family pointAt old Γ candidate) : ∃ first : ℕ, first ≤ candidate ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first := taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_candidate hq_candidate (taoSection7Case3_afterTriangleHit_of_blackPivotStep hstep) theorem taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_activeBlackPivotStep {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A q : ℕ} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hq_candidate : q < candidate) (hstep : TaoSection7Case3ActiveBlackPivotStep black family pointAt A q old Γ candidate) : ∃ first : ℕ, first ≤ candidate ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first := taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_blackPivotStep hq_candidate hstep.pivot_step theorem taoSection7Case3_afterTriangleHit_new_triangle {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {t : ℕ} (hhit : TaoSection7Case3AfterTriangleHit pointAt family old t) : ∃ Γ : TaoSection7Triangle, Γ ∈ family ∧ Γ.Mem (pointAt t) ∧ Γ ≠ old := by rcases hhit with ⟨hexit_height, Γ, hΓ, hmemΓ⟩ refine ⟨Γ, hΓ, hmemΓ, ?_⟩ intro hΓ_eq subst hΓ_eq exact taoSection7Case3_not_mem_of_cornerL_lt hexit_height hmemΓ theorem taoSection7Case3_firstAfterTriangleHitFrom_new_triangle {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {q first : ℕ} (hfirst : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first) : ∃ Γ : TaoSection7Triangle, Γ ∈ family ∧ Γ.Mem (pointAt first) ∧ Γ ≠ old := taoSection7Case3_afterTriangleHit_new_triangle hfirst.2.1 theorem taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {q first : ℕ} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hfirst : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first) : black (pointAt first) := by exact (hcover (pointAt first)).mpr hfirst.2.1.2 theorem taoSection7Case3_exists_firstAfterTriangleHitFromWithNewTriangle_le_of_activeBlackPivotStep {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A q : ℕ} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hq_candidate : q < candidate) (hstep : TaoSection7Case3ActiveBlackPivotStep black family pointAt A q old Γ candidate) : ∃ first : ℕ, first ≤ candidate ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first ∧ ∃ Γ' : TaoSection7Triangle, Γ' ∈ family ∧ Γ'.Mem (pointAt first) ∧ Γ' ≠ old := by rcases taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_activeBlackPivotStep hq_candidate hstep with ⟨first, hfirst_le, hfirst⟩ exact ⟨first, hfirst_le, hfirst, taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst⟩ /-- Source-data-level wrapper: the current source-path data and active-triangle state produce a bounded later candidate, and from that candidate a first post-pivot triangle hit no later than the candidate. -/ theorem taoSection7Case3_exists_firstAfterTriangleHitFromWithNewTriangle_of_sourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) ∧ ¬ W candidate ∧ taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P ∧ candidate - q ≤ pres.length ∧ ∃ first : ℕ, first ≤ candidate ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first ∧ ∃ Γ' : TaoSection7Triangle, Γ' ∈ family ∧ Γ'.Mem (pointAt first) ∧ Γ' ≠ old := by rcases taoSection7Case3_exists_activeBlackPivotStep_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, Γ, hstep⟩ have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict omega rcases taoSection7Case3_exists_firstAfterTriangleHitFromWithNewTriangle_le_of_activeBlackPivotStep hq_candidate hstep with ⟨first, hfirst_le, hfirst, hnew⟩ exact ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, first, hfirst_le, hfirst, hnew⟩ /-- Source-data-level wrapper with the blackness of the minimized first hit exposed. This uses the bidirectional `TaoSection7TriangleFamilyCoverBlack` hypothesis; it does not assume blackness separately at `first`. -/ theorem taoSection7Case3_exists_firstAfterTriangleHitFromWithBlack_of_sourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) ∧ ¬ W candidate ∧ taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P ∧ candidate - q ≤ pres.length ∧ ∃ first : ℕ, first ≤ candidate ∧ first < P ∧ taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) ∧ TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first ∧ ∃ Γ' : TaoSection7Triangle, Γ' ∈ family ∧ Γ'.Mem (pointAt first) ∧ Γ' ≠ old := by rcases taoSection7Case3_exists_firstAfterTriangleHitFromWithNewTriangle_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, first, hfirst_le, hfirst, hnew⟩ have hfirst_ltP : first < P := lt_of_le_of_lt hfirst_le hltP have hblack : taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst exact ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, first, hfirst_le, hfirst_ltP, hblack, hfirst, hnew⟩ /-- Accepted transition at the minimized post-pivot first hit. This is narrower than `TaoSection7Case3AcceptedTriangleTransition`: the transition target is the first triangle hit after `q`, not merely the later black candidate used to prove existence. -/ structure TaoSection7Case3FirstHitAcceptedTransition (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) (q : ℕ) (old Γ : TaoSection7Triangle) (first : ℕ) : Prop where first_hit : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first first_black : black (pointAt first) new_mem_family : Γ ∈ family new_mem : Γ.Mem (pointAt first) new_ne_old : Γ ≠ old accepted : Accepts q old first Γ theorem taoSection7Case3_exists_firstHitAcceptedTransition_of_firstHitFrom {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old : TaoSection7Triangle} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hfirst : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first) (haccept : ∀ Γ : TaoSection7Triangle, Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → black (pointAt first) → Accepts q old first Γ) : ∃ Γ : TaoSection7Triangle, TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first := by have hblack : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst rcases taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst with ⟨Γ, hΓ, hmem, hne⟩ exact ⟨Γ, hfirst, hblack, hΓ, hmem, hne, haccept Γ hΓ hmem hne hblack⟩ /-- Package a q-relative first-hit accepted transition as the start-based strict bounded advance consumed by the Case 3 one-step event adapter. The start-based recurrence step and budget remain explicit hypotheses; they are not consequences of first-hit acceptance alone. -/ theorem taoSection7Case3_strictBoundedAdvance_of_firstHitAcceptedTransition {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {B : ℕ → ℕ} {Step : ℕ → ℕ → Prop} {start threshold q first : ℕ} {old Γ : TaoSection7Triangle} (hq : q ∈ Finset.Icc start (start + threshold)) (hacc : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first) (hstep : Step start first) (hbound : first ≤ B start) : ∃ next : ℕ, taoSection7Case3StrictBoundedAdvance B Step start next := by refine ⟨first, hstep, ?_, hbound⟩ exact lt_of_le_of_lt (Finset.mem_Icc.mp hq).1 hacc.first_hit.1 /-- Source-data-level accepted-transition wrapper at the minimized first hit. The caller still supplies the actual stopping/iteration acceptance theorem; this wrapper only ensures that theorem is applied to the q-relative first post-pivot triangle hit and carries the candidate bound that produced it. -/ theorem taoSection7Case3_exists_firstHitAcceptedTransition_of_sourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) ∧ ¬ W candidate ∧ taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P ∧ candidate - q ≤ pres.length ∧ ∃ first Γ, first ≤ candidate ∧ first < P ∧ TaoSection7Case3FirstHitAcceptedTransition (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts q old Γ first := by rcases taoSection7Case3_exists_firstAfterTriangleHitFromWithBlack_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, first, hfirst_le, hfirst_ltP, hblack, hfirst, hnew⟩ rcases hnew with ⟨Γ, hΓ, hΓmem, hΓne⟩ refine ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, first, Γ, hfirst_le, hfirst_ltP, ?_⟩ exact ⟨hfirst, hblack, hΓ, hΓmem, hΓne, haccept first Γ hfirst_ltP hblack hfirst hΓ hΓmem hΓne⟩ /-- Active black-pivot package that retains the later-search/window facts returned by the source-data producer. This avoids treating the bare active step as if it still remembered the candidate bounds needed by later first-stopping and `t_R <= P` plumbing. -/ structure TaoSection7Case3ActiveBlackPivotStepInLaterWindow (W : ℕ → Prop) (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (A threshold P q sourceLength : ℕ) (old Γ : TaoSection7Triangle) (candidate : ℕ) : Prop where mem_later_window : candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) not_white : ¬ W candidate strict_exit : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate candidate_ltP : candidate < P source_length_bound : candidate - q ≤ sourceLength active_step : TaoSection7Case3ActiveBlackPivotStep black family pointAt A q old Γ candidate theorem taoSection7Case3_activeBlackPivotStepInLaterWindow_q_lt_candidate {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A threshold P q sourceLength : ℕ} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindow W black family pointAt A threshold P q sourceLength old Γ candidate) : q < candidate := by have hstrict := hstep.strict_exit dsimp [taoSection7Case3HeightExitBound] at hstrict omega /-- Generic-scale version of `TaoSection7Case3ActiveBlackPivotStepInLaterWindow`. The later-window and strict-exit facts are measured with the caller-supplied `gapBound`, while the active triangle threshold is measured with the caller-supplied real `bound`. -/ structure TaoSection7Case3ActiveBlackPivotStepInLaterWindowWithBound (W : ℕ → Prop) (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (bound : ℕ → ℝ) (gapBound : ℕ → ℕ) (threshold P q sourceLength : ℕ) (old Γ : TaoSection7Triangle) (candidate : ℕ) : Prop where mem_later_window : candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound q) (taoSection7Case3LaterSearchStart gapBound q + threshold) not_white : ¬ W candidate strict_exit : taoSection7Case3HeightExitBound gapBound q < candidate candidate_ltP : candidate < P source_length_bound : candidate - q ≤ sourceLength active_step : TaoSection7Case3ActiveBlackPivotStepWithBound black family pointAt bound q old Γ candidate theorem taoSection7Case3_activeBlackPivotStepInLaterWindowWithBound_q_lt_candidate {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {threshold P q sourceLength : ℕ} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindowWithBound W black family pointAt bound gapBound threshold P q sourceLength old Γ candidate) : q < candidate := by have hstrict := hstep.strict_exit dsimp [taoSection7Case3HeightExitBound] at hstrict omega theorem taoSection7Case3_exists_activeBlackPivotStepInLaterWindow_of_sourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate := by rcases taoSection7Case3_exists_activeBlackPivotStep_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, Γ, hstep⟩ exact ⟨candidate, Γ, hmem, hnotW, hstrict, hltP, hlen, hstep⟩ theorem taoSection7Case3_exists_activeBlackPivotStepInLaterWindow_of_sourcePathData_scale (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindowWithBound W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt bound gapBound threshold P q pres.length old Γ candidate := by rcases taoSection7Case3_exists_activeBlackPivotStep_of_sourcePathData_scale (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover with ⟨candidate, hmem, hnotW, hstrict, hltP, hlen, Γ, hstep⟩ exact ⟨candidate, Γ, hmem, hnotW, hstrict, hltP, hlen, hstep⟩ theorem taoSection7Case3_exists_firstHitAcceptedTransition_of_activeBlackPivotStepInLaterWindow {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {A threshold P q sourceLength : ℕ} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindow W black family pointAt A threshold P q sourceLength old Γ candidate) (haccept : ∀ first Γ', first ≤ candidate → first < P → black (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ' ∈ family → Γ'.Mem (pointAt first) → Γ' ≠ old → Accepts q old first Γ') : ∃ first Γ', first ≤ candidate ∧ first < P ∧ TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ' first := by have hq_candidate : q < candidate := taoSection7Case3_activeBlackPivotStepInLaterWindow_q_lt_candidate hstep rcases taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_activeBlackPivotStep hq_candidate hstep.active_step with ⟨first, hfirst_le, hfirst⟩ have hfirst_ltP : first < P := lt_of_le_of_lt hfirst_le hstep.candidate_ltP have hblack : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst rcases taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst with ⟨Γ', hΓ', hmem, hne⟩ exact ⟨first, Γ', hfirst_le, hfirst_ltP, hfirst, hblack, hΓ', hmem, hne, haccept first Γ' hfirst_le hfirst_ltP hblack hfirst hΓ' hmem hne⟩ /-- Source provenance for the minimized q-relative first hit. This is intentionally stronger than the current source-path data interface: the point equality and suffix-length bound are recorded at the minimized `first`, not only at the later candidate used to prove existence. -/ structure TaoSection7Case3FirstHitSourceProvenance (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (j : ℕ+) (s : ℕ) (pre : List ℕ) (pres : List (List ℕ)) (P q : ℕ) (old Γ : TaoSection7Triangle) (candidate first : ℕ) : Prop where first_le_candidate : first ≤ candidate first_ltP : first < P first_source : pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint first_source_length : first - q ≤ pres.length first_hit : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first first_black : black (pointAt first) new_mem_family : Γ ∈ family new_mem : Γ.Mem (pointAt first) new_ne_old : Γ ≠ old theorem taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindow {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A threshold P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindow W black family pointAt A threshold P q pres.length old Γ candidate) (hfirst_source : ∀ first : ℕ, q < first → first ≤ candidate → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ' candidate first := by have hq_candidate : q < candidate := taoSection7Case3_activeBlackPivotStepInLaterWindow_q_lt_candidate hstep rcases taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_activeBlackPivotStep hq_candidate hstep.active_step with ⟨first, hfirst_le, hfirst⟩ have hfirst_ltP : first < P := lt_of_le_of_lt hfirst_le hstep.candidate_ltP have hblack : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst have hsource : pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint := hfirst_source first hfirst.1 hfirst_le have hlen : first - q ≤ pres.length := by exact (Nat.sub_le_sub_right hfirst_le q).trans hstep.source_length_bound rcases taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst with ⟨Γ', hΓ', hmem, hne⟩ exact ⟨first, Γ', hfirst_le, hfirst_ltP, hsource, hlen, hfirst, hblack, hΓ', hmem, hne⟩ theorem taoSection7Case3_firstHit_source_length_of_ltP {P q first : ℕ} {pres : List (List ℕ)} (hfirst_ltP : first < P) (hcoverage : taoSection7SourceSuffixCoversWindow q P pres) : first - q ≤ pres.length := taoSection7_candidate_sub_le_sourceSuffixLength_of_lt_window hfirst_ltP hcoverage /-- Variant of first-hit source provenance where the suffix-length bound at `first` comes from the whole-window suffix coverage, not from the later candidate. The only genuinely new source-facing premise is the point equality at the minimized first hit for post-`q` points below `P`. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindow_coverage {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A threshold P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindow W black family pointAt A threshold P q pres.length old Γ candidate) (hcoverage : taoSection7SourceSuffixCoversWindow q P pres) (hfirst_source : ∀ first : ℕ, q < first → first < P → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ' candidate first := by have hq_candidate : q < candidate := taoSection7Case3_activeBlackPivotStepInLaterWindow_q_lt_candidate hstep rcases taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_activeBlackPivotStep hq_candidate hstep.active_step with ⟨first, hfirst_le, hfirst⟩ have hfirst_ltP : first < P := lt_of_le_of_lt hfirst_le hstep.candidate_ltP have hblack : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst have hsource : pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint := hfirst_source first hfirst.1 hfirst_ltP have hlen : first - q ≤ pres.length := taoSection7Case3_firstHit_source_length_of_ltP hfirst_ltP hcoverage rcases taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst with ⟨Γ', hΓ', hmem, hne⟩ exact ⟨first, Γ', hfirst_le, hfirst_ltP, hsource, hlen, hfirst, hblack, hΓ', hmem, hne⟩ /-- Generic-scale first-hit provenance from a later-window active black pivot. This is the first-stopping analogue of the generic active-transition package: it keeps the minimized first hit tied to the source path while preserving the caller-supplied `bound/gapBound` window data. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindowWithBound_coverage {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {threshold P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate : ℕ} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3ActiveBlackPivotStepInLaterWindowWithBound W black family pointAt bound gapBound threshold P q pres.length old Γ candidate) (hcoverage : taoSection7SourceSuffixCoversWindow q P pres) (hfirst_source : ∀ first : ℕ, q < first → first < P → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ' candidate first := by have hq_candidate : q < candidate := taoSection7Case3_activeBlackPivotStepInLaterWindowWithBound_q_lt_candidate hstep rcases taoSection7Case3_exists_firstAfterTriangleHitFrom_le_of_blackPivotStep hq_candidate hstep.active_step.pivot_step with ⟨first, hfirst_le, hfirst⟩ have hfirst_ltP : first < P := lt_of_le_of_lt hfirst_le hstep.candidate_ltP have hblack : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hfirst have hsource : pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint := hfirst_source first hfirst.1 hfirst_ltP have hlen : first - q ≤ pres.length := taoSection7Case3_firstHit_source_length_of_ltP hfirst_ltP hcoverage rcases taoSection7Case3_firstAfterTriangleHitFrom_new_triangle hfirst with ⟨Γ', hΓ', hmem, hne⟩ exact ⟨first, Γ', hfirst_le, hfirst_ltP, hsource, hlen, hfirst, hblack, hΓ', hmem, hne⟩ /-- Source-data wrapper for first-hit provenance with an explicit wider source equality premise for the minimized hit. The current `TaoSection7Case3SourcePathData` fields cover the later-search candidate. This wrapper shows the extra hypothesis needed to rebase at the minimized first hit without assuming it lies in the later-search block. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfirst_source : ∀ candidate first : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → ¬ W candidate → taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate → candidate < P → candidate - q ≤ pres.length → q < first → first ≤ candidate → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ' candidate first := by rcases taoSection7Case3_exists_activeBlackPivotStepInLaterWindow_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, Γ, hstep⟩ have hfirst_source_local : ∀ first : ℕ, q < first → first ≤ candidate → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint := by intro first hq_first hfirst_le exact hfirst_source candidate first hstep.mem_later_window hstep.not_white hstep.strict_exit hstep.candidate_ltP hstep.source_length_bound hq_first hfirst_le rcases taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindow hcover hstep hfirst_source_local with ⟨first, Γ', hfirstProv⟩ exact ⟨candidate, Γ, hstep, first, Γ', hfirstProv⟩ /-- Source-data wrapper with the sharper source-facing obligation: to rebase at the minimized first hit, it is enough to identify `pointAt first` with the source path for every post-`q` offset below `P`. The finite suffix-length bound is then supplied by `TaoSection7Case3SourcePathData.coverage`. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_coverage (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfirst_source : ∀ first : ℕ, q < first → first < P → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ' candidate first := by rcases taoSection7Case3_exists_activeBlackPivotStepInLaterWindow_of_sourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover with ⟨candidate, Γ, hstep⟩ rcases taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindow_coverage hcover hstep hdata.coverage hfirst_source with ⟨first, Γ', hfirstProv⟩ exact ⟨candidate, Γ, hstep, first, Γ', hfirstProv⟩ /-- Generic-scale source-data wrapper for first-hit provenance. This is the checked stopping socket for the source-pivot route before the outer strict-advance producer: it consumes the base-aware/generic source data and returns a later-window active black pivot plus minimized first-hit source provenance under the same `bound/gapBound`. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfirst_source : ∀ first : ℕ, q < first → first < P → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindowWithBound W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt bound gapBound threshold P q pres.length old Γ candidate ∧ ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ' candidate first := by rcases taoSection7Case3_exists_activeBlackPivotStepInLaterWindow_of_sourcePathData_scale (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover with ⟨candidate, Γ, hstep⟩ rcases taoSection7Case3_exists_firstHitSourceProvenance_of_activeBlackPivotStepInLaterWindowWithBound_coverage hcover hstep hdata.coverage hfirst_source with ⟨first, Γ', hfirstProv⟩ exact ⟨candidate, Γ, hstep, first, Γ', hfirstProv⟩ /-- Start-based strict-advance socket from generic first-hit source provenance plus explicit start-step/budget compatibility. This is still not the final `hadvance` producer: the caller must prove the actual recurrence step and budget for the minimized first hit. -/ theorem taoSection7Case3_exists_strictBoundedAdvance_of_sourcePathData_scale_coverage_startStepBudget (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {bound : ℕ → ℝ} {gapBound B : ℕ → ℕ} {Step : ℕ → ℕ → Prop} {P start threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hq : q ∈ Finset.Icc start (start + threshold)) (hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfirst_source : ∀ first : ℕ, q < first → first < P → pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) (haccept : ∀ candidate first Γ, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first → Accepts q old first Γ) (hcompat : ∀ candidate first Γ, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first → Accepts q old first Γ → Step start first ∧ first ≤ B start) : ∃ next : ℕ, taoSection7Case3StrictBoundedAdvance B Step start next := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfirst_source with ⟨candidate, _candidateΓ, _hwindow, first, Γ, hsrc⟩ have haccepted : Accepts q old first Γ := haccept candidate first Γ hsrc have hacc : TaoSection7Case3FirstHitAcceptedTransition (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts q old Γ first := ⟨hsrc.first_hit, hsrc.first_black, hsrc.new_mem_family, hsrc.new_mem, hsrc.new_ne_old, haccepted⟩ rcases hcompat candidate first Γ hsrc haccepted with ⟨hstep, hbound⟩ exact taoSection7Case3_strictBoundedAdvance_of_firstHitAcceptedTransition hq hacc hstep hbound /-- The later-search right endpoint controls the minimized first hit whenever the producing candidate lies in the later-search window. This is only budget arithmetic; it does not choose the recurrence relation `Step`. -/ theorem taoSection7Case3_first_le_startBudget_of_laterWindow_rightBound {B gapBound : ℕ → ℕ} {start threshold q candidate first : ℕ} (hmono : Monotone gapBound) (hq : q ∈ Finset.Icc start (start + threshold)) (hmem : candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound q) (taoSection7Case3LaterSearchStart gapBound q + threshold)) (hfirst : first ≤ candidate) (hright : taoSection7Case3LaterSearchBound gapBound threshold (start + threshold) ≤ B start) : first ≤ B start := by have hcandidate_le : candidate ≤ taoSection7Case3LaterSearchBound gapBound threshold q := by simpa [taoSection7Case3LaterSearchBound] using (Finset.mem_Icc.mp hmem).2 exact hfirst.trans (hcandidate_le.trans ((taoSection7Case3_laterSearchBound_le_right_of_mem_window hmono hq).trans hright)) /-- Producer-facing source-path data with full post-`q` point equality on the whole `P` window. The current production `TaoSection7Case3SourcePathData` only asks for point equality on the later-search block. This stronger package is the shape needed if the minimized q-relative first hit may occur before that block but still needs to be used as a source-path rebase point. -/ structure TaoSection7Case3FullWindowSourcePathData (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J A : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold q : ℕ) (j : ℕ+) (s : ℕ) (pre : List ℕ) (pres : List (List ℕ)) : Prop where coverage : taoSection7SourceSuffixCoversWindow q P pres point_q : pointAt q = (taoSection7SourceHitPoint j s pre).toPoint point_window : ∀ t : ℕ, q < t → t < P → pointAt t = (taoSection7SourceHitPathPoint j s pre pres (t - q)).toPoint white_iff_later : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate)) domain_later : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (pointAt candidate) /-- Generic-gap full-window source-path data for the stateful Case 3 recurrence route. This is the full-window analogue of `TaoSection7Case3SourcePathDataWithGap`; it keeps enough point equality to rebase at the minimized first hit, while the later-search color/domain facts use the caller-supplied `gapBound`. -/ structure TaoSection7Case3FullWindowSourcePathDataWithGap (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold q : ℕ) (gapBound : ℕ → ℕ) (j : ℕ+) (s : ℕ) (pre : List ℕ) (pres : List (List ℕ)) : Prop where coverage : taoSection7SourceSuffixCoversWindow q P pres point_q : pointAt q = (taoSection7SourceHitPoint j s pre).toPoint point_window : ∀ t : ℕ, q < t → t < P → pointAt t = (taoSection7SourceHitPathPoint j s pre pres (t - q)).toPoint white_iff_later : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound q) (taoSection7Case3LaterSearchStart gapBound q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate)) domain_later : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound q) (taoSection7Case3LaterSearchStart gapBound q + threshold) → taoSection7SourcePointInDomain J (pointAt candidate) theorem TaoSection7Case3FullWindowSourcePathData.to_sourcePathData {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres := by refine { coverage := hfull.coverage point_q := hfull.point_q point_candidate := ?_ white_iff := hfull.white_iff_later domain := hfull.domain_later } intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega exact hfull.point_window candidate hq_candidate hstrict_ltP.2 theorem TaoSection7Case3FullWindowSourcePathDataWithGap.to_sourcePathDataWithGap {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := by refine { coverage := hfull.coverage point_q := hfull.point_q point_candidate := ?_ white_iff := hfull.white_iff_later domain := hfull.domain_later } intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound gapBound q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega exact hfull.point_window candidate hq_candidate hstrict_ltP.2 /-- Current horizontal scale for Lemma 7.10 at a Section 7 point. -/ def taoSection7Lemma710CurrentM (n : ℕ) (p : TaoSection7Point) : ℝ := ((n / 2 - (p.j : ℕ) : ℕ) : ℝ) /-- Nat-subtraction nontruncation for Tao's current horizontal scale. -/ theorem taoSection7Lemma710CurrentMNat_add_currentJ_eq_cutoff {cutoff : ℕ} {p : TaoSection7Point} (hp : taoSection7SourcePointInDomain cutoff p) : (p.j : ℕ) + (cutoff - (p.j : ℕ)) = cutoff := Nat.add_sub_of_le hp /-- Real form of the nontruncation identity for `taoSection7Lemma710CurrentM`. -/ theorem taoSection7Lemma710CurrentM_add_currentJ_eq_cutoff {n : ℕ} {p : TaoSection7Point} (hp : taoSection7SourcePointInDomain (n / 2) p) : ((p.j : ℕ) : ℝ) + taoSection7Lemma710CurrentM n p = ((n / 2 : ℕ) : ℝ) := by unfold taoSection7Lemma710CurrentM exact_mod_cast taoSection7Lemma710CurrentMNat_add_currentJ_eq_cutoff hp /-- Current-`m` wrapper for Tao's `(7.52)` height bound from a named real right-edge strip hypothesis. -/ theorem taoSection7Lemma710_verticalDepth_le_log9_div_log2_mul_currentM_of_domain {n : ℕ} {old : TaoSection7Triangle} {p : TaoSection7Point} (hmem : old.Mem p) (hdomain : taoSection7SourcePointInDomain (n / 2) p) (hright : TaoSection7TriangleRightEdgeInStrip ((n / 2 : ℕ) : ℝ) old) : ((old.verticalDepth p : ℤ) : ℝ) ≤ (Real.log 9 / Real.log 2) * taoSection7Lemma710CurrentM n p := by refine TaoSection7Triangle.verticalDepth_le_log9_div_log2_mul_rightGap_of_mem_rightEdge (old := old) (p := p) (S := ((old.verticalDepth p : ℤ) : ℝ)) (M := taoSection7Lemma710CurrentM n p) (cutoff := ((n / 2 : ℕ) : ℝ)) hmem rfl hright ?_ unfold taoSection7Lemma710CurrentM dsimp [TaoSection7Point.jReal] rw [Nat.cast_sub hdomain] /-- Active-triangle form of the `(7.52)` right-edge current-`m` bound. -/ theorem taoSection7Lemma710_verticalDepth_le_log9_div_log2_mul_currentM_of_active_rightEdge {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n A q : ℕ} {old : TaoSection7Triangle} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hqdom : taoSection7SourcePointInDomain (n / 2) (pointAt q)) (hright : TaoSection7TriangleFamilyRightEdgeInStrip ((n / 2 : ℕ) : ℝ) family) : ((old.verticalDepth (pointAt q) : ℤ) : ℝ) ≤ (Real.log 9 / Real.log 2) * taoSection7Lemma710CurrentM n (pointAt q) := taoSection7Lemma710_verticalDepth_le_log9_div_log2_mul_currentM_of_domain hactive.pivot_mem hqdom (TaoSection7Case3ActiveTriangleAt.rightEdgeInStrip hactive hright) /-- Current source-scale controls for Tao Lemma 7.10. This packages the current gap `M = floor(n/2)-j`, the old crossing height `S`, the log-domain/lower-scale hypotheses supplied by later source work, and the checked `(7.52)` upper scale bound. It is not endpoint production. -/ structure TaoSection7Lemma710CurrentScaleControls (n : ℕ) (old : TaoSection7Triangle) (p : TaoSection7Point) (M S : ℝ) : Prop where M_def : M = taoSection7Lemma710CurrentM n p S_def : S = ((old.verticalDepth p : ℤ) : ℝ) domain : taoSection7SourcePointInDomain (n / 2) p M_large : 1 < M logM_pos : 0 < Real.log M S_nonneg : 0 ≤ S hMlower : M / (Real.log M)^2 ≤ S upper_752 : S ≤ (Real.log 9 / Real.log 2) * M theorem taoSection7Lemma710CurrentScaleControls_of_domain_rightEdge {n : ℕ} {old : TaoSection7Triangle} {p : TaoSection7Point} {M S : ℝ} (hmem : old.Mem p) (hdomain : taoSection7SourcePointInDomain (n / 2) p) (hright : TaoSection7TriangleRightEdgeInStrip ((n / 2 : ℕ) : ℝ) old) (hM : M = taoSection7Lemma710CurrentM n p) (hS : S = ((old.verticalDepth p : ℤ) : ℝ)) (hMlarge : 1 < M) (hMlower : M / (Real.log M)^2 ≤ S) : TaoSection7Lemma710CurrentScaleControls n old p M S := by refine { M_def := hM S_def := hS domain := hdomain M_large := hMlarge logM_pos := Real.log_pos hMlarge S_nonneg := ?_ hMlower := hMlower upper_752 := ?_ } · rw [hS] exact_mod_cast TaoSection7Triangle.verticalDepth_nonneg_of_mem hmem · rw [hS, hM] exact taoSection7Lemma710_verticalDepth_le_log9_div_log2_mul_currentM_of_domain hmem hdomain hright theorem taoSection7Lemma710CurrentScaleControls_of_active_rightEdge {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n A q : ℕ} {old : TaoSection7Triangle} {M S : ℝ} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hqdom : taoSection7SourcePointInDomain (n / 2) (pointAt q)) (hright : TaoSection7TriangleFamilyRightEdgeInStrip ((n / 2 : ℕ) : ℝ) family) (hM : M = taoSection7Lemma710CurrentM n (pointAt q)) (hS : S = ((old.verticalDepth (pointAt q) : ℤ) : ℝ)) (hMlarge : 1 < M) (hMlower : M / (Real.log M)^2 ≤ S) : TaoSection7Lemma710CurrentScaleControls n old (pointAt q) M S := taoSection7Lemma710CurrentScaleControls_of_domain_rightEdge hactive.pivot_mem hqdom (TaoSection7Case3ActiveTriangleAt.rightEdgeInStrip hactive hright) hM hS hMlarge hMlower namespace TaoSection7Lemma710CurrentScaleControls /-- Consume the current-scale ledger to convert an `S^(3/5)` horizontal error to `(C*M)^(3/5)`. -/ theorem jerr_le_log9_div_log2_mul_currentM_rpow {n : ℕ} {old : TaoSection7Triangle} {p : TaoSection7Point} {M S Jerr CJ : ℝ} (hscale : TaoSection7Lemma710CurrentScaleControls n old p M S) (hCJ0 : 0 ≤ CJ) (hJ : Jerr ≤ CJ * S ^ (3 / 5 : ℝ)) : Jerr ≤ CJ * ((Real.log 9 / Real.log 2) * M) ^ (3 / 5 : ℝ) := TaoSection7Lemma710.jerr_mpow_of_s_rpow_bound_of_upper752 hscale.S_nonneg hCJ0 hscale.upper_752 hJ /-- Consumer form that leaves the constant absorption from `((log 9 / log 2) * M)^(3/5)` to `M^(3/5)` explicit. -/ theorem jerr_le_mpow_of_s_rpow_bound_absorb {n : ℕ} {old : TaoSection7Triangle} {p : TaoSection7Point} {M S Jerr CJ CJm : ℝ} (hscale : TaoSection7Lemma710CurrentScaleControls n old p M S) (hCJ0 : 0 ≤ CJ) (hJ : Jerr ≤ CJ * S ^ (3 / 5 : ℝ)) (habsorb : CJ * ((Real.log 9 / Real.log 2) * M) ^ (3 / 5 : ℝ) ≤ CJm * M ^ (3 / 5 : ℝ)) : Jerr ≤ CJm * M ^ (3 / 5 : ℝ) := le_trans (hscale.jerr_le_log9_div_log2_mul_currentM_rpow hCJ0 hJ) habsorb /-- Current-scale consumer for the scalar row-margin premise. The horizontal estimate may be supplied in the old-height scale `S`; the conversion to an `M^(3/5)` bound and final large-`m` absorption remain explicit. -/ theorem error_margin_of_s_rpow_bound_absorb {n : ℕ} {old : TaoSection7Triangle} {p : TaoSection7Point} {M S B Lerr Jerr CJS CJm CL CB : ℝ} (hscale : TaoSection7Lemma710CurrentScaleControls n old p M S) (hCL_nonneg : 0 ≤ CL) (hCJS_nonneg : 0 ≤ CJS) (hJ : Jerr ≤ CJS * S ^ (3 / 5 : ℝ)) (hJabsorb : CJS * ((Real.log 9 / Real.log 2) * M) ^ (3 / 5 : ℝ) ≤ CJm * M ^ (3 / 5 : ℝ)) (hL : Lerr ≤ CL * B) (hB : B ≤ CB * M ^ (3 / 5 : ℝ)) (habsorb : CJm * M ^ (3 / 5 : ℝ) + (Real.log 2 / Real.log 9) * (CL * CB * M ^ (3 / 5 : ℝ)) + 1 ≤ (M / (Real.log M)^2) * (Real.log 2 / Real.log 9 - (1 / 4 : ℝ))) : Jerr + ((Real.log 2 / Real.log 9) * Lerr + 1) ≤ S * (Real.log 2 / Real.log 9 - (1 / 4 : ℝ)) := by have hJ_m : Jerr ≤ CJm * M ^ (3 / 5 : ℝ) := hscale.jerr_le_mpow_of_s_rpow_bound_absorb hCJS_nonneg hJ hJabsorb exact TaoSection7Lemma710.lemma710_error_margin_of_sourceScale hCL_nonneg hscale.hMlower hJ_m hL hB habsorb end TaoSection7Lemma710CurrentScaleControls /-- The full-window source package identifies the pivot index with the shifted source index. -/ theorem TaoSection7Case3FullWindowSourcePathData.point_q_j_eq {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) : (pointAt q).j = taoSection7ShiftIndex j pre.length := by have h := congrArg TaoSection7Point.j hfull.point_q simpa [taoSection7SourceHitPoint, TaoSection7RenewalPoint.toPoint] using h /-- Current horizontal scale `m = floor(n/2) - j` using the shifted pivot index. -/ theorem TaoSection7Case3FullWindowSourcePathData.currentM_eq_shifted {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) : n / 2 - ((pointAt q).j : ℕ) = n / 2 - ((taoSection7ShiftIndex j pre.length : ℕ+) : ℕ) := by rw [TaoSection7Case3FullWindowSourcePathData.point_q_j_eq hfull] /-- The source-domain guard identifies the shifted pivot index as a nontruncated source index below the current cutoff `n / 2`. -/ theorem TaoSection7Case3FullWindowSourcePathData.shiftedIndex_le_cutoff {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n A P threshold q : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n (n / 2) A xi epsilon P threshold q j s pre pres) (hqdom : taoSection7SourcePointInDomain (n / 2) (pointAt q)) : ((taoSection7ShiftIndex j pre.length : ℕ+) : ℕ) ≤ n / 2 := by simpa [taoSection7SourcePointInDomain, TaoSection7Case3FullWindowSourcePathData.point_q_j_eq hfull] using hqdom /-- Shifted-index form of the current-scale Nat nontruncation identity. -/ theorem TaoSection7Case3FullWindowSourcePathData.shiftedIndex_add_currentM_eq_cutoff {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n A P threshold q : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n (n / 2) A xi epsilon P threshold q j s pre pres) (hqdom : taoSection7SourcePointInDomain (n / 2) (pointAt q)) : ((taoSection7ShiftIndex j pre.length : ℕ+) : ℕ) + (n / 2 - ((taoSection7ShiftIndex j pre.length : ℕ+) : ℕ)) = n / 2 := Nat.add_sub_of_le (TaoSection7Case3FullWindowSourcePathData.shiftedIndex_le_cutoff hfull hqdom) /-- The full-window source package identifies the pivot height with the source hit height. -/ theorem TaoSection7Case3FullWindowSourcePathData.point_q_l_eq {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) : (pointAt q).l = Int.ofNat (s + pre.sum + 3) := by have h := congrArg TaoSection7Point.l hfull.point_q simpa [taoSection7SourceHitPoint, TaoSection7RenewalPoint.toPoint] using h /-- Source-height adapter for the old crossing height used by Lemma 7.10's row-control margin. This is only an identity; scale lower bounds remain a separate source obligation. -/ theorem TaoSection7Case3FullWindowSourcePathData.oldCrossingHeight_eq {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) : ((old.cornerL - (pointAt q).l : ℤ) : ℝ) = ((old.cornerL - Int.ofNat (s + pre.sum + 3) : ℤ) : ℝ) := by have hq_l : (pointAt q).l = Int.ofNat (s + pre.sum + 3) := TaoSection7Case3FullWindowSourcePathData.point_q_l_eq hfull simp [hq_l] /-- Real-subtraction form of `oldCrossingHeight_eq` for later scale estimates. -/ theorem TaoSection7Case3FullWindowSourcePathData.oldCrossingHeight_realSub_eq {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) : ((old.cornerL - (pointAt q).l : ℤ) : ℝ) = (old.cornerL : ℝ) - ((s + pre.sum + 3 : ℕ) : ℝ) := by rw [TaoSection7Case3FullWindowSourcePathData.oldCrossingHeight_eq hfull] rw [Int.cast_sub] congr 1 /-- Split a suffix list at a positive one-based offset. -/ theorem taoSection7Case3_exists_sourceSuffix_split_at_positive_offset {α : Type*} (xs : List α) {d : ℕ} (hpos : 0 < d) (hle : d ≤ xs.length) : ∃ before x after, xs = before ++ x :: after ∧ before.length + 1 = d := by cases d with | zero => omega | succ k => have hk : k < xs.length := by omega refine ⟨xs.take k, xs[k], xs.drop (k + 1), ?_, ?_⟩ · calc xs = xs.take k ++ xs.drop k := by rw [List.take_append_drop] _ = xs.take k ++ xs[k] :: xs.drop (k + 1) := by rw [← List.cons_getElem_drop_succ (l := xs) (n := k) (h := hk)] · have htake : (xs.take k).length = k := by simp [List.length_take, Nat.min_eq_left (Nat.le_of_lt hk)] omega theorem taoSection7Case3_exists_sourceSuffix_split_at_firstHit {α : Type*} (xs : List α) {q first : ℕ} (hq_first : q < first) (hfirst_len : first - q ≤ xs.length) : ∃ before x after, xs = before ++ x :: after ∧ before.length + 1 = first - q := by exact taoSection7Case3_exists_sourceSuffix_split_at_positive_offset xs (by omega) hfirst_len theorem taoSection7Case3_sourceSuffixCoversWindow_rebase {q first P : ℕ} {pres before after : List (List ℕ)} {next : List ℕ} (hcoverage : taoSection7SourceSuffixCoversWindow q P pres) (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) (hq_first : q < first) : taoSection7SourceSuffixCoversWindow first P after := by rw [taoSection7SourceSuffixCoversWindow] at hcoverage ⊢ rw [hsplit] at hcoverage simp at hcoverage omega theorem taoSection7Case3_sourceHitPathPoint_rebase_at_split (j : ℕ+) (s : ℕ) (pre : List ℕ) {pres before after : List (List ℕ)} {next : List ℕ} {q first : ℕ} (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) : taoSection7SourceHitPathPoint j s pre pres (first - q) = taoSection7SourceHitPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next := by rw [hsplit, ← hidx] exact taoSection7SourceHitPathPoint_after_prefix (j := j) (s := s) (first := pre) (before := before) (pre := next) (pres := after) theorem taoSection7Case3_sourceHitPathPoint_rebase_after_split (j : ℕ+) (s : ℕ) (pre : List ℕ) {pres before after : List (List ℕ)} {next : List ℕ} {q first t : ℕ} (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) (hq_first : q < first) (hfirst_t : first < t) : taoSection7SourceHitPathPoint j s pre pres (t - q) = taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after (t - first) := by rw [hsplit] have htq : t - q = before.length + (t - first) + 1 := by omega rw [htq] exact taoSection7SourceHitPathPoint_after_prefix_add (j := j) (s := s) (first := pre) (before := before) (pre := next) (pres := after) (n := t - first) theorem taoSection7Case3_pointAt_rebase_first (pointAt : ℕ → TaoSection7Point) {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres before after : List (List ℕ)} {next : List ℕ} {q first : ℕ} (hfirst_source : pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint) (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) : pointAt first = (taoSection7SourceHitPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next).toPoint := by calc pointAt first = (taoSection7SourceHitPathPoint j s pre pres (first - q)).toPoint := hfirst_source _ = (taoSection7SourceHitPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next).toPoint := by rw [taoSection7Case3_sourceHitPathPoint_rebase_at_split (j := j) (s := s) (pre := pre) hsplit hidx] theorem taoSection7Case3_pointAt_rebase_window (pointAt : ℕ → TaoSection7Point) {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres before after : List (List ℕ)} {next : List ℕ} {P q first : ℕ} (hpoint_window : ∀ t : ℕ, q < t → t < P → pointAt t = (taoSection7SourceHitPathPoint j s pre pres (t - q)).toPoint) (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) (hq_first : q < first) : ∀ t : ℕ, first < t → t < P → pointAt t = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after (t - first)).toPoint := by intro t hfirst_t htP have hq_t : q < t := by omega calc pointAt t = (taoSection7SourceHitPathPoint j s pre pres (t - q)).toPoint := hpoint_window t hq_t htP _ = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after (t - first)).toPoint := by rw [taoSection7Case3_sourceHitPathPoint_rebase_after_split (j := j) (s := s) (pre := pre) hsplit hidx hq_first hfirst_t] theorem taoSection7Case3_fullWindowSourcePathData_rebase_of_split {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first : ℕ} {j : ℕ+} {s : ℕ} {pre next : List ℕ} {pres before after : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) (hq_first : q < first) (hfirst_ltP : first < P) (hwhite_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate))) (hdomain_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first + threshold) → taoSection7SourcePointInDomain J (pointAt candidate)) : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by refine { coverage := ?_ point_q := ?_ point_window := ?_ white_iff_later := hwhite_new domain_later := hdomain_new } · exact taoSection7Case3_sourceSuffixCoversWindow_rebase hfull.coverage hsplit hidx hq_first · exact taoSection7Case3_pointAt_rebase_first pointAt (hfull.point_window first hq_first hfirst_ltP) hsplit hidx · exact taoSection7Case3_pointAt_rebase_window pointAt hfull.point_window hsplit hidx hq_first theorem taoSection7Case3_fullWindowSourcePathDataWithGap_rebase_of_split {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre next : List ℕ} {pres before after : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hsplit : pres = before ++ next :: after) (hidx : before.length + 1 = first - q) (hq_first : q < first) (hfirst_ltP : first < P) (hwhite_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound first) (taoSection7Case3LaterSearchStart gapBound first + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate))) (hdomain_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound first) (taoSection7Case3LaterSearchStart gapBound first + threshold) → taoSection7SourcePointInDomain J (pointAt candidate)) : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by refine { coverage := ?_ point_q := ?_ point_window := ?_ white_iff_later := hwhite_new domain_later := hdomain_new } · exact taoSection7Case3_sourceSuffixCoversWindow_rebase hfull.coverage hsplit hidx hq_first · exact taoSection7Case3_pointAt_rebase_first pointAt (hfull.point_window first hq_first hfirst_ltP) hsplit hidx · exact taoSection7Case3_pointAt_rebase_window pointAt hfull.point_window hsplit hidx hq_first theorem taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hwhite_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate))) (hdomain_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) first + threshold) → taoSection7SourcePointInDomain J (pointAt candidate)) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by have hq_first : q < first := hsrc.first_hit.1 rcases taoSection7Case3_exists_sourceSuffix_split_at_firstHit pres hq_first hsrc.first_source_length with ⟨before, next, after, hsplit, hidx⟩ refine ⟨before, next, after, hsplit, hidx, ?_⟩ exact taoSection7Case3_fullWindowSourcePathData_rebase_of_split (hfull := hfull) hsplit hidx hq_first hsrc.first_ltP hwhite_new hdomain_new theorem taoSection7Case3_exists_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hwhite_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound first) (taoSection7Case3LaterSearchStart gapBound first + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate))) (hdomain_new : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound first) (taoSection7Case3LaterSearchStart gapBound first + threshold) → taoSection7SourcePointInDomain J (pointAt candidate)) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by have hq_first : q < first := hsrc.first_hit.1 rcases taoSection7Case3_exists_sourceSuffix_split_at_firstHit pres hq_first hsrc.first_source_length with ⟨before, next, after, hsplit, hidx⟩ refine ⟨before, next, after, hsplit, hidx, ?_⟩ exact taoSection7Case3_fullWindowSourcePathDataWithGap_rebase_of_split (hfull := hfull) hsplit hidx hq_first hsrc.first_ltP hwhite_new hdomain_new theorem taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance_allWindowFacts {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first) (hwhite_all : ∀ t : ℕ, t < P → (W t ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt t))) (hdomain_all : ∀ t : ℕ, t < P → taoSection7SourcePointInDomain J (pointAt t)) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by refine taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) ?_ ?_ · intro t ht exact hwhite_all t (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom_first ht).2 · intro t ht exact hdomain_all t (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom_first ht).2 theorem taoSection7Case3_exists_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance_allWindowFacts {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hwhite_all : ∀ t : ℕ, t < P → (W t ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt t))) (hdomain_all : ∀ t : ℕ, t < P → taoSection7SourcePointInDomain J (pointAt t)) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by refine taoSection7Case3_exists_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) ?_ ?_ · intro t ht exact hwhite_all t (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom_first ht).2 · intro t ht exact hdomain_all t (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom_first ht).2 /-- Source facts available on the whole finite `P` window. This is a support token for rebasing after a first hit. It does not produce the facts from the source process or from Tao's recursion. -/ structure TaoSection7Case3AllWindowSourceFacts (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P : ℕ) : Prop where white_all : ∀ t : ℕ, t < P → (W t ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt t)) domain_all : ∀ t : ℕ, t < P → taoSection7SourcePointInDomain J (pointAt t) theorem taoSection7Case3_allWindowSourceFacts_of_sourceCutoffPoint_alignment (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (halign : ∀ t : ℕ, t < P → (W t ↔ taoSection7Case3SourceCutoffPointW pointAt n xi epsilon J t)) (hdomain : ∀ t : ℕ, t < P → taoSection7SourcePointInDomain J (pointAt t)) : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P := by refine { white_all := ?_ domain_all := hdomain } intro t htP exact taoSection7Case3_rawWhite_iff_of_sourceCutoffPoint_alignment W pointAt halign htP (hdomain t htP) namespace TaoSection7Case3AllWindowSourceFacts theorem white_iff_later {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A P threshold base : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P base) : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) base) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) base + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate)) := by intro candidate hcandidate exact hfacts.white_all candidate (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate).2 theorem domain_later {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A P threshold base : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P base) : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) base) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) base + threshold) → taoSection7SourcePointInDomain J (pointAt candidate) := by intro candidate hcandidate exact hfacts.domain_all candidate (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate).2 theorem white_iff_later_withGap {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J P threshold base : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {gapBound : ℕ → ℕ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P base) : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound base) (taoSection7Case3LaterSearchStart gapBound base + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt candidate)) := by intro candidate hcandidate exact hfacts.white_all candidate (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate).2 theorem domain_later_withGap {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J P threshold base : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {gapBound : ℕ → ℕ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P base) : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart gapBound base) (taoSection7Case3LaterSearchStart gapBound base + threshold) → taoSection7SourcePointInDomain J (pointAt candidate) := by intro candidate hcandidate exact hfacts.domain_all candidate (taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate).2 /-- Domain projection for the actual current q-point of the all-window source package. -/ theorem q_sourceDomain {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J P q : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hqP : q < P) : taoSection7SourcePointInDomain J (pointAt q) := hfacts.domain_all q hqP theorem blackInDomain_iff_sourceBlackPoint {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J P p : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hpP : p < P) : (taoSection7SourceBlackInDomain n xi epsilon J (pointAt p) ↔ taoSection7SourceBlackPoint n xi epsilon (pointAt p)) := by constructor · intro hblack exact hblack.2 · intro hblack exact ⟨hfacts.domain_all p hpP, hblack⟩ theorem positiveWhiteCount_eq_sourceWhitePoint_sum {W : ℕ → Prop} [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) {t : ℕ} (ht : t < P) : taoSection7Case3PositiveWhiteCount W t = (Finset.Icc 1 t).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0) := by unfold taoSection7Case3PositiveWhiteCount refine Finset.sum_congr rfl ?_ intro p hp have hpP : p < P := lt_of_le_of_lt (Finset.mem_Icc.1 hp).2 ht have hwhite : W p ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt p) := hfacts.white_all p hpP by_cases hW : W p · have hsource : taoSection7SourceWhitePoint n xi epsilon (pointAt p) := hwhite.mp hW simp [hW, hsource] · have hsource : ¬ taoSection7SourceWhitePoint n xi epsilon (pointAt p) := by intro hsource exact hW (hwhite.mpr hsource) simp [hW, hsource] theorem threshold_lt_positiveWhiteCount_of_sourceWhitePoint_sum {W : ℕ → Prop} [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) {t threshold : ℕ} (ht : t < P) (hsource : threshold < (Finset.Icc 1 t).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0)) : threshold < taoSection7Case3PositiveWhiteCount W t := by rwa [hfacts.positiveWhiteCount_eq_sourceWhitePoint_sum ht] end TaoSection7Case3AllWindowSourceFacts theorem taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance_allWindowPackage {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by exact taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (TaoSection7Case3AllWindowSourceFacts.white_iff_later hfacts hroom_first) (TaoSection7Case3AllWindowSourceFacts.domain_later hfacts hroom_first) /-- Full-window source-path data feeds the existing later-black route and also supplies the point equality needed by the q-relative minimized first hit. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_fullWindowSourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ' candidate first := by have hdata : TaoSection7Case3SourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres := hfull.to_sourcePathData hroom exact taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hlow hroom hactive hcover hfull.point_window /-- Global-prefix producer-facing full-window source provenance. This is the full-window analogue of `TaoSection7Case3GlobalSourceSuffixProvenance`: it keeps the existing global split coordinates, but strengthens `pointAt` equality from `{q} ∪ laterSearchIcc` to every strict post-`q` offset below `P`. -/ structure TaoSection7Case3GlobalFullWindowSourceSuffixProvenance (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J A : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold q : ℕ) (blocks before : List (List ℕ)) (pre : List ℕ) (pres : List (List ℕ)) : Prop where blocks_eq : blocks = before ++ pre :: pres q_eq : q = before.length window_le : P ≤ blocks.length noThree_before : ∀ block ∈ before, taoSection7NoThree block noThree_pre : taoSection7NoThree pre noThree_pres : ∀ block ∈ pres, taoSection7NoThree block point_q : pointAt q = (taoSection7SourceHitPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre).toPoint pointAt_source_window : ∀ t : ℕ, q < t → t < P → pointAt t = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres (t - q)).toPoint white_iff_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres (candidate - q)).toPoint) domain_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres (candidate - q)).toPoint theorem TaoSection7Case3GlobalFullWindowSourceSuffixProvenance.to_fullWindowSourcePathData {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3GlobalFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks before pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres := by refine { coverage := ?_ point_q := hglobal.point_q point_window := hglobal.pointAt_source_window white_iff_later := ?_ domain_later := ?_ } · exact taoSection7SourceSuffixCoversWindow_of_blockSplit hglobal.blocks_eq hglobal.q_eq hglobal.window_le · intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega have hpoint : pointAt candidate = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres (candidate - q)).toPoint := hglobal.pointAt_source_window candidate hq_candidate hstrict_ltP.2 simpa [hpoint] using hglobal.white_iff_source candidate hcandidate · intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega have hpoint : pointAt candidate = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres (candidate - q)).toPoint := hglobal.pointAt_source_window candidate hq_candidate hstrict_ltP.2 simpa [hpoint] using hglobal.domain_source candidate hcandidate theorem TaoSection7Case3GlobalFullWindowSourceSuffixProvenance.to_globalSourceSuffixProvenance {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3GlobalFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks before pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) : TaoSection7Case3GlobalSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks before pre pres := by refine { blocks_eq := hglobal.blocks_eq q_eq := hglobal.q_eq window_le := hglobal.window_le noThree_before := hglobal.noThree_before noThree_pre := hglobal.noThree_pre noThree_pres := hglobal.noThree_pres pointAt_source := ?_ white_iff_source := hglobal.white_iff_source domain_source := hglobal.domain_source } intro t ht rcases Finset.mem_insert.mp ht with htq | ht_later · subst t simpa using hglobal.point_q · have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < t ∧ t < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom ht_later have hq_t : q < t := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega exact hglobal.pointAt_source_window t hq_t hstrict_ltP.2 theorem taoSection7Case3_exists_firstHitSourceProvenance_of_globalFullWindowSourceSuffixProvenance (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3GlobalFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks before pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ first Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt (taoSection7SourceBlockStartJ before) (taoSection7SourceBlockStartS before) pre pres P q old Γ' candidate first := by exact taoSection7Case3_exists_firstHitSourceProvenance_of_fullWindowSourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := taoSection7SourceBlockStartJ before) (s := taoSection7SourceBlockStartS before) (pre := pre) (pres := pres) (hglobal.to_fullWindowSourcePathData hroom) hlow hroom hactive hcover /-- Distinguished-first producer-facing full-window source provenance. This is the full-window analogue of `TaoSection7Case3DistinguishedSourceSuffixProvenance`, using the `first :: before` origin convention. -/ structure TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J A : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold q : ℕ) (blocks : List (List ℕ)) (first : List ℕ) (before : List (List ℕ)) (pre : List ℕ) (pres : List (List ℕ)) : Prop where blocks_eq : blocks = first :: before ++ pre :: pres q_eq : q = before.length + 1 window_le : P ≤ blocks.length noThree_first : taoSection7NoThree first noThree_before : ∀ block ∈ before, taoSection7NoThree block noThree_pre : taoSection7NoThree pre noThree_pres : ∀ block ∈ pres, taoSection7NoThree block point_q : pointAt q = (taoSection7SourceHitPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre).toPoint pointAt_source_window : ∀ t : ℕ, q < t → t < P → pointAt t = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (t - q)).toPoint white_iff_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint) domain_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint theorem TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance.to_fullWindowSourcePathData {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hsrc : TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres := by refine { coverage := ?_ point_q := hsrc.point_q point_window := hsrc.pointAt_source_window white_iff_later := ?_ domain_later := ?_ } · exact taoSection7SourceSuffixCoversWindow_of_distinguishedBlockSplit hsrc.blocks_eq hsrc.q_eq hsrc.window_le · intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega have hpoint : pointAt candidate = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint := hsrc.pointAt_source_window candidate hq_candidate hstrict_ltP.2 simpa [hpoint] using hsrc.white_iff_source candidate hcandidate · intro candidate hcandidate have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < candidate ∧ candidate < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom hcandidate have hq_candidate : q < candidate := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega have hpoint : pointAt candidate = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint := hsrc.pointAt_source_window candidate hq_candidate hstrict_ltP.2 simpa [hpoint] using hsrc.domain_source candidate hcandidate theorem TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance.to_distinguishedSourceSuffixProvenance {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hsrc : TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) : TaoSection7Case3DistinguishedSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres := by refine { blocks_eq := hsrc.blocks_eq q_eq := hsrc.q_eq window_le := hsrc.window_le noThree_first := hsrc.noThree_first noThree_before := hsrc.noThree_before noThree_pre := hsrc.noThree_pre noThree_pres := hsrc.noThree_pres pointAt_source := ?_ white_iff_source := hsrc.white_iff_source domain_source := hsrc.domain_source } intro t ht rcases Finset.mem_insert.mp ht with htq | ht_later · subst t simpa using hsrc.point_q · have hstrict_ltP : taoSection7Case3HeightExitBound (taoSection7Case3ExitGapBound A) q < t ∧ t < P := taoSection7Case3_laterSearch_mem_strict_exit_and_lt_window hroom ht_later have hq_t : q < t := by dsimp [taoSection7Case3HeightExitBound] at hstrict_ltP omega exact hsrc.pointAt_source_window t hq_t hstrict_ltP.2 theorem taoSection7Case3_exists_firstHitSourceProvenance_of_distinguishedFullWindowSourceSuffixProvenance (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hsrc : TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ hit Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres P q old Γ' candidate hit := by exact taoSection7Case3_exists_firstHitSourceProvenance_of_fullWindowSourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := taoSection7SourceBlockStartJ (first :: before)) (s := taoSection7SourceBlockStartS (first :: before)) (pre := pre) (pres := pres) (hsrc.to_fullWindowSourcePathData hroom) hlow hroom hactive hcover /-- The canonical `pointAt` function obtained by reading all offsets from one global source-hit path. -/ def taoSection7SourcePointAtOfBlocks (first : List ℕ) (rest : List (List ℕ)) : ℕ → TaoSection7Point := fun t => (taoSection7SourceHitPathPoint (1 : ℕ+) 0 first rest t).toPoint /-- Distinguished-first global path model for full-window source provenance. This packages the strongest natural producer shape: `pointAt` agrees with the global source-hit path from the initial block `first` throughout the inspected `P` window. The suffix full-window provenance then follows from the checked prefix algebra `taoSection7SourceHitPathPoint_after_prefix_add`. -/ structure TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance (W : ℕ → Prop) (pointAt : ℕ → TaoSection7Point) (n J A : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold q : ℕ) (blocks : List (List ℕ)) (first : List ℕ) (before : List (List ℕ)) (pre : List ℕ) (pres : List (List ℕ)) : Prop where blocks_eq : blocks = first :: before ++ pre :: pres q_eq : q = before.length + 1 window_le : P ≤ blocks.length noThree_first : taoSection7NoThree first noThree_before : ∀ block ∈ before, taoSection7NoThree block noThree_pre : taoSection7NoThree pre noThree_pres : ∀ block ∈ pres, taoSection7NoThree block point_q : pointAt q = (taoSection7SourceHitPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre).toPoint pointAt_global : ∀ t : ℕ, t < P → pointAt t = (taoSection7SourceHitPathPoint (1 : ℕ+) 0 first (before ++ pre :: pres) t).toPoint white_iff_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint) domain_source : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint theorem TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance.of_sourcePointAtOfBlocks (W : ℕ → Prop) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hblocks : blocks = first :: before ++ pre :: pres) (hq : q = before.length + 1) (hP : P ≤ blocks.length) (hnoThree_first : taoSection7NoThree first) (hnoThree_before : ∀ block ∈ before, taoSection7NoThree block) (hnoThree_pre : taoSection7NoThree pre) (hnoThree_pres : ∀ block ∈ pres, taoSection7NoThree block) (hwhite : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint)) (hdomain : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint) : TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance W (taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) n J A xi epsilon P threshold q blocks first before pre pres := by refine { blocks_eq := hblocks q_eq := hq window_le := hP noThree_first := hnoThree_first noThree_before := hnoThree_before noThree_pre := hnoThree_pre noThree_pres := hnoThree_pres point_q := ?_ pointAt_global := ?_ white_iff_source := hwhite domain_source := hdomain } · subst q have hstartJ : taoSection7SourceBlockStartJFrom (1 : ℕ+) (first :: before) = taoSection7SourceBlockStartJ (first :: before) := by apply Subtype.ext simp [taoSection7SourceBlockStartJFrom, taoSection7SourceBlockStartJ, taoSection7ShiftIndex] have hstartS : taoSection7SourceBlockStartSFrom 0 (first :: before) = taoSection7SourceBlockStartS (first :: before) := by simp [taoSection7SourceBlockStartSFrom, taoSection7SourceBlockStartS] have hprefix := taoSection7SourceHitPathPoint_after_prefix (j := (1 : ℕ+)) (s := 0) (first := first) (before := before) (pre := pre) (pres := pres) simpa [taoSection7SourcePointAtOfBlocks, hstartJ, hstartS] using congrArg TaoSection7RenewalPoint.toPoint hprefix · intro t _htP rfl theorem TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance.to_distinguishedFullWindowSourceSuffixProvenance {W : ℕ → Prop} {pointAt : ℕ → TaoSection7Point} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres) : TaoSection7Case3DistinguishedFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres := by refine { blocks_eq := hglobal.blocks_eq q_eq := hglobal.q_eq window_le := hglobal.window_le noThree_first := hglobal.noThree_first noThree_before := hglobal.noThree_before noThree_pre := hglobal.noThree_pre noThree_pres := hglobal.noThree_pres point_q := hglobal.point_q pointAt_source_window := ?_ white_iff_source := hglobal.white_iff_source domain_source := hglobal.domain_source } · intro t hq_t htP have hpoint := hglobal.pointAt_global t htP have ht_eq : t = before.length + (t - q) + 1 := by rw [hglobal.q_eq] rw [hglobal.q_eq] at hq_t omega have hidx_eq : before.length + (t - q) + 1 - q = t - q := by rw [hglobal.q_eq] at hq_t ⊢ omega have hstartJ : taoSection7SourceBlockStartJFrom (1 : ℕ+) (first :: before) = taoSection7SourceBlockStartJ (first :: before) := by apply Subtype.ext simp [taoSection7SourceBlockStartJFrom, taoSection7SourceBlockStartJ, taoSection7ShiftIndex] have hstartS : taoSection7SourceBlockStartSFrom 0 (first :: before) = taoSection7SourceBlockStartS (first :: before) := by simp [taoSection7SourceBlockStartSFrom, taoSection7SourceBlockStartS] have hprefix : taoSection7SourceHitPathPoint (1 : ℕ+) 0 first (before ++ pre :: pres) (before.length + (t - q) + 1) = taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (t - q) := by have h := taoSection7SourceHitPathPoint_after_prefix_add (j := (1 : ℕ+)) (s := 0) (first := first) (before := before) (pre := pre) (pres := pres) (n := t - q) simpa [hstartJ, hstartS] using h calc pointAt t = (taoSection7SourceHitPathPoint (1 : ℕ+) 0 first (before ++ pre :: pres) t).toPoint := hpoint _ = (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (t - q)).toPoint := by rw [ht_eq] simpa [hidx_eq] using congrArg TaoSection7RenewalPoint.toPoint hprefix theorem taoSection7Case3_exists_firstHitSourceProvenance_of_distinguishedGlobalPathFullWindowProvenance (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance W pointAt n J A xi epsilon P threshold q blocks first before pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt A threshold P q pres.length old Γ candidate ∧ ∃ hit Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres P q old Γ' candidate hit := by exact taoSection7Case3_exists_firstHitSourceProvenance_of_distinguishedFullWindowSourceSuffixProvenance (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := first) (before := before) (pre := pre) (pres := pres) (hglobal.to_distinguishedFullWindowSourceSuffixProvenance) hlow hroom hactive hcover /-- Direct source-defined full-window producer for the distinguished block split. This is the composed worker target: when `pointAt` is the canonical global source-hit path from `first :: before ++ pre :: pres`, the checked prefix algebra supplies full-window source provenance and hence the q-relative first-hit source-provenance consumer. -/ theorem taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePointAtOfBlocks (W : ℕ → Prop) [DecidablePred W] {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {first : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hblocks : blocks = first :: before ++ pre :: pres) (hq : q = before.length + 1) (hP : P ≤ blocks.length) (hnoThree_first : taoSection7NoThree first) (hnoThree_before : ∀ block ∈ before, taoSection7NoThree block) (hnoThree_pre : taoSection7NoThree pre) (hnoThree_pres : ∀ block ∈ pres, taoSection7NoThree block) (hwhite : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint)) (hdomain : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres (candidate - q)).toPoint) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family (taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate Γ, TaoSection7Case3ActiveBlackPivotStepInLaterWindow W (taoSection7SourceBlackInDomain n xi epsilon J) family (taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) A threshold P q pres.length old Γ candidate ∧ ∃ hit Γ', TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) (taoSection7SourceBlockStartJ (first :: before)) (taoSection7SourceBlockStartS (first :: before)) pre pres P q old Γ' candidate hit := by have hglobal : TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance W (taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) n J A xi epsilon P threshold q blocks first before pre pres := TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance.of_sourcePointAtOfBlocks (W := W) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := first) (before := before) (pre := pre) (pres := pres) hblocks hq hP hnoThree_first hnoThree_before hnoThree_pre hnoThree_pres hwhite hdomain exact taoSection7Case3_exists_firstHitSourceProvenance_of_distinguishedGlobalPathFullWindowProvenance (W := W) (pointAt := taoSection7SourcePointAtOfBlocks first (before ++ pre :: pres)) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := first) (before := before) (pre := pre) (pres := pres) hglobal hlow hroom hactive hcover /-- One recursive encounter step with both accepted-transition and source-rebase provenance at the q-relative first hit. This is still a supplied-step package: it records the data future source and stopping-process producers must supply, but does not prove that such a step exists from Tao's recursive process. -/ structure TaoSection7Case3EncounterStep (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) (P q : ℕ) (old Γ : TaoSection7Triangle) (first : ℕ) : Prop where accepted : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first first_ltP : first < P source_support : ∃ candidate : ℕ, ∃ j : ℕ+, ∃ s : ℕ, ∃ pre : List ℕ, ∃ pres : List (List ℕ), TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first /-- Finite chain of source-supported q-relative encounter steps. This is the closest support-only surface to Tao's `t_1,...,t_R` bookkeeping: each step is an accepted first hit below the `P` window with source-rebase provenance. The chain still does not prove that Tao's actual process reaches this many steps. -/ inductive TaoSection7Case3RecursiveEncounterChain (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) (P : ℕ) : ℕ → TaoSection7Triangle → List (ℕ × TaoSection7Triangle) → Prop | nil (q : ℕ) (old : TaoSection7Triangle) : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old [] | cons {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hstep : TaoSection7Case3EncounterStep black family pointAt Accepts P q old Γ first) (htail : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P first Γ rest) : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest) namespace TaoSection7Case3RecursiveEncounterChain theorem head_step {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest)) : TaoSection7Case3EncounterStep black family pointAt Accepts P q old Γ first := by cases hchain with | cons hstep _htail => exact hstep theorem tail_chain {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest)) : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P first Γ rest := by cases hchain with | cons _hstep htail => exact htail theorem head_first_ltP {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest)) : first < P := (head_step hchain).first_ltP theorem head_q_lt_first {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest)) : q < first := (head_step hchain).accepted.first_hit.1 theorem head_source_support {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old ((first, Γ) :: rest)) : ∃ candidate : ℕ, ∃ j : ℕ+, ∃ s : ℕ, ∃ pre : List ℕ, ∃ pres : List (List ℕ), TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first := (head_step hchain).source_support end TaoSection7Case3RecursiveEncounterChain /-- Finite chain of q-relative accepted first-hit transitions. This is a bookkeeping surface for Tao's later `t_i`/`r` bridge. The list records only accepted first hits already supplied by the local first-stopping API; it does not prove that the chain is maximal, that it is Tao's actual recursive source process, or that it reaches length `R`. -/ inductive TaoSection7Case3RecursiveFirstHitChain (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) : ℕ → TaoSection7Triangle → List (ℕ × TaoSection7Triangle) → Prop | nil (q : ℕ) (old : TaoSection7Triangle) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old [] | cons {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hstep : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first) (htail : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts first Γ rest) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old ((first, Γ) :: rest) namespace TaoSection7Case3RecursiveFirstHitChain /-- Number of accepted first-hit transitions in the finite chain. -/ def length (steps : List (ℕ × TaoSection7Triangle)) : ℕ := steps.length /-- The optional `R`th transition time, using one-based `R` to match Tao's notation `t_R`. -/ def tR? : List (ℕ × TaoSection7Triangle) → ℕ → Option ℕ | [], _ => none | _step :: _rest, 0 => none | step :: _rest, 1 => some step.1 | _step :: rest, R + 2 => tR? rest (R + 1) /-- The finite chain has at least `R` accepted transitions. -/ def HasAtLeast (steps : List (ℕ × TaoSection7Triangle)) (R : ℕ) : Prop := R ≤ steps.length theorem head_accepted {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old ((first, Γ) :: rest)) : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first := by cases hchain with | cons hstep _htail => exact hstep theorem tail_chain {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old ((first, Γ) :: rest)) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts first Γ rest := by cases hchain with | cons _hstep htail => exact htail theorem head_q_lt_first {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old ((first, Γ) :: rest)) : q < first := by exact (head_accepted hchain).first_hit.1 theorem head_new_ne_old {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old ((first, Γ) :: rest)) : Γ ≠ old := by exact (head_accepted hchain).new_ne_old end TaoSection7Case3RecursiveFirstHitChain /-- A triangle hit in the restarted source process, before selecting which triangle will be the current stopping triangle. -/ def TaoSection7Case3TriangleHit (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (t : ℕ) : Prop := ∃ Γ : TaoSection7Triangle, Γ ∈ family ∧ Γ.Mem (pointAt t) /-- First triangle hit after a start index. For Tao's `t_1` in the restarted process the intended start is `0`. The later `t_i` use the stricter post-exit predicate `TaoSection7Case3FirstAfterTriangleHitFrom` instead. -/ def TaoSection7Case3FirstTriangleHitFrom (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (start t : ℕ) : Prop := start < t ∧ TaoSection7Case3TriangleHit pointAt family t ∧ ∀ s : ℕ, start < s → s < t → ¬ TaoSection7Case3TriangleHit pointAt family s /-- The initial `t_1` stopping step in Tao's restarted process. -/ structure TaoSection7Case3InitialStoppingStep (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (first : ℕ) (Γ : TaoSection7Triangle) : Prop where first_hit : TaoSection7Case3FirstTriangleHitFrom pointAt family 0 first mem_family : Γ ∈ family mem : Γ.Mem (pointAt first) /-- One later stopping transition, from a previous stopping triangle to the first later triangle hit after strict vertical exit from that previous triangle. -/ structure TaoSection7Case3StoppingTransition (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (q : ℕ) (old : TaoSection7Triangle) (first : ℕ) (Γ : TaoSection7Triangle) : Prop where first_after_exit : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first new_mem_family : Γ ∈ family new_mem : Γ.Mem (pointAt first) new_ne_old : Γ ≠ old theorem taoSection7Case3_firstTriangleHitFrom_time_eq {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {start first first' : ℕ} (hfirst : TaoSection7Case3FirstTriangleHitFrom pointAt family start first) (hfirst' : TaoSection7Case3FirstTriangleHitFrom pointAt family start first') : first = first' := by by_contra hne by_cases hlt : first < first' · exact hfirst'.2.2 first hfirst.1 hlt hfirst.2.1 · have hgt : first' < first := by omega exact hfirst.2.2 first' hfirst'.1 hgt hfirst'.2.1 theorem taoSection7Case3_firstAfterTriangleHitFrom_time_eq {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {q first first' : ℕ} (hfirst : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first) (hfirst' : TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first') : first = first' := by by_contra hne by_cases hlt : first < first' · exact hfirst'.2.2 first hfirst.1 hlt hfirst.2.1 · have hgt : first' < first := by omega exact hfirst.2.2 first' hfirst'.1 hgt hfirst'.2.1 theorem taoSection7Triangle_eq_of_familyPairwiseDisjoint_mem {family : Set TaoSection7Triangle} {Δ Γ : TaoSection7Triangle} {p : TaoSection7Point} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hΔ : Δ ∈ family) (hΓ : Γ ∈ family) (hΔ_mem : Δ.Mem p) (hΓ_mem : Γ.Mem p) : Δ = Γ := by by_contra hne exact TaoSection7TriangleFamilyPairwiseDisjoint.no_common_mem hpair hΔ hΓ hne hΔ_mem hΓ_mem namespace TaoSection7Case3StoppingTransition theorem pair_eq_of_pairwiseDisjoint {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q : ℕ} {old Γ Γ' : TaoSection7Triangle} {first first' : ℕ} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ) (hstep' : TaoSection7Case3StoppingTransition pointAt family q old first' Γ') : (first, Γ) = (first', Γ') := by have hfirst_eq : first = first' := taoSection7Case3_firstAfterTriangleHitFrom_time_eq hstep.first_after_exit hstep'.first_after_exit subst hfirst_eq have hΓ_eq : Γ = Γ' := taoSection7Triangle_eq_of_familyPairwiseDisjoint_mem hpair hstep.new_mem_family hstep'.new_mem_family hstep.new_mem hstep'.new_mem subst hΓ_eq rfl end TaoSection7Case3StoppingTransition namespace TaoSection7Case3InitialStoppingStep theorem pair_eq_of_pairwiseDisjoint {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Γ Γ' : TaoSection7Triangle} {first first' : ℕ} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hstep : TaoSection7Case3InitialStoppingStep pointAt family first Γ) (hstep' : TaoSection7Case3InitialStoppingStep pointAt family first' Γ') : (first, Γ) = (first', Γ') := by have hfirst_eq : first = first' := taoSection7Case3_firstTriangleHitFrom_time_eq hstep.first_hit hstep'.first_hit subst hfirst_eq have hΓ_eq : Γ = Γ' := taoSection7Triangle_eq_of_familyPairwiseDisjoint_mem hpair hstep.mem_family hstep'.mem_family hstep.mem hstep'.mem subst hΓ_eq rfl end TaoSection7Case3InitialStoppingStep namespace TaoSection7Case3FirstHitSourceProvenance theorem to_stoppingTransition {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) : TaoSection7Case3StoppingTransition pointAt family q old first Γ := ⟨hsrc.first_hit, hsrc.new_mem_family, hsrc.new_mem, hsrc.new_ne_old⟩ def to_rawOutsideEprimeEndpointData {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {K B : ℝ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hbridge : TaoSection7Lemma710.LowTipBridge old Γ K B) (hl_close : |(pointAt first).lReal - (old.cornerL : ℝ)| ≤ K * B) : TaoSection7Lemma710.RawOutsideEprimeEndpointData old Γ (pointAt first) K B := { endpoint_mem := hsrc.new_mem lower_bridge := hbridge l_close := hl_close } /-- First-hit source-side controls for Lemma 7.10's lower-tip row witness. This is the producer-side socket for the explicit row-budget data in `Section7Geometry`: the first-hit provenance supplies the new-triangle endpoint membership, while future outside-`E'` and large-`m` source estimates must fill the endpoint bounds and old-row budgets here. -/ structure OutsideEprimeBadLowerTipRowControls (pointAt : ℕ → TaoSection7Point) (old : TaoSection7Triangle) (q first : ℕ) (S Lerr Jerr gap : ℝ) : Prop where base_mem_old : old.Mem (pointAt q) s_def : S = ((old.cornerL - (pointAt q).l : ℤ) : ℝ) endpoint_l_lower : (old.cornerL : ℝ) ≤ (pointAt first).lReal endpoint_l_upper : (pointAt first).lReal - (old.cornerL : ℝ) ≤ Lerr endpoint_j_close : |(pointAt first).jReal - ((pointAt q).jReal + S / 4)| ≤ Jerr gap_nonneg : 0 ≤ gap Lerr_nonneg : 0 ≤ Lerr Jerr_nonneg : 0 ≤ Jerr old_left_budget : Jerr + ((Real.log 2 / Real.log 9) * Lerr + 1) ≤ S / 4 old_right_budget : (Jerr + ((Real.log 2 / Real.log 9) * Lerr + 1)) * Real.log 9 ≤ S * (Real.log 2 - Real.log 9 / 4) /-- Build the row-control socket from cheap active/first-hit provenance plus explicit analytic endpoint and scalar-margin estimates. This still leaves the true outside-`E'` estimates `(7.63)/(7.64)` and the large-`m` source-scale margin as hypotheses. -/ theorem outsideEprimeBadLowerTipRowControls_of_active_firstHit_estimates {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {Lerr Jerr gap : ℝ} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hL : (pointAt first).lReal - (old.cornerL : ℝ) ≤ Lerr) (hJ : |(pointAt first).jReal - ((pointAt q).jReal + (((old.cornerL - (pointAt q).l : ℤ) : ℝ)) / 4)| ≤ Jerr) (hgap : 0 ≤ gap) (hLerr : 0 ≤ Lerr) (hJerr : 0 ≤ Jerr) (hE : Jerr + ((Real.log 2 / Real.log 9) * Lerr + 1) ≤ (((old.cornerL - (pointAt q).l : ℤ) : ℝ)) * (Real.log 2 / Real.log 9 - (1 / 4 : ℝ))) : OutsideEprimeBadLowerTipRowControls pointAt old q first (((old.cornerL - (pointAt q).l : ℤ) : ℝ)) Lerr Jerr gap := by have hS_nonneg : 0 ≤ (((old.cornerL - (pointAt q).l : ℤ) : ℝ)) := by have hheight : (pointAt q).l ≤ old.cornerL := TaoSection7Triangle.mem_height_le hactive.pivot_mem have hint : 0 ≤ old.cornerL - (pointAt q).l := sub_nonneg.mpr hheight exact_mod_cast hint have hbudgets := TaoSection7Lemma710.lemma710_oldRowBudgets_of_error_margin hS_nonneg hE refine { base_mem_old := hactive.pivot_mem s_def := rfl endpoint_l_lower := ?_ endpoint_l_upper := hL endpoint_j_close := ?_ gap_nonneg := hgap Lerr_nonneg := hLerr Jerr_nonneg := hJerr old_left_budget := hbudgets.1 old_right_budget := hbudgets.2 } · have hheight_lt : old.cornerL < (pointAt first).l := hsrc.first_hit.2.1.1 have hheight_le : old.cornerL ≤ (pointAt first).l := le_of_lt hheight_lt change ((old.cornerL : ℤ) : ℝ) ≤ (((pointAt first).l : ℤ) : ℝ) exact_mod_cast hheight_le · simpa using hJ /-- Current-scale variant of the row-control constructor. The future outside-`E'` producer may supply the horizontal error at `S^(3/5)`; this theorem consumes the checked current-scale ledger and explicit absorption premises to fill the old-row scalar margin. -/ theorem outsideEprimeBadLowerTipRowControls_of_active_firstHit_currentScale_estimates {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n A P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {M S Lerr Jerr gap CJS CJm : ℝ} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hscale : TaoSection7Lemma710CurrentScaleControls n old (pointAt q) M S) (hL_endpoint : (pointAt first).lReal - (old.cornerL : ℝ) ≤ Lerr) (hJ_endpoint : |(pointAt first).jReal - ((pointAt q).jReal + S / 4)| ≤ Jerr) (hgap : 0 ≤ gap) (hLerr : 0 ≤ Lerr) (hJerr : 0 ≤ Jerr) (hCJS_nonneg : 0 ≤ CJS) (hJ_scale : Jerr ≤ CJS * S ^ (3 / 5 : ℝ)) (hJabsorb : CJS * ((Real.log 9 / Real.log 2) * M) ^ (3 / 5 : ℝ) ≤ CJm * M ^ (3 / 5 : ℝ)) (hmargin : CJm * M ^ (3 / 5 : ℝ) + ((Real.log 2 / Real.log 9) * Lerr + 1) ≤ S * (Real.log 2 / Real.log 9 - (1 / 4 : ℝ))) : OutsideEprimeBadLowerTipRowControls pointAt old q first S Lerr Jerr gap := by have hJ_m : Jerr ≤ CJm * M ^ (3 / 5 : ℝ) := hscale.jerr_le_mpow_of_s_rpow_bound_absorb hCJS_nonneg hJ_scale hJabsorb have hE : Jerr + ((Real.log 2 / Real.log 9) * Lerr + 1) ≤ S * (Real.log 2 / Real.log 9 - (1 / 4 : ℝ)) := by linarith have hbudgets := TaoSection7Lemma710.lemma710_oldRowBudgets_of_error_margin hscale.S_nonneg hE refine { base_mem_old := hactive.pivot_mem s_def := ?_ endpoint_l_lower := ?_ endpoint_l_upper := hL_endpoint endpoint_j_close := hJ_endpoint gap_nonneg := hgap Lerr_nonneg := hLerr Jerr_nonneg := hJerr old_left_budget := hbudgets.1 old_right_budget := hbudgets.2 } · simpa [TaoSection7Triangle.verticalDepth] using hscale.S_def · have hheight_lt : old.cornerL < (pointAt first).l := hsrc.first_hit.2.1.1 have hheight_le : old.cornerL ≤ (pointAt first).l := le_of_lt hheight_lt change ((old.cornerL : ℤ) : ℝ) ≤ (((pointAt first).l : ℤ) : ℝ) exact_mod_cast hheight_le theorem to_outsideEprimeBadLowerTipRowData {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {S Lerr Jerr gap : ℝ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hcontrols : OutsideEprimeBadLowerTipRowControls pointAt old q first S Lerr Jerr gap) : TaoSection7Lemma710.OutsideEprimeBadLowerTipRowData old Γ (pointAt q) (pointAt first) S Lerr Jerr gap := { base_mem_old := hcontrols.base_mem_old endpoint_mem_new := hsrc.new_mem s_def := hcontrols.s_def endpoint_l_lower := hcontrols.endpoint_l_lower endpoint_l_upper := hcontrols.endpoint_l_upper endpoint_j_close := hcontrols.endpoint_j_close gap_nonneg := hcontrols.gap_nonneg Lerr_nonneg := hcontrols.Lerr_nonneg Jerr_nonneg := hcontrols.Jerr_nonneg old_left_budget := hcontrols.old_left_budget old_right_budget := hcontrols.old_right_budget } /-- Convert checked lower-tip row controls into the raw common-point endpoint data consumed by the deterministic `NearSigma` bridge. This is still not endpoint production: the row controls and the `K*B` absorptions are explicit inputs. -/ theorem to_rawOutsideEprimeCommonPointEndpointData_from_rowControls {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {S Lerr Jerr gap K B : ℝ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hcontrols : OutsideEprimeBadLowerTipRowControls pointAt old q first S Lerr Jerr gap) (hgap : gap ≤ K * B) (hLerr : Lerr ≤ K * B) : TaoSection7Lemma710.RawOutsideEprimeCommonPointEndpointData old Γ (pointAt first) gap K B := by have hrow : TaoSection7Lemma710.OutsideEprimeBadLowerTipRowData old Γ (pointAt q) (pointAt first) S Lerr Jerr gap := hsrc.to_outsideEprimeBadLowerTipRowData hcontrols have hrowBridge : TaoSection7Lemma710.LowTipRowCommonPointBridge old Γ gap := TaoSection7Lemma710.lowTipRowCommonPointBridge_of_outsideEprimeBadLowerTipRowData hrow have hcommon : TaoSection7Lemma710.LowTipCommonPointBridge old Γ gap := TaoSection7Lemma710.lowTipCommonPointBridge_of_rowCommonPointBridge hrowBridge have hKB_nonneg : 0 ≤ K * B := le_trans hcontrols.Lerr_nonneg hLerr have hlower : -(K * B) ≤ (pointAt first).lReal - (old.cornerL : ℝ) := by have hdiff_nonneg : 0 ≤ (pointAt first).lReal - (old.cornerL : ℝ) := by linarith [hcontrols.endpoint_l_lower] linarith have hupper : (pointAt first).lReal - (old.cornerL : ℝ) ≤ K * B := le_trans hcontrols.endpoint_l_upper hLerr exact { endpoint_mem := hsrc.new_mem lower_common_bridge := hcommon gap_le := hgap l_close := abs_le.mpr ⟨hlower, hupper⟩ } /-- Stopping-side deterministic localization for Lemma 7.10: once raw common-point endpoint data is available, first-hit provenance supplies the family facts needed by the geometry bridge to `NearSigma`. This is not probability or endpoint production; the raw endpoint data and radius bounds remain explicit inputs. -/ theorem to_center_nearSigma_of_rawOutsideEprimeCommonPointEndpointData {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {sMin gap K B J R : ℝ} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hsize : sMin ≤ Γ.size) (hJ : (Real.log 2 / Real.log 9) * (K * B + K * B) ≤ J) (hradius : J ^ 2 + (K * B) ^ 2 ≤ R ^ 2) (hdata : TaoSection7Lemma710.RawOutsideEprimeCommonPointEndpointData old Γ (pointAt first) gap K B) : ∃ c : TaoSection7Lemma710.Center family old sMin K B, c.triangle = Γ ∧ TaoSection7Lemma710.NearSigma R (TaoSection7Lemma710.Sigma family old sMin K B) (pointAt first) := TaoSection7Lemma710.rawOutsideEprimeCommonPointEndpointData_to_center_nearSigma hpair hactive.old_mem_family hsrc.new_mem_family hsrc.new_ne_old hsize hJ hradius hdata /-- Stopping-side pointwise source-hit package for the outside-`E'` center event. All analytic endpoint estimates remain explicit row-control and absorption hypotheses. -/ theorem to_outsideEprimeSourceHitData_from_rowControls {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {A P q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} {candidate first : ℕ} {sMin S Lerr Jerr gap K B J R : ℝ} (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hsize : sMin ≤ Γ.size) (hJ : (Real.log 2 / Real.log 9) * (K * B + K * B) ≤ J) (hradius : J ^ 2 + (K * B) ^ 2 ≤ R ^ 2) (hcontrols : OutsideEprimeBadLowerTipRowControls pointAt old q first S Lerr Jerr gap) (hgap : gap ≤ K * B) (hLerr : Lerr ≤ K * B) : TaoSection7Lemma710.Lemma710OutsideEprimeSourceHitData family old Γ (pointAt first) sMin gap K B J R where pairwiseDisjoint := hpair old_mem_family := hactive.old_mem_family new_mem_family := hsrc.new_mem_family new_ne_old := hsrc.new_ne_old size_ge := hsize j_close_budget := hJ radius_absorbs := hradius raw_endpoint := hsrc.to_rawOutsideEprimeCommonPointEndpointData_from_rowControls hcontrols hgap hLerr theorem to_center_distSq_of_outsideEprimeSourceHitData {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {old Γ : TaoSection7Triangle} {first : ℕ} {sMin gap K B J R : ℝ} (hdata : TaoSection7Lemma710.Lemma710OutsideEprimeSourceHitData family old Γ (pointAt first) sMin gap K B J R) : ∃ c : TaoSection7Lemma710.Center family old sMin K B, c.triangle = Γ ∧ (pointAt first).distSq c.point ≤ R ^ 2 := TaoSection7Lemma710.lemma710_outsideEprimeSourceHitData_to_center_distSq hdata theorem to_firstHitAcceptedTransition {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (haccept : Accepts q old first Γ) : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first := ⟨hsrc.first_hit, hsrc.first_black, hsrc.new_mem_family, hsrc.new_mem, hsrc.new_ne_old, haccept⟩ theorem to_encounterStep {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {P q : ℕ} {old Γ : TaoSection7Triangle} {candidate first : ℕ} (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (haccept : Accepts q old first Γ) : TaoSection7Case3EncounterStep black family pointAt Accepts P q old Γ first := { accepted := hsrc.to_firstHitAcceptedTransition haccept first_ltP := hsrc.first_ltP source_support := ⟨candidate, j, s, pre, pres, hsrc⟩ } end TaoSection7Case3FirstHitSourceProvenance theorem taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : Accepts q old first Γ) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3EncounterStep black family pointAt Accepts P q old Γ first ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by rcases taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance_allWindowPackage (hfull := hfull) (hsrc := hsrc) hroom_first hfacts with ⟨before, next, after, hsplit, hidx, hrebase⟩ exact ⟨before, next, after, hsplit, hidx, hsrc.to_encounterStep haccept, hrebase⟩ theorem taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first candidate : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old Γ candidate first) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : Accepts q old first Γ) : ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3EncounterStep black family pointAt Accepts P q old Γ first ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by rcases taoSection7Case3_exists_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance_allWindowFacts (hfull := hfull) (hsrc := hsrc) hroom_first hfacts.white_all hfacts.domain_all with ⟨before, next, after, hsplit, hidx, hrebase⟩ exact ⟨before, next, after, hsplit, hidx, hsrc.to_encounterStep haccept, hrebase⟩ theorem taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathData_of_fullWindowSourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ candidate : ℕ, ∃ first : ℕ, ∃ Γ : TaoSection7Triangle, ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first ∧ TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ first ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_fullWindowSourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hfull hlow hroom hactive hcover with ⟨candidate, _pivot, _hpivot, first, Γ, hsrc⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have haccept_step : Accepts q old first Γ := haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_step) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hstep, hrebase⟩ theorem taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ candidate : ℕ, ∃ first : ℕ, ∃ Γ : TaoSection7Triangle, ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first ∧ TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ first ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, _hpivot, first, Γ, hsrc⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have haccept_step : Accepts q old first Γ := haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_step) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hstep, hrebase⟩ /-- State-preserving generic one-step support with the start-budget consequence. The theorem carries the encounter step and rebased source state forward, and derives `first <= B start` from the shared later-search right endpoint. It still does not choose or prove an abstract event-layer `Step start first`. -/ theorem taoSection7Case3_exists_encounterStep_rebasedWithGap_and_budget_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound B : ℕ → ℕ} {P start threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hq : q ∈ Finset.Icc start (start + threshold)) (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hright : taoSection7Case3LaterSearchBound gapBound threshold (start + threshold) ≤ B start) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ candidate : ℕ, ∃ first : ℕ, ∃ Γ : TaoSection7Triangle, ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first ∧ TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ first ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after ∧ first ≤ B start := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, hwindow, first, Γ, hsrc⟩ have hfirst_budget : first ≤ B start := taoSection7Case3_first_le_startBudget_of_laterWindow_rightBound hscale.gap_mono hq hwindow.mem_later_window hsrc.first_le_candidate hright have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have haccept_step : Accepts q old first Γ := haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_step) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hstep, hrebase, hfirst_budget⟩ /-- One-step producer-facing successor state for the generic `WithGap` recurrence. This packages the checked first-hit/rebase step with the successor active-triangle state obtained from the not-large-triangle event. It still assumes the acceptance and room facts needed by the local first-hit step; the future source-derived indexed recurrence must supply those hypotheses at every iteration. -/ theorem taoSection7Case3_exists_nextActiveSourceStateWithGap_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound B : ℕ → ℕ} {allowed : Set ℕ} {P start threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hq : q ∈ Finset.Icc start (start + threshold)) (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hright : taoSection7Case3LaterSearchBound gapBound threshold (start + threshold) ≤ B start) (hallowed_first : ∀ first : ℕ, q < first → first < P → first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ first : ℕ, ∃ Γ : TaoSection7Triangle, ∃ jNext : ℕ+, ∃ sNext : ℕ, ∃ preNext : List ℕ, ∃ presNext : List (List ℕ), q < first ∧ first < P ∧ first ≤ B start ∧ taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first ∧ TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ first ∧ TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound first Γ ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound jNext sNext preNext presNext := by rcases taoSection7Case3_exists_encounterStep_rebasedWithGap_and_budget_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAt) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (B := B) (P := P) (start := start) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) (old := old) hq hfull hscale hlow hroom hactive hcover hroom_next hfacts hright haccept with ⟨_candidate, first, Γ, before, next, after, _hsplit, _hidx, hsrc, hstep, hrebase, hbudget⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have hallowed : first ∈ allowed := hallowed_first first hsrc.first_hit.1 hsrc.first_ltP have hsmall : Γ.size < bound first := taoSection7Case3_triangle_size_lt_of_not_large_union (pointAt := pointAt) (family := family) (allowed := allowed) (bound := bound) hnotLarge hallowed hsrc.new_mem_family hsrc.new_mem refine ⟨first, Γ, taoSection7SourceBlockStartJFrom j (pre :: before), taoSection7SourceBlockStartSFrom s (pre :: before), next, after, hsrc.first_hit.1, hsrc.first_ltP, hbudget, hroom_first, hstep, ?_, hrebase⟩ exact { old_mem_family := hsrc.new_mem_family pivot_mem := hsrc.new_mem small_size := hsmall } theorem taoSection7Case3_exists_stoppingTransition_and_rebasedFullWindowSourcePathData_of_fullWindowSourcePathData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold q j s pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) : ∃ candidate : ℕ, ∃ first : ℕ, ∃ Γ : TaoSection7Triangle, ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres = before ++ next :: after ∧ before.length + 1 = first - q ∧ TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first ∧ TaoSection7Case3StoppingTransition pointAt family q old first Γ ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold first (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_fullWindowSourcePathData (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hfull hlow hroom hactive hcover with ⟨candidate, _pivot, _hpivot, first, Γ, hsrc⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP rcases taoSection7Case3_exists_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance_allWindowPackage (hfull := hfull) (hsrc := hsrc) hroom_first hfacts with ⟨before, next, after, hsplit, hidx, hrebase⟩ exact ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hsrc.to_stoppingTransition, hrebase⟩ namespace TaoSection7Case3FirstHitAcceptedTransition theorem to_stoppingTransition {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} (hstep : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first) : TaoSection7Case3StoppingTransition pointAt family q old first Γ := ⟨hstep.first_hit, hstep.new_mem_family, hstep.new_mem, hstep.new_ne_old⟩ end TaoSection7Case3FirstHitAcceptedTransition theorem taoSection7Case3_exists_stoppingTransition_of_sourcePointAtOfBlocks (W : ℕ → Prop) [DecidablePred W] {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {firstBlock : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hblocks : blocks = firstBlock :: before ++ pre :: pres) (hq : q = before.length + 1) (hP : P ≤ blocks.length) (hnoThree_first : taoSection7NoThree firstBlock) (hnoThree_before : ∀ block ∈ before, taoSection7NoThree block) (hnoThree_pre : taoSection7NoThree pre) (hnoThree_pres : ∀ block ∈ pres, taoSection7NoThree block) (hwhite : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (firstBlock :: before)) (taoSection7SourceBlockStartS (firstBlock :: before)) pre pres (candidate - q)).toPoint)) (hdomain : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (firstBlock :: before)) (taoSection7SourceBlockStartS (firstBlock :: before)) pre pres (candidate - q)).toPoint) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) : ∃ candidate hit Γ, candidate < P ∧ hit ≤ candidate ∧ hit < P ∧ TaoSection7Case3StoppingTransition (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) family q old hit Γ ∧ TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) (taoSection7SourceBlockStartJ (firstBlock :: before)) (taoSection7SourceBlockStartS (firstBlock :: before)) pre pres P q old Γ candidate hit := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePointAtOfBlocks (W := W) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := firstBlock) (before := before) (pre := pre) (pres := pres) hblocks hq hP hnoThree_first hnoThree_before hnoThree_pre hnoThree_pres hwhite hdomain hlow hroom hactive hcover with ⟨candidate, _candidateTriangle, hstep, hit, Γ, hsrc⟩ exact ⟨candidate, hit, Γ, hstep.candidate_ltP, hsrc.first_le_candidate, hsrc.first_ltP, hsrc.to_stoppingTransition, hsrc⟩ theorem taoSection7Case3_exists_encounterStep_of_sourcePointAtOfBlocks (W : ℕ → Prop) [DecidablePred W] (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {firstBlock : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hblocks : blocks = firstBlock :: before ++ pre :: pres) (hq : q = before.length + 1) (hP : P ≤ blocks.length) (hnoThree_first : taoSection7NoThree firstBlock) (hnoThree_before : ∀ block ∈ before, taoSection7NoThree block) (hnoThree_pre : taoSection7NoThree pre) (hnoThree_pres : ∀ block ∈ pres, taoSection7NoThree block) (hwhite : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → (W candidate ↔ taoSection7SourceWhitePoint n xi epsilon (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (firstBlock :: before)) (taoSection7SourceBlockStartS (firstBlock :: before)) pre pres (candidate - q)).toPoint)) (hdomain : ∀ candidate : ℕ, candidate ∈ Finset.Icc (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q) (taoSection7Case3LaterSearchStart (taoSection7Case3ExitGapBound A) q + threshold) → taoSection7SourcePointInDomain J (taoSection7SourceHitPathPoint (taoSection7SourceBlockStartJ (firstBlock :: before)) (taoSection7SourceBlockStartS (firstBlock :: before)) pre pres (candidate - q)).toPoint) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (haccept : ∀ hit Γ, hit < P → taoSection7SourceBlackInDomain n xi epsilon J ((taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) hit) → TaoSection7Case3FirstAfterTriangleHitFrom (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) family old q hit → Γ ∈ family → Γ.Mem ((taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) hit) → Γ ≠ old → Accepts q old hit Γ) : ∃ hit Γ, TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family (taoSection7SourcePointAtOfBlocks firstBlock (before ++ pre :: pres)) Accepts P q old Γ hit := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePointAtOfBlocks (W := W) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := firstBlock) (before := before) (pre := pre) (pres := pres) hblocks hq hP hnoThree_first hnoThree_before hnoThree_pre hnoThree_pres hwhite hdomain hlow hroom hactive hcover with ⟨_candidate, _pivotΓ, _hpivot, hit, Γ, hsrc⟩ refine ⟨hit, Γ, ?_⟩ exact hsrc.to_encounterStep (haccept hit Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old) theorem taoSection7Case3_exists_encounterStep_of_globalFullWindowSourceSuffixProvenance (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hglobal : TaoSection7Case3GlobalFullWindowSourceSuffixProvenance W pointAt n J A xi epsilon P threshold q blocks before pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ first Γ, TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ first := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_globalFullWindowSourceSuffixProvenance (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (before := before) (pre := pre) (pres := pres) hglobal hlow hroom hactive hcover with ⟨_candidate, _pivotΓ, _hpivot, first, Γ, hsrc⟩ refine ⟨first, Γ, ?_⟩ exact hsrc.to_encounterStep (haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old) theorem taoSection7Case3_exists_encounterStep_of_distinguishedGlobalPathFullWindowProvenance (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {P threshold q : ℕ} {blocks : List (List ℕ)} {firstBlock : List ℕ} {before : List (List ℕ)} {pre : List ℕ} {pres : List (List ℕ)} (hsrc : TaoSection7Case3DistinguishedGlobalPathFullWindowProvenance W pointAt n J A xi epsilon P threshold q blocks firstBlock before pre pres) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAt family pointAt A q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ hit Γ, TaoSection7Case3EncounterStep (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P q old Γ hit := by rcases taoSection7Case3_exists_firstHitSourceProvenance_of_distinguishedGlobalPathFullWindowProvenance (W := W) (pointAt := pointAt) (n := n) (J := J) (A := A) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (P := P) (threshold := threshold) (q := q) (blocks := blocks) (first := firstBlock) (before := before) (pre := pre) (pres := pres) hsrc hlow hroom hactive hcover with ⟨_candidate, _pivotΓ, _hpivot, hit, Γ, hfirst⟩ refine ⟨hit, Γ, ?_⟩ exact hfirst.to_encounterStep (haccept hit Γ hfirst.first_ltP hfirst.first_black hfirst.first_hit hfirst.new_mem_family hfirst.new_mem hfirst.new_ne_old) namespace TaoSection7Case3StoppingTransition theorem first_black_of_cover {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q first : ℕ} {old Γ : TaoSection7Triangle} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ) : black (pointAt first) := taoSection7Case3_firstAfterTriangleHitFrom_black_of_cover hcover hstep.first_after_exit theorem to_firstHitAcceptedTransition {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q first : ℕ} {old Γ : TaoSection7Triangle} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ) (haccept : black (pointAt first) → Accepts q old first Γ) : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts q old Γ first := by have hblack : black (pointAt first) := first_black_of_cover hcover hstep exact ⟨hstep.first_after_exit, hblack, hstep.new_mem_family, hstep.new_mem, hstep.new_ne_old, haccept hblack⟩ end TaoSection7Case3StoppingTransition /-- Tail of a Tao-style stopping trace after a current stopping triangle. Each cons step advances from `(q, old)` to the next first post-exit triangle hit `(first, Γ)`, then continues from that new state. -/ inductive TaoSection7Case3StoppingTail (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) : ℕ → TaoSection7Triangle → List (ℕ × TaoSection7Triangle) → Prop | nil (q : ℕ) (old : TaoSection7Triangle) : TaoSection7Case3StoppingTail pointAt family q old [] | cons {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ) (htail : TaoSection7Case3StoppingTail pointAt family first Γ rest) : TaoSection7Case3StoppingTail pointAt family q old ((first, Γ) :: rest) /-- Finite prefix of Tao's source stopping trace. The empty prefix is allowed as a support object. A complete run below adds a terminal condition saying whether no first hit exists or no later black point exists after exiting the last triangle. -/ inductive TaoSection7Case3StoppingPrefix (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) : List (ℕ × TaoSection7Triangle) → Prop | nil : TaoSection7Case3StoppingPrefix pointAt family [] | cons {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hinit : TaoSection7Case3InitialStoppingStep pointAt family first Γ) (htail : TaoSection7Case3StoppingTail pointAt family first Γ rest) : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest) /-- No initial triangle hit exists after `start`. -/ def TaoSection7Case3NoInitialTriangleHitAfter (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (start : ℕ) : Prop := ∀ t : ℕ, start < t → ¬ TaoSection7Case3TriangleHit pointAt family t /-- Terminal condition after the last constructed stopping triangle: once the path strictly exits that triangle vertically, no later point is black. -/ def TaoSection7Case3NoLaterBlackAfterExit (black : TaoSection7Point → Prop) (pointAt : ℕ → TaoSection7Point) (old : TaoSection7Triangle) : Prop := ∀ t : ℕ, 0 < t → old.cornerL < (pointAt t).l → ¬ black (pointAt t) /-- If every post-exit time has left the finite source domain horizontally, then there are no later post-exit triangle hits for a family covering the domain-gated source black set. -/ theorem taoSection7Case3_noAfterTriangleHit_of_postExit_sourceDomainExit {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hout : ∀ t : ℕ, 0 < t → old.cornerL < (pointAt t).l → J < ((pointAt t).j : ℕ)) : ∀ t : ℕ, 0 < t → ¬ TaoSection7Case3AfterTriangleHit pointAt family old t := by intro t ht hhit rcases hhit with ⟨hexit, hmem⟩ have hblack : taoSection7SourceBlackInDomain n xi epsilon J (pointAt t) := (hcover (pointAt t)).mpr hmem exact (Nat.not_lt_of_ge (taoSection7SourceBlackInDomain_domain hblack)) (hout t ht hexit) namespace TaoSection7Case3NoLaterBlackAfterExit /-- Terminality producer from the absence of any later post-exit triangle hit. The family cover converts a hypothetical later black point into a later triangle hit after `old`, contradicting `hnoAfter`. -/ theorem of_noAfterTriangleHit {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hnoAfter : ∀ t : ℕ, 0 < t → ¬ TaoSection7Case3AfterTriangleHit pointAt family old t) : TaoSection7Case3NoLaterBlackAfterExit black pointAt old := by intro t ht hexit hblack exact hnoAfter t ht ⟨hexit, (hcover (pointAt t)).mp hblack⟩ /-- Terminality producer from a source-domain exit certificate after every post-exit time. -/ theorem of_postExit_sourceDomainExit {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {old : TaoSection7Triangle} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hout : ∀ t : ℕ, 0 < t → old.cornerL < (pointAt t).l → J < ((pointAt t).j : ℕ)) : TaoSection7Case3NoLaterBlackAfterExit (taoSection7SourceBlackInDomain n xi epsilon J) pointAt old := of_noAfterTriangleHit hcover (taoSection7Case3_noAfterTriangleHit_of_postExit_sourceDomainExit hcover hout) end TaoSection7Case3NoLaterBlackAfterExit /-- Last `(time, triangle)` pair of a finite stopping trace, if nonempty. -/ def TaoSection7Case3StoppingLast? : List (ℕ × TaoSection7Triangle) → Option (ℕ × TaoSection7Triangle) | [] => none | [step] => some step | _step :: rest => TaoSection7Case3StoppingLast? rest /-- The terminal/maximality condition corresponding to Tao's definition of `r`. For an empty trace, no initial triangle hit exists. For a nonempty trace, the last stopping triangle admits no later black point after strict vertical exit. -/ def TaoSection7Case3StoppingTerminal (black : TaoSection7Point → Prop) (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) : List (ℕ × TaoSection7Triangle) → Prop | [] => TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0 | steps => match TaoSection7Case3StoppingLast? steps with | none => TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0 | some last => TaoSection7Case3NoLaterBlackAfterExit black pointAt last.2 namespace TaoSection7Case3StoppingTerminal theorem nil {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} (hterminal : TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0) : TaoSection7Case3StoppingTerminal black pointAt family [] := by simpa [TaoSection7Case3StoppingTerminal] using hterminal /-- Build the nonempty terminal/maximality condition from the explicit last stopping triangle and the no-later-black-after-exit certificate. -/ theorem of_last?_eq_some_noLaterBlackAfterExit {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {last : ℕ × TaoSection7Triangle} (hlast : TaoSection7Case3StoppingLast? steps = some last) (hterminal : TaoSection7Case3NoLaterBlackAfterExit black pointAt last.2) : TaoSection7Case3StoppingTerminal black pointAt family steps := by cases steps with | nil => simp [TaoSection7Case3StoppingLast?] at hlast | cons step rest => dsimp [TaoSection7Case3StoppingTerminal] rw [hlast] exact hterminal theorem singleton_of_noLaterBlackAfterExit {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {step : ℕ × TaoSection7Triangle} (hterminal : TaoSection7Case3NoLaterBlackAfterExit black pointAt step.2) : TaoSection7Case3StoppingTerminal black pointAt family [step] := of_last?_eq_some_noLaterBlackAfterExit (by simp [TaoSection7Case3StoppingLast?]) hterminal theorem noInitialTriangleHitAfter_of_nil {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} (hterminal : TaoSection7Case3StoppingTerminal black pointAt family []) : TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0 := by simpa [TaoSection7Case3StoppingTerminal] using hterminal theorem noLaterBlackAfterExit_of_last?_eq_some {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {last : ℕ × TaoSection7Triangle} (hterminal : TaoSection7Case3StoppingTerminal black pointAt family steps) (hlast : TaoSection7Case3StoppingLast? steps = some last) : TaoSection7Case3NoLaterBlackAfterExit black pointAt last.2 := by cases steps with | nil => simp [TaoSection7Case3StoppingLast?] at hlast | cons step rest => dsimp [TaoSection7Case3StoppingTerminal] at hterminal rw [hlast] at hterminal exact hterminal end TaoSection7Case3StoppingTerminal /-- Complete finite source stopping run: a finite prefix together with the maximality/terminal condition that defines the stopping count `r`. -/ structure TaoSection7Case3SourceStoppingRun (black : TaoSection7Point → Prop) (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (steps : List (ℕ × TaoSection7Triangle)) : Prop where trace_prefix : TaoSection7Case3StoppingPrefix pointAt family steps terminal_condition : TaoSection7Case3StoppingTerminal black pointAt family steps namespace TaoSection7Case3RecursiveEncounterChain theorem to_stoppingTail {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old steps) : TaoSection7Case3StoppingTail pointAt family q old steps := by induction hchain with | nil q old => exact TaoSection7Case3StoppingTail.nil q old | cons hstep _htail ih => exact TaoSection7Case3StoppingTail.cons hstep.accepted.to_stoppingTransition ih theorem all_stop_ltP {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P q old steps) : ∀ step ∈ steps, step.1 < P := by induction hchain with | nil q old => intro step hmem simp at hmem | cons hstep _htail ih => intro step hmem simp at hmem rcases hmem with hstep_eq | hmem_tail · cases hstep_eq exact hstep.first_ltP · exact ih step hmem_tail end TaoSection7Case3RecursiveEncounterChain namespace TaoSection7Case3StoppingTail theorem head_step {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (htail : TaoSection7Case3StoppingTail pointAt family q old ((first, Γ) :: rest)) : TaoSection7Case3StoppingTransition pointAt family q old first Γ := by cases htail with | cons hstep _htail => exact hstep theorem tail {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (htail : TaoSection7Case3StoppingTail pointAt family q old ((first, Γ) :: rest)) : TaoSection7Case3StoppingTail pointAt family first Γ rest := by cases htail with | cons _hstep htail => exact htail theorem head_time_gt {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q first : ℕ} {old Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (htail : TaoSection7Case3StoppingTail pointAt family q old ((first, Γ) :: rest)) : q < first := (head_step htail).first_after_exit.1 theorem to_recursiveFirstHitChain {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (htail : TaoSection7Case3StoppingTail pointAt family q old steps) (haccept : ∀ {q first : ℕ} {old Γ : TaoSection7Triangle}, TaoSection7Case3StoppingTransition pointAt family q old first Γ → black (pointAt first) → Accepts q old first Γ) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts q old steps := by induction htail with | nil q old => exact TaoSection7Case3RecursiveFirstHitChain.nil q old | cons hstep htail ih => have haccepted : TaoSection7Case3FirstHitAcceptedTransition black family pointAt Accepts _ _ _ _ := TaoSection7Case3StoppingTransition.to_firstHitAcceptedTransition hcover hstep (haccept hstep) exact TaoSection7Case3RecursiveFirstHitChain.cons haccepted ih theorem take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (htail : TaoSection7Case3StoppingTail pointAt family q old steps) (R : ℕ) : TaoSection7Case3StoppingTail pointAt family q old (steps.take R) := by induction htail generalizing R with | nil q old => simpa using TaoSection7Case3StoppingTail.nil (pointAt := pointAt) (family := family) q old | cons hstep _htail ih => cases R with | zero => simpa using TaoSection7Case3StoppingTail.nil (pointAt := pointAt) (family := family) _ _ | succ Rtail => simpa using TaoSection7Case3StoppingTail.cons hstep (ih Rtail) end TaoSection7Case3StoppingTail namespace TaoSection7Case3StoppingPrefix theorem head_initial {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hprefix : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest)) : TaoSection7Case3InitialStoppingStep pointAt family first Γ := by cases hprefix with | cons hinit _htail => exact hinit theorem tail {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hprefix : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest)) : TaoSection7Case3StoppingTail pointAt family first Γ rest := by cases hprefix with | cons _hinit htail => exact htail theorem head_time_pos {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hprefix : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest)) : 0 < first := (head_initial hprefix).first_hit.1 theorem tail_recursiveFirstHitChain {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hcover : TaoSection7TriangleFamilyCoverBlack black family) (hprefix : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest)) (haccept : ∀ {q first : ℕ} {old Γ : TaoSection7Triangle}, TaoSection7Case3StoppingTransition pointAt family q old first Γ → black (pointAt first) → Accepts q old first Γ) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts first Γ rest := TaoSection7Case3StoppingTail.to_recursiveFirstHitChain hcover (tail hprefix) haccept theorem take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (R : ℕ) : TaoSection7Case3StoppingPrefix pointAt family (steps.take R) := by cases hprefix with | nil => simpa using TaoSection7Case3StoppingPrefix.nil (pointAt := pointAt) (family := family) | cons hinit htail => cases R with | zero => simpa using TaoSection7Case3StoppingPrefix.nil (pointAt := pointAt) (family := family) | succ Rtail => simpa using TaoSection7Case3StoppingPrefix.cons hinit (TaoSection7Case3StoppingTail.take htail Rtail) end TaoSection7Case3StoppingPrefix namespace TaoSection7Case3SourceStoppingRun theorem of_prefix_terminal {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hterminal : TaoSection7Case3StoppingTerminal black pointAt family steps) : TaoSection7Case3SourceStoppingRun black pointAt family steps := ⟨hprefix, hterminal⟩ theorem nil_of_noInitialTriangleHitAfter {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} (hterminal : TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0) : TaoSection7Case3SourceStoppingRun black pointAt family [] := of_prefix_terminal TaoSection7Case3StoppingPrefix.nil (TaoSection7Case3StoppingTerminal.nil hterminal) theorem of_prefix_last?_noLaterBlackAfterExit {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {last : ℕ × TaoSection7Triangle} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hlast : TaoSection7Case3StoppingLast? steps = some last) (hterminal : TaoSection7Case3NoLaterBlackAfterExit black pointAt last.2) : TaoSection7Case3SourceStoppingRun black pointAt family steps := of_prefix_terminal hprefix (TaoSection7Case3StoppingTerminal.of_last?_eq_some_noLaterBlackAfterExit hlast hterminal) theorem cons_of_initial_tail_terminal {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hinit : TaoSection7Case3InitialStoppingStep pointAt family first Γ) (htail : TaoSection7Case3StoppingTail pointAt family first Γ rest) (hterminal : TaoSection7Case3StoppingTerminal black pointAt family ((first, Γ) :: rest)) : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest) := of_prefix_terminal (TaoSection7Case3StoppingPrefix.cons hinit htail) hterminal theorem cons_of_initial_tail_last?_noLaterBlackAfterExit {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} {last : ℕ × TaoSection7Triangle} (hinit : TaoSection7Case3InitialStoppingStep pointAt family first Γ) (htail : TaoSection7Case3StoppingTail pointAt family first Γ rest) (hlast : TaoSection7Case3StoppingLast? ((first, Γ) :: rest) = some last) (hterminal : TaoSection7Case3NoLaterBlackAfterExit black pointAt last.2) : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest) := of_prefix_last?_noLaterBlackAfterExit (TaoSection7Case3StoppingPrefix.cons hinit htail) hlast hterminal /-- The finite stopping count represented by a complete run. -/ def r (steps : List (ℕ × TaoSection7Triangle)) : ℕ := steps.length /-- One-based accessor for Tao's `t_R`. -/ def tR? : List (ℕ × TaoSection7Triangle) → ℕ → Option ℕ | [], _ => none | _step :: _rest, 0 => none | step :: _rest, 1 => some step.1 | _step :: rest, R + 2 => tR? rest (R + 1) /-- The run has at least `R` constructed stopping times. -/ def HasAtLeast (steps : List (ℕ × TaoSection7Triangle)) (R : ℕ) : Prop := R ≤ r steps /-- Endpoint package needed by the Case 3 counting layer. Future source iteration work must prove this from the complete stopping run and low-window hypotheses; this definition itself is only bookkeeping. -/ def HasTWithinWindow (steps : List (ℕ × TaoSection7Triangle)) (R P : ℕ) : Prop := ∃ t : ℕ, tR? steps R = some t ∧ t < P /-- Finite prefix package for the deterministic Case 3 endpoint route. This deliberately does not require the terminal/maximality condition of a full `TaoSection7Case3SourceStoppingRun`: it is the producer surface for proving an `R`-long stopping prefix inside the finite `P` window before the final no-later-black argument is available. -/ structure PrefixWindowData (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (steps : List (ℕ × TaoSection7Triangle)) (R P : ℕ) : Prop where trace_prefix : TaoSection7Case3StoppingPrefix pointAt family steps hasAtLeast : HasAtLeast steps R all_stop_ltP : ∀ step ∈ steps, step.1 < P theorem prefixWindowData_of_prefix_hasAtLeast_all_ltP {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hR : HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) : PrefixWindowData pointAt family steps R P := ⟨hprefix, hR, hall⟩ theorem tR?_some_length_le {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (htR : tR? steps R = some tR) : R ≤ steps.length := by induction steps generalizing R tR with | nil => cases R <;> simp [tR?] at htR | cons step rest ih => cases R with | zero => simp | succ Rpred => cases Rpred with | zero => simp | succ Rtail => have hrest : Rtail + 1 ≤ rest.length := ih (R := Rtail + 1) (tR := tR) (by simpa [tR?] using htR) simp omega theorem tR?_isSome_pos {steps : List (ℕ × TaoSection7Triangle)} {R : ℕ} (h : (tR? steps R).isSome = true) : 0 < R := by cases R with | zero => cases steps <;> simp [tR?] at h | succ R => exact Nat.succ_pos R theorem tR?_take_eq_of_tR? {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (htR : tR? steps R = some tR) : tR? (steps.take R) R = some tR := by induction steps generalizing R tR with | nil => cases R <;> simp [tR?] at htR | cons step rest ih => cases R with | zero => simp [tR?] at htR | succ Rpred => cases Rpred with | zero => simpa [tR?] using htR | succ Rtail => have hrest : tR? (rest.take (Rtail + 1)) (Rtail + 1) = some tR := ih (R := Rtail + 1) (tR := tR) (by simpa [tR?] using htR) simpa [tR?] using hrest theorem tR?_take_self (steps : List (ℕ × TaoSection7Triangle)) (R : ℕ) : tR? (steps.take R) R = tR? steps R := by induction steps generalizing R with | nil => cases R <;> simp [tR?] | cons step rest ih => cases R with | zero => simp [tR?] | succ Rpred => cases Rpred with | zero => simp [tR?] | succ Rtail => simpa [tR?] using ih (Rtail + 1) /-- If a finite stopping list has at least `R` entries and every recorded stopping time lies below `P`, then the one-based `t_R` accessor lies inside the same P-window. This is only list bookkeeping. Future source iteration work must still prove `0 < R`, `HasAtLeast steps R`, and the per-stop window bound. -/ theorem hasTWithinWindow_of_hasAtLeast_all_ltP {steps : List (ℕ × TaoSection7Triangle)} {R P : ℕ} (hRpos : 0 < R) (hR : HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) : HasTWithinWindow steps R P := by induction steps generalizing R with | nil => simp [HasAtLeast, r] at hR omega | cons step rest ih => cases R with | zero => omega | succ Rpred => cases Rpred with | zero => refine ⟨step.1, ?_, ?_⟩ · simp [tR?] · exact hall step (by simp) | succ Rtail => have hRtail_pos : 0 < Rtail + 1 := by omega have hRtail : HasAtLeast rest (Rtail + 1) := by simp [HasAtLeast, r] at hR ⊢ omega have hall_tail : ∀ tailStep ∈ rest, tailStep.1 < P := by intro tailStep htailStep exact hall tailStep (by simp [htailStep]) rcases ih hRtail_pos hRtail hall_tail with ⟨t, htR, htP⟩ refine ⟨t, ?_, htP⟩ simpa [tR?] using htR theorem stoppingTail_base_lt_tR_of_tR? {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (htail : TaoSection7Case3StoppingTail pointAt family q old steps) (htR : tR? steps R = some tR) : q < tR := by induction steps generalizing q old R tR with | nil => cases htail cases R <;> simp [tR?] at htR | cons step rest ih => cases step with | mk first Γ => cases htail with | cons hstep htail => cases R with | zero => simp [tR?] at htR | succ Rpred => cases Rpred with | zero => simp [tR?] at htR cases htR exact hstep.first_after_exit.1 | succ Rtail => have hrest : tR? rest (Rtail + 1) = some tR := by simpa [tR?] using htR exact lt_trans hstep.first_after_exit.1 (ih (q := first) (old := Γ) (R := Rtail + 1) (tR := tR) htail hrest) theorem stoppingTail_all_stop_le_tR_of_mem_take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (htail : TaoSection7Case3StoppingTail pointAt family q old steps) (htR : tR? steps R = some tR) : ∀ step ∈ steps.take R, step.1 ≤ tR := by induction steps generalizing q old R tR with | nil => intro step hmem simp at hmem | cons head rest ih => cases head with | mk first Γ => cases htail with | cons hstep htail => cases R with | zero => intro step hmem simp at hmem | succ Rpred => cases Rpred with | zero => intro step hmem have hstep_eq : step = (first, Γ) := by simpa using hmem cases hstep_eq simp [tR?] at htR cases htR exact le_rfl | succ Rtail => have hrest : tR? rest (Rtail + 1) = some tR := by simpa [tR?] using htR intro step hmem simp at hmem rcases hmem with hhead | htailmem · cases hhead exact le_of_lt (stoppingTail_base_lt_tR_of_tR? htail hrest) · exact ih (q := first) (old := Γ) (R := Rtail + 1) (tR := tR) htail hrest step htailmem theorem stoppingPrefix_all_stop_le_tR_of_mem_take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (htR : tR? steps R = some tR) : ∀ step ∈ steps.take R, step.1 ≤ tR := by induction steps generalizing R tR with | nil => intro step hmem simp at hmem | cons head rest _ih => cases head with | mk first Γ => cases hprefix with | cons _hinit htail => cases R with | zero => intro step hmem simp at hmem | succ Rpred => cases Rpred with | zero => intro step hmem have hstep_eq : step = (first, Γ) := by simpa using hmem cases hstep_eq simp [tR?] at htR cases htR exact le_rfl | succ Rtail => have hrest : tR? rest (Rtail + 1) = some tR := by simpa [tR?] using htR intro step hmem simp at hmem rcases hmem with hhead | htailmem · cases hhead exact le_of_lt (stoppingTail_base_lt_tR_of_tR? htail hrest) · exact stoppingTail_all_stop_le_tR_of_mem_take htail hrest step htailmem theorem prefixWindowData_of_tR_ltP_take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR P : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (htR : tR? steps R = some tR) (htR_ltP : tR < P) : PrefixWindowData pointAt family (steps.take R) R P := by have hlen : R ≤ steps.length := tR?_some_length_le htR refine { trace_prefix := TaoSection7Case3StoppingPrefix.take hprefix R hasAtLeast := ?_ all_stop_ltP := ?_ } · simp [HasAtLeast, r, List.length_take, Nat.min_eq_left hlen] · intro step hmem exact lt_of_le_of_lt (stoppingPrefix_all_stop_le_tR_of_mem_take hprefix htR step hmem) htR_ltP namespace PrefixWindowData theorem hasTWithinWindow {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P : ℕ} (hdata : PrefixWindowData pointAt family steps R P) (hRpos : 0 < R) : HasTWithinWindow steps R P := hasTWithinWindow_of_hasAtLeast_all_ltP hRpos hdata.hasAtLeast hdata.all_stop_ltP theorem not_low_window_of_many_positive (W : ℕ → Prop) [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (hdata : PrefixWindowData pointAt family steps R P) (hRpos : 0 < R) (hmany : ∀ t : ℕ, tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := by rcases hdata.hasTWithinWindow hRpos with ⟨t, htR, htP⟩ exact taoSection7Case3_many_positive_offsets_not_low_window W htP (hmany t htR) end PrefixWindowData /-- Endpoint bridge from a source stopping-run accessor to the existing finite P-window counting contradiction. Future Lemma 7.9 work should supply the positive-offset lower bound at the actual `t_R`; future source iteration work should supply `HasTWithinWindow`. This lemma only performs the endpoint/count-window transfer. -/ theorem not_low_window_of_hasTWithinWindow_many_positive (W : ℕ → Prop) [DecidablePred W] {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (hwithin : HasTWithinWindow steps R P) (hmany : ∀ t : ℕ, tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := by rcases hwithin with ⟨t, htR, htP⟩ exact taoSection7Case3_many_positive_offsets_not_low_window W htP (hmany t htR) /-- Pointwise form of the same endpoint/count-window bridge when the caller has already extracted the actual `t_R`. -/ theorem not_low_window_of_tR_many_positive (W : ℕ → Prop) [DecidablePred W] {steps : List (ℕ × TaoSection7Triangle)} {R P threshold t : ℕ} (htR : tR? steps R = some t) (htP : t < P) (hmany : threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := not_low_window_of_hasTWithinWindow_many_positive (W := W) (steps := steps) (R := R) (P := P) (threshold := threshold) ⟨t, htR, htP⟩ (fun t' ht' => by rw [htR] at ht' cases ht' exact hmany) /-- Combined assumption-consuming adapter: if future source iteration work proves the run has at least `R` stops and all recorded stop times are inside the P-window, and future Lemma 7.9 work supplies the positive-offset lower bound at the extracted `t_R`, then the low P-window event is impossible. -/ theorem not_low_window_of_hasAtLeast_all_ltP_many_positive (W : ℕ → Prop) [DecidablePred W] {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (hRpos : 0 < R) (hR : HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) (hmany : ∀ t : ℕ, tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := not_low_window_of_hasTWithinWindow_many_positive (W := W) (steps := steps) (R := R) (P := P) (threshold := threshold) (hasTWithinWindow_of_hasAtLeast_all_ltP hRpos hR hall) hmany theorem not_low_window_of_run_hasTWithinWindow_many_positive (W : ℕ → Prop) [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (_hrun : TaoSection7Case3SourceStoppingRun black pointAt family steps) (hwithin : HasTWithinWindow steps R P) (hmany : ∀ t : ℕ, tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := not_low_window_of_hasTWithinWindow_many_positive (W := W) (steps := steps) (R := R) (P := P) (threshold := threshold) hwithin hmany theorem not_low_window_of_run_hasAtLeast_all_ltP_many_positive (W : ℕ → Prop) [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family steps) (hRpos : 0 < R) (hR : HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) (hmany : ∀ t : ℕ, tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := not_low_window_of_run_hasTWithinWindow_many_positive (W := W) (steps := steps) (R := R) (P := P) (threshold := threshold) hrun (hasTWithinWindow_of_hasAtLeast_all_ltP hRpos hR hall) hmany theorem head_initial_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest)) : TaoSection7Case3InitialStoppingStep pointAt family first Γ := TaoSection7Case3StoppingPrefix.head_initial hrun.trace_prefix theorem tail_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest)) : TaoSection7Case3StoppingTail pointAt family first Γ rest := TaoSection7Case3StoppingPrefix.tail hrun.trace_prefix theorem head_time_pos_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest)) : 0 < first := TaoSection7Case3StoppingPrefix.head_time_pos hrun.trace_prefix theorem tail_recursiveFirstHitChain_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family ((first, Γ) :: rest)) (hcover : TaoSection7TriangleFamilyCoverBlack black family) (haccept : ∀ {q first : ℕ} {old Γ : TaoSection7Triangle}, TaoSection7Case3StoppingTransition pointAt family q old first Γ → black (pointAt first) → Accepts q old first Γ) : TaoSection7Case3RecursiveFirstHitChain black family pointAt Accepts first Γ rest := TaoSection7Case3StoppingPrefix.tail_recursiveFirstHitChain hcover hrun.trace_prefix haccept theorem trace_prefix_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family steps) : TaoSection7Case3StoppingPrefix pointAt family steps := hrun.trace_prefix theorem terminal_condition_of_run {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (hrun : TaoSection7Case3SourceStoppingRun black pointAt family steps) : TaoSection7Case3StoppingTerminal black pointAt family steps := hrun.terminal_condition end TaoSection7Case3SourceStoppingRun /-- Producer-facing data for Tao's constructed source stopping run at a fixed cutoff. This is still only a socket: future source-iteration work must produce the run, the `R <= r` evidence, the concrete `t_R`, and the Lemma 7.9 count identity for the same `steps`. -/ structure TaoSection7Case3ConstructedSourceStoppingRunData (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (steps : List (ℕ × TaoSection7Triangle)) (R tR lemma79Count : ℕ) : Prop where source_run : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps R_pos : 0 < R R_le_r : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R tR_eq : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR count_identity : lemma79Count = taoSection7Case3PositiveWhiteCount W tR /-- Event-free source identity for Tao's restarted Case 3 stopping run. This records that the finite stopping run and the low-window predicate are tied to one restarted source trajectory. It deliberately does not contain outside-`E*` evidence, `F*`/Markov evidence, endpoint-window production, or probability budgets. -/ structure TaoSection7Case3ActualRunSourceIdentity (sourceAt : ℕ → TaoSection7Point) (restart : ℕ) (W : ℕ → Prop) [DecidablePred W] (black : TaoSection7Point → Prop) (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P : ℕ) (steps : List (ℕ × TaoSection7Triangle)) (R tR lemma79Count : ℕ) : Prop where restart_pointAt : ∀ p : ℕ, p < P → pointAt p = sourceAt (restart + p) sameW : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P black_iff_source : ∀ p : ℕ, p < P → (black (pointAt p) ↔ taoSection7SourceBlackPoint n xi epsilon (pointAt p)) source_run : TaoSection7Case3SourceStoppingRun black pointAt family steps R_pos : 0 < R R_le_r : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R tR_eq : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR count_identity : lemma79Count = taoSection7Case3PositiveWhiteCount W tR namespace TaoSection7Case3ActualRunSourceIdentity theorem of_sourceBlackInDomain_sourceRun {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (hrestart : ∀ p : ℕ, p < P → pointAt p = sourceAt (restart + p)) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hRpos : 0 < R) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R) (htR : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR) (hcount : lemma79Count = taoSection7Case3PositiveWhiteCount W tR) : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps R tR lemma79Count where restart_pointAt := hrestart sameW := hfacts black_iff_source := by intro p hp exact hfacts.blackInDomain_iff_sourceBlackPoint hp source_run := hrun R_pos := hRpos R_le_r := hR tR_eq := htR count_identity := hcount theorem sourceWhite_iff {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count p : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (hp : p < P) : W p ↔ taoSection7SourceWhitePoint n xi epsilon (pointAt p) := h.sameW.white_all p hp theorem sourceDomain {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count p : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (hp : p < P) : taoSection7SourcePointInDomain J (pointAt p) := h.sameW.domain_all p hp theorem sourceAt_of_restart {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count p : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (hp : p < P) : sourceAt (restart + p) = pointAt p := (h.restart_pointAt p hp).symm theorem tracePrefix {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) : TaoSection7Case3StoppingPrefix pointAt family steps := h.source_run.trace_prefix /-- The terminal condition is relative to the restarted `pointAt` path carried by the actual-run identity. Converting from an unrestarted ambient source path is a separate suffix-reindexing obligation. -/ theorem terminalCondition {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) : TaoSection7Case3StoppingTerminal black pointAt family steps := h.source_run.terminal_condition theorem R_le_r_steps {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) : R ≤ TaoSection7Case3SourceStoppingRun.r steps := by simpa [TaoSection7Case3SourceStoppingRun.HasAtLeast] using h.R_le_r theorem hasTWithinWindow_of_tR_ltP {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htR_ltP : tR < P) : TaoSection7Case3SourceStoppingRun.HasTWithinWindow steps R P := ⟨tR, h.tR_eq, htR_ltP⟩ theorem prefixWindowData_of_all_stop_ltP {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (hall : ∀ step ∈ steps, step.1 < P) : TaoSection7Case3SourceStoppingRun.PrefixWindowData pointAt family steps R P := { trace_prefix := h.source_run.trace_prefix hasAtLeast := h.R_le_r all_stop_ltP := hall } theorem prefixWindowData_of_tR_ltP_take {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htR_ltP : tR < P) : TaoSection7Case3SourceStoppingRun.PrefixWindowData pointAt family (steps.take R) R P := TaoSection7Case3SourceStoppingRun.prefixWindowData_of_tR_ltP_take h.source_run.trace_prefix h.tR_eq htR_ltP /-- Endpoint-window extractor for actual-run identities. If later source iteration work proves that every recorded stop of the actual run lies below the finite window `P`, then the accessed `t_R` carried by the identity also lies below `P`. This removes the endpoint half of the outside-low-white certificate from callers that can already produce the per-stop window bound. -/ theorem tR_ltP_of_all_stop_ltP {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (hall : ∀ step ∈ steps, step.1 < P) : tR < P := by rcases TaoSection7Case3SourceStoppingRun.hasTWithinWindow_of_hasAtLeast_all_ltP h.R_pos h.R_le_r hall with ⟨t, htR, htP⟩ rw [h.tR_eq] at htR cases htR exact htP /-- Count-window consumer for the actual-run identity. The endpoint fact `tR < P` is an explicit downstream input, so this lemma does not produce endpoint-window credit. -/ theorem not_low_window_of_tR_ltP_many_positive {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count threshold : ℕ} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htR_ltP : tR < P) (hmany : threshold < lemma79Count) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := by exact TaoSection7Case3SourceStoppingRun.not_low_window_of_tR_many_positive (W := W) (steps := steps) (R := R) (P := P) (threshold := threshold) (t := tR) h.tR_eq htR_ltP (by simpa [h.count_identity] using hmany) end TaoSection7Case3ActualRunSourceIdentity namespace TaoSection7Case3ConstructedSourceStoppingRunData /-- Build constructed source-run data from explicit terminal/maximality evidence. This removes the opaque `source_run` premise, but still leaves the real producer obligations visible: the stopping prefix, the last-triangle no-later-black certificate, the cutoff evidence, the `t_R` accessor equality, and the Lemma 7.9 count identity. -/ theorem of_prefix_last?_noLaterBlackAfterExit {W : ℕ → Prop} [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {last : ℕ × TaoSection7Triangle} {R tR lemma79Count : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hlast : TaoSection7Case3StoppingLast? steps = some last) (hterminal : TaoSection7Case3NoLaterBlackAfterExit (taoSection7SourceBlackInDomain n xi epsilon J) pointAt last.2) (hRpos : 0 < R) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R) (htR : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR) (hcount : lemma79Count = taoSection7Case3PositiveWhiteCount W tR) : TaoSection7Case3ConstructedSourceStoppingRunData W pointAt family n J xi epsilon steps R tR lemma79Count where source_run := TaoSection7Case3SourceStoppingRun.of_prefix_last?_noLaterBlackAfterExit hprefix hlast hterminal R_pos := hRpos R_le_r := hR tR_eq := htR count_identity := hcount /-- Constructed source-run data closes the actual-run source identity specialized to Tao's domain-gated source black predicate. This remains a producer socket: the data itself still has to supply the source run, the cutoff evidence, the `t_R` accessor equality, and the Lemma 7.9 count identity for the same `steps`. -/ theorem to_actualRunSourceIdentity {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : List (ℕ × TaoSection7Triangle)} {R tR lemma79Count : ℕ} (hdata : TaoSection7Case3ConstructedSourceStoppingRunData W pointAt family n J xi epsilon steps R tR lemma79Count) (hrestart : ∀ p : ℕ, p < P → pointAt p = sourceAt (restart + p)) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps R tR lemma79Count := TaoSection7Case3ActualRunSourceIdentity.of_sourceBlackInDomain_sourceRun hrestart hfacts hdata.source_run hdata.R_pos hdata.R_le_r hdata.tR_eq hdata.count_identity end TaoSection7Case3ConstructedSourceStoppingRunData /-- Samplewise actual-run identities feed the deterministic inner `(7.56)` subset socket. The remaining outside-bad-event work is explicit in `houtside`: for every sample outside both `E*_used` and `F*`, it must provide the endpoint-window fact `t_R < P` and the Lemma 7.9/Markov many-white lower bound at that same `t_R`. This theorem only performs the low-window contradiction step and then uses the generic `LowWhite ⊆ E*_used ∪ F*` wrapper. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {black : Ω → TaoSection7Point → Prop} {pointAt : Ω → ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P R threshold : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (black ω) (pointAt ω) family n J xi epsilon P (steps ω) R (tR ω) (lemma79Count ω)) (houtside : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → tR ω < P ∧ threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhite_subset_EStarUsed_union_FSlack ?_ intro ω hlow hnotE hnotF rcases houtside ω hnotE hnotF with ⟨htR_ltP, hmany⟩ have hnotLow : ¬ taoSection7Case3WindowWhiteCount (W ω) P ≤ threshold := (hidentity ω).not_low_window_of_tR_ltP_many_positive htR_ltP hmany exact hnotLow (by simpa [taoSection7Case3LowWhiteEvent, taoSection7Case3LowWhite] using hlow) /-- Variant of the actual-run low-white subset bridge whose endpoint half is derived from a per-sample all-stops-below-`P` certificate. The many-white lower bound remains explicit. The endpoint producer is now the more source-facing statement that all stops in the actual run remain inside the finite `P` window. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_all_stop_ltP {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {black : Ω → TaoSection7Point → Prop} {pointAt : Ω → ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P R threshold : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (black ω) (pointAt ω) family n J xi epsilon P (steps ω) R (tR ω) (lemma79Count ω)) (hall_stop : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → ∀ step ∈ steps ω, step.1 < P) (hmany : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity (hidentity := hidentity) ?_ intro ω hnotE hnotF exact ⟨(hidentity ω).tR_ltP_of_all_stop_ltP (hall_stop ω hnotE hnotF), hmany ω hnotE hnotF⟩ namespace TaoSection7Case3StoppingPrefix /-- Prefix-named alias for the finite R-prefix window package. The underlying definition lives near the `t_R` accessors, but this namespace makes clear that the data does not require terminal source-run maximality. -/ abbrev WindowData (pointAt : ℕ → TaoSection7Point) (family : Set TaoSection7Triangle) (steps : List (ℕ × TaoSection7Triangle)) (R P : ℕ) : Prop := TaoSection7Case3SourceStoppingRun.PrefixWindowData pointAt family steps R P theorem windowData_of_prefix_hasAtLeast_all_ltP {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) : WindowData pointAt family steps R P := TaoSection7Case3SourceStoppingRun.prefixWindowData_of_prefix_hasAtLeast_all_ltP hprefix hR hall theorem hasTWithinWindow_of_prefix_hasAtLeast_all_ltP {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hRpos : 0 < R) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) : TaoSection7Case3SourceStoppingRun.HasTWithinWindow steps R P := (windowData_of_prefix_hasAtLeast_all_ltP hprefix hR hall).hasTWithinWindow hRpos theorem not_low_window_of_prefix_hasAtLeast_all_ltP_many_positive (W : ℕ → Prop) [DecidablePred W] {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R P threshold : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hRpos : 0 < R) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast steps R) (hall : ∀ step ∈ steps, step.1 < P) (hmany : ∀ t : ℕ, TaoSection7Case3SourceStoppingRun.tR? steps R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := TaoSection7Case3SourceStoppingRun.PrefixWindowData.not_low_window_of_many_positive (W := W) (windowData_of_prefix_hasAtLeast_all_ltP hprefix hR hall) hRpos hmany theorem all_stop_le_tR_of_mem_take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (htR : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR) : ∀ step ∈ steps.take R, step.1 ≤ tR := TaoSection7Case3SourceStoppingRun.stoppingPrefix_all_stop_le_tR_of_mem_take hprefix htR theorem windowData_of_tR_ltP_take {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} {R tR P : ℕ} (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (htR : TaoSection7Case3SourceStoppingRun.tR? steps R = some tR) (htR_ltP : tR < P) : WindowData pointAt family (steps.take R) R P := TaoSection7Case3SourceStoppingRun.prefixWindowData_of_tR_ltP_take hprefix htR htR_ltP theorem windowData_of_initial_and_encounterChain {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hinit : TaoSection7Case3InitialStoppingStep pointAt family first Γ) (hfirst_ltP : first < P) (hchain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P first Γ rest) (hR : TaoSection7Case3SourceStoppingRun.HasAtLeast ((first, Γ) :: rest) R) : WindowData pointAt family ((first, Γ) :: rest) R P := by have htail : TaoSection7Case3StoppingTail pointAt family first Γ rest := TaoSection7Case3RecursiveEncounterChain.to_stoppingTail hchain have hprefix : TaoSection7Case3StoppingPrefix pointAt family ((first, Γ) :: rest) := TaoSection7Case3StoppingPrefix.cons hinit htail have htail_ltP : ∀ step ∈ rest, step.1 < P := TaoSection7Case3RecursiveEncounterChain.all_stop_ltP hchain have hall : ∀ step ∈ ((first, Γ) :: rest), step.1 < P := by intro step hmem simp at hmem rcases hmem with hstep_eq | hmem_tail · cases hstep_eq exact hfirst_ltP · exact htail_ltP step hmem_tail exact windowData_of_prefix_hasAtLeast_all_ltP hprefix hR hall end TaoSection7Case3StoppingPrefix /-- Assumption socket for the finite R-prefix layer. This deliberately bundles an already-produced initial stopping step, recursive encounter chain, and length lower bound. It is not the source-derived R-prefix producer. -/ structure TaoSection7Case3PrefixRInputs (black : TaoSection7Point → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop) (P R first : ℕ) (Γ : TaoSection7Triangle) (rest : List (ℕ × TaoSection7Triangle)) : Prop where initial : TaoSection7Case3InitialStoppingStep pointAt family first Γ first_ltP : first < P encounter_chain : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P first Γ rest hasAtLeast : TaoSection7Case3SourceStoppingRun.HasAtLeast ((first, Γ) :: rest) R namespace TaoSection7Case3PrefixRInputs theorem windowData {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hdata : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) : TaoSection7Case3StoppingPrefix.WindowData pointAt family ((first, Γ) :: rest) R P := TaoSection7Case3StoppingPrefix.windowData_of_initial_and_encounterChain hdata.initial hdata.first_ltP hdata.encounter_chain hdata.hasAtLeast theorem hasTWithinWindow {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hdata : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) (hRpos : 0 < R) : TaoSection7Case3SourceStoppingRun.HasTWithinWindow ((first, Γ) :: rest) R P := hdata.windowData.hasTWithinWindow hRpos theorem all_stop_ltP {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hdata : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) : ∀ step ∈ ((first, Γ) :: rest), step.1 < P := hdata.windowData.all_stop_ltP theorem not_low_window_of_many_positive (W : ℕ → Prop) [DecidablePred W] {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R threshold first : ℕ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (hdata : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) (hRpos : 0 < R) (hmany : ∀ t : ℕ, TaoSection7Case3SourceStoppingRun.tR? ((first, Γ) :: rest) R = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := hdata.windowData.not_low_window_of_many_positive W hRpos hmany end TaoSection7Case3PrefixRInputs /-- Outside-event form of the all-stops producer supplied by finite R-prefix inputs. This is the checked producer for the endpoint half of the current outside-low-white certificate when the actual-run list has already been identified with the constructed prefix list. -/ theorem taoSection7Case3_hall_stop_of_prefixRInputs {Ω : Type*} {black : Ω → TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P R : ℕ} {first : Ω → ℕ} {Γ : Ω → TaoSection7Triangle} {rest steps : Ω → List (ℕ × TaoSection7Triangle)} {EStarUsed FSlack : Set Ω} (hprefix : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3PrefixRInputs (black ω) family (pointAt ω) (Accepts ω) P R (first ω) (Γ ω) (rest ω)) (hsteps : ∀ ω : Ω, steps ω = (first ω, Γ ω) :: rest ω) : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → ∀ step ∈ steps ω, step.1 < P := by intro ω hnotE hnotF step hmem have hmem_prefix : step ∈ (first ω, Γ ω) :: rest ω := by simpa [hsteps ω] using hmem exact (hprefix ω hnotE hnotF).all_stop_ltP step hmem_prefix namespace TaoSection7Case3ActualRunSourceIdentity /-- Endpoint-window extractor from the source-facing finite R-prefix package. This removes the raw all-stops-below-`P` function when the caller has already assembled `TaoSection7Case3PrefixRInputs` for the actual stopping prefix. -/ theorem tR_ltP_of_prefixRInputs {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J P R first tR lemma79Count : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {Γ : TaoSection7Triangle} {rest : List (ℕ × TaoSection7Triangle)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P ((first, Γ) :: rest) R tR lemma79Count) (hprefix : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) : tR < P := h.tR_ltP_of_all_stop_ltP hprefix.all_stop_ltP /-- Transport the actual-run `t_R` accessor to an identified finite prefix. This is the endpoint-safe form needed when Tao's run has at least `R` stops: the constructed prefix is `steps.take R`, not necessarily the whole terminal run. -/ theorem prefix_tR_eq_of_take_eq {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {n J P R first tR lemma79Count : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {Γ : TaoSection7Triangle} {steps rest : List (ℕ × TaoSection7Triangle)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htake : steps.take R = (first, Γ) :: rest) : TaoSection7Case3SourceStoppingRun.tR? ((first, Γ) :: rest) R = some tR := by have htR_take : TaoSection7Case3SourceStoppingRun.tR? (steps.take R) R = some tR := TaoSection7Case3SourceStoppingRun.tR?_take_eq_of_tR? h.tR_eq simpa [htake] using htR_take /-- Endpoint-window extractor from a prefix identified as `steps.take R`. Unlike `tR_ltP_of_prefixRInputs`, this does not require the actual terminal run to be exactly the constructed prefix. It only uses the first `R` stops of the full actual run. -/ theorem tR_ltP_of_prefixRInputs_take_eq {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {pointAt : ℕ → TaoSection7Point} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J P R first tR lemma79Count : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {Γ : TaoSection7Triangle} {steps rest : List (ℕ × TaoSection7Triangle)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htake : steps.take R = (first, Γ) :: rest) (hprefix : TaoSection7Case3PrefixRInputs black family pointAt Accepts P R first Γ rest) : tR < P := by have htR_prefix : TaoSection7Case3SourceStoppingRun.tR? ((first, Γ) :: rest) R = some tR := h.prefix_tR_eq_of_take_eq htake rcases hprefix.hasTWithinWindow h.R_pos with ⟨t, htR, htP⟩ rw [htR_prefix] at htR cases htR exact htP end TaoSection7Case3ActualRunSourceIdentity /-- Variant of the actual-run low-white subset bridge whose endpoint half is derived from the checked finite R-prefix input package. This is the source-iteration-shaped successor of the raw all-stops-below-`P` bridge. It still leaves the many-white lower bound to the future Lemma 7.9/Markov producer. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_prefixRInputs {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {black : Ω → TaoSection7Point → Prop} {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {n J P R threshold : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {first : Ω → ℕ} {Γ : Ω → TaoSection7Triangle} {rest : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (black ω) (pointAt ω) family n J xi epsilon P ((first ω, Γ ω) :: rest ω) R (tR ω) (lemma79Count ω)) (hprefix : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3PrefixRInputs (black ω) family (pointAt ω) (Accepts ω) P R (first ω) (Γ ω) (rest ω)) (hmany : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity (hidentity := hidentity) ?_ intro ω hnotE hnotF exact ⟨(hidentity ω).tR_ltP_of_prefixRInputs (hprefix ω hnotE hnotF), hmany ω hnotE hnotF⟩ /-- Take-prefix variant of the actual-run low-white subset bridge. The actual run may have more than `R` stops. Outside the exceptional sets the caller identifies the first `R` stops as the checked finite prefix via `steps.take R`, and this theorem derives the endpoint half from that prefix without asserting terminality at the prefix. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_prefixRInputs_take_eq {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {black : Ω → TaoSection7Point → Prop} {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {n J P R threshold : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps rest : Ω → List (ℕ × TaoSection7Triangle)} {first : Ω → ℕ} {Γ : Ω → TaoSection7Triangle} {tR lemma79Count : Ω → ℕ} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (black ω) (pointAt ω) family n J xi epsilon P (steps ω) R (tR ω) (lemma79Count ω)) (htake : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → (steps ω).take R = (first ω, Γ ω) :: rest ω) (hprefix : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3PrefixRInputs (black ω) family (pointAt ω) (Accepts ω) P R (first ω) (Γ ω) (rest ω)) (hmany : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity (hidentity := hidentity) ?_ intro ω hnotE hnotF exact ⟨(hidentity ω).tR_ltP_of_prefixRInputs_take_eq (htake ω hnotE hnotF) (hprefix ω hnotE hnotF), hmany ω hnotE hnotF⟩ /-- Tail list for an indexed sequence of stopping times and triangles. The element at offset `k + m + 1` is recorded as the next stopping pair. This is an assembler for already-supplied encounter steps, not a source producer. -/ def taoSection7Case3IndexedEncounterTail (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) : ℕ → ℕ → List (ℕ × TaoSection7Triangle) | _, 0 => [] | k, n + 1 => (t (k + 1), tri (k + 1)) :: taoSection7Case3IndexedEncounterTail t tri (k + 1) n theorem taoSection7Case3IndexedEncounterTail_length (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) : ∀ k n : ℕ, (taoSection7Case3IndexedEncounterTail t tri k n).length = n := by intro k n induction n generalizing k with | zero => simp [taoSection7Case3IndexedEncounterTail] | succ n ih => simp [taoSection7Case3IndexedEncounterTail, ih] theorem taoSection7Case3_recursiveEncounterChain_of_indexedTail {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P : ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) : ∀ k n : ℕ, (∀ i : ℕ, k ≤ i → i < k + n → TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1))) → TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P (t k) (tri k) (taoSection7Case3IndexedEncounterTail t tri k n) := by intro k n induction n generalizing k with | zero => intro _hstep simpa [taoSection7Case3IndexedEncounterTail] using TaoSection7Case3RecursiveEncounterChain.nil (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) (t k) (tri k) | succ n ih => intro hstep have hhead : TaoSection7Case3EncounterStep black family pointAt Accepts P (t k) (tri k) (tri (k + 1)) (t (k + 1)) := hstep k le_rfl (by omega) have htail : TaoSection7Case3RecursiveEncounterChain black family pointAt Accepts P (t (k + 1)) (tri (k + 1)) (taoSection7Case3IndexedEncounterTail t tri (k + 1) n) := ih (k + 1) (by intro i hki hi exact hstep i (by omega) (by omega)) simpa [taoSection7Case3IndexedEncounterTail] using TaoSection7Case3RecursiveEncounterChain.cons hhead htail theorem taoSection7Case3_hasAtLeast_cons_indexedTail (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (k Rtail : ℕ) : TaoSection7Case3SourceStoppingRun.HasAtLeast ((t k, tri k) :: taoSection7Case3IndexedEncounterTail t tri k Rtail) (Rtail + 1) := by simp [TaoSection7Case3SourceStoppingRun.HasAtLeast, TaoSection7Case3SourceStoppingRun.r, taoSection7Case3IndexedEncounterTail_length] theorem taoSection7Case3_tR?_cons_indexedTail_last (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) : ∀ k Rtail : ℕ, TaoSection7Case3SourceStoppingRun.tR? ((t k, tri k) :: taoSection7Case3IndexedEncounterTail t tri k Rtail) (Rtail + 1) = some (t (k + Rtail)) := by intro k Rtail induction Rtail generalizing k with | zero => simp [TaoSection7Case3SourceStoppingRun.tR?] | succ Rtail ih => have hidx : k + (Rtail + 1) = k + 1 + Rtail := by omega simp [taoSection7Case3IndexedEncounterTail, TaoSection7Case3SourceStoppingRun.tR?, ih (k + 1), hidx] theorem taoSection7Case3StoppingLast?_cons_indexedTail (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) : ∀ k Rtail : ℕ, TaoSection7Case3StoppingLast? ((t k, tri k) :: taoSection7Case3IndexedEncounterTail t tri k Rtail) = some (t (k + Rtail), tri (k + Rtail)) := by intro k Rtail induction Rtail generalizing k with | zero => simp [taoSection7Case3IndexedEncounterTail, TaoSection7Case3StoppingLast?] | succ Rtail ih => have hidx : k + (Rtail + 1) = k + 1 + Rtail := by omega simp [taoSection7Case3IndexedEncounterTail, TaoSection7Case3StoppingLast?, ih (k + 1), hidx] theorem taoSection7Case3_prefixRInputs_of_indexedEncounterTail {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {P : ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) {Rtail : ℕ} (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hsteps : ∀ i : ℕ, i < Rtail → TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1))) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) := by refine { initial := hinit first_ltP := hfirst_ltP encounter_chain := ?_ hasAtLeast := ?_ } · exact taoSection7Case3_recursiveEncounterChain_of_indexedTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri 0 Rtail (by intro i _h0 hi exact hsteps i (by simpa using hi)) · exact taoSection7Case3_hasAtLeast_cons_indexedTail t tri 0 Rtail theorem taoSection7Case3_prefixRInputs_and_rebasedStates_of_indexedFirstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (candidate : ℕ → ℕ) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hfull : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold (t i) (j i) (s i) (pre i) (pres i)) (hsrc : ∀ i : ℕ, i < Rtail → TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) (candidate i) (t (i + 1))) (hroom_next : ∀ i : ℕ, i < Rtail → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBound A) threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtail → Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i < Rtail → ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres i = before ++ next :: after ∧ before.length + 1 = t (i + 1) - t i ∧ TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold (t (i + 1)) (taoSection7SourceBlockStartJFrom (j i) (pre i :: before)) (taoSection7SourceBlockStartSFrom (s i) (pre i :: before)) next after := by constructor · exact taoSection7Case3_prefixRInputs_of_indexedEncounterTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri (Rtail := Rtail) hinit hfirst_ltP (by intro i hi exact (hsrc i hi).to_encounterStep (haccept i hi)) · intro i hi exact taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathData_of_firstHitSourceProvenance (hfull := hfull i hi) (hsrc := hsrc i hi) (hroom_first := hroom_next i hi) (hfacts := hfacts) (haccept := haccept i hi) theorem taoSection7Case3_prefixRInputs_and_rebasedStatesWithGap_of_indexedFirstHitSourceProvenance {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (candidate : ℕ → ℕ) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hfull : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i)) (hsrc : ∀ i : ℕ, i < Rtail → TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) (candidate i) (t (i + 1))) (hroom_next : ∀ i : ℕ, i < Rtail → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtail → Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i < Rtail → ∃ before : List (List ℕ), ∃ next : List ℕ, ∃ after : List (List ℕ), pres i = before ++ next :: after ∧ before.length + 1 = t (i + 1) - t i ∧ TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t (i + 1)) gapBound (taoSection7SourceBlockStartJFrom (j i) (pre i :: before)) (taoSection7SourceBlockStartSFrom (s i) (pre i :: before)) next after := by constructor · exact taoSection7Case3_prefixRInputs_of_indexedEncounterTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri (Rtail := Rtail) hinit hfirst_ltP (by intro i hi exact (hsrc i hi).to_encounterStep (haccept i hi)) · intro i hi exact taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull i hi) (hsrc := hsrc i hi) (hroom_first := hroom_next i hi) (hfacts := hfacts) (haccept := haccept i hi) theorem taoSection7Case3_prefixRInputs_and_stateInvariant_of_indexedSourceStateUpdates {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J A : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold (t 0) (j 0) (s 0) (pre 0) (pres 0)) (hupdates : ∀ i : ℕ, i < Rtail → TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) ∧ TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold (t (i + 1)) (j (i + 1)) (s (i + 1)) (pre (i + 1)) (pres (i + 1))) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathData W pointAt n J A xi epsilon P threshold (t i) (j i) (s i) (pre i) (pres i) := by constructor · exact taoSection7Case3_prefixRInputs_of_indexedEncounterTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri (Rtail := Rtail) hinit hfirst_ltP (by intro i hi exact (hupdates i hi).1) · intro i hi cases i with | zero => exact hstate0 | succ k => have hk : k < Rtail := by omega simpa using (hupdates k hk).2 theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedSourceStateUpdates {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hupdates : ∀ i : ℕ, i < Rtail → TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t (i + 1)) gapBound (j (i + 1)) (s (i + 1)) (pre (i + 1)) (pres (i + 1))) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) := by constructor · exact taoSection7Case3_prefixRInputs_of_indexedEncounterTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri (Rtail := Rtail) hinit hfirst_ltP (by intro i hi exact (hupdates i hi).1) · intro i hi cases i with | zero => exact hstate0 | succ k => have hk : k < Rtail := by omega simpa using (hupdates k hk).2 /-- One indexed source-state update after rebasing at the next first hit. The future source-derived recurrence producer should fill this packet for each `i`. The equalities make explicit that the next indexed state is exactly the rebased source block returned by the one-step first-hit construction. -/ structure TaoSection7Case3IndexedRebasedSourceUpdateWithGap {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (i : ℕ) : Type where before : List (List ℕ) next : List ℕ after : List (List ℕ) split : pres i = before ++ next :: after j_next : j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: before) s_next : s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: before) pre_next : pre (i + 1) = next pres_next : pres (i + 1) = after encounter : TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) rebased : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t (i + 1)) gapBound (taoSection7SourceBlockStartJFrom (j i) (pre i :: before)) (taoSection7SourceBlockStartSFrom (s i) (pre i :: before)) next after namespace TaoSection7Case3IndexedRebasedSourceUpdateWithGap theorem to_sourceStateUpdate {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {t : ℕ → ℕ} {tri : ℕ → TaoSection7Triangle} {j : ℕ → ℕ+} {s : ℕ → ℕ} {pre : ℕ → List ℕ} {pres : ℕ → List (List ℕ)} {i : ℕ} (hupdate : TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i) : TaoSection7Case3EncounterStep black family pointAt Accepts P (t i) (tri i) (tri (i + 1)) (t (i + 1)) ∧ TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t (i + 1)) gapBound (j (i + 1)) (s (i + 1)) (pre (i + 1)) (pres (i + 1)) := by constructor · exact hupdate.encounter · rw [hupdate.j_next, hupdate.s_next, hupdate.pre_next, hupdate.pres_next] exact hupdate.rebased end TaoSection7Case3IndexedRebasedSourceUpdateWithGap /-- Existential one-step rebased source update, before committing to indexed successor arrays. This is the local packet shape produced from the current full-window source state: it chooses the first hit, source split, and rebased successor state. -/ structure TaoSection7Case3OneStepRebasedSourceUpdateWithGap {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {gapBound : ℕ → ℕ} (j : ℕ+) (s : ℕ) (pre : List ℕ) (pres : List (List ℕ)) (old : TaoSection7Triangle) : Type where candidate : ℕ first : ℕ triangle : TaoSection7Triangle before : List (List ℕ) next : List ℕ after : List (List ℕ) split : pres = before ++ next :: after index : before.length + 1 = first - q source : TaoSection7Case3FirstHitSourceProvenance black family pointAt j s pre pres P q old triangle candidate first encounter : TaoSection7Case3EncounterStep black family pointAt Accepts P q old triangle first rebased : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold first gapBound (taoSection7SourceBlockStartJFrom j (pre :: before)) (taoSection7SourceBlockStartSFrom s (pre :: before)) next after namespace TaoSection7Case3OneStepRebasedSourceUpdateWithGap theorem q_lt_first {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) : q < hupdate.first := hupdate.source.first_hit.1 theorem first_ltP {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) : hupdate.first < P := hupdate.source.first_ltP theorem pair_eq_of_pairwiseDisjoint_stoppingTransition {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold q first : ℕ} {gapBound : ℕ → ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old Γ : TaoSection7Triangle} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ) : (first, Γ) = (hupdate.first, hupdate.triangle) := TaoSection7Case3StoppingTransition.pair_eq_of_pairwiseDisjoint hpair hstep hupdate.source.to_stoppingTransition def to_indexedRebasedSourceUpdateWithGap {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {t : ℕ → ℕ} {tri : ℕ → TaoSection7Triangle} {j : ℕ → ℕ+} {s : ℕ → ℕ} {pre : ℕ → List ℕ} {pres : ℕ → List (List ℕ)} {i : ℕ} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := t i) (gapBound := gapBound) (j i) (s i) (pre i) (pres i) (tri i)) (ht_next : t (i + 1) = hupdate.first) (htri_next : tri (i + 1) = hupdate.triangle) (hj_next : j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: hupdate.before)) (hs_next : s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: hupdate.before)) (hpre_next : pre (i + 1) = hupdate.next) (hpres_next : pres (i + 1) = hupdate.after) : TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i := by refine { before := hupdate.before next := hupdate.next after := hupdate.after split := hupdate.split j_next := hj_next s_next := hs_next pre_next := hpre_next pres_next := hpres_next encounter := ?_ rebased := ?_ } · simpa [ht_next, htri_next] using hupdate.encounter · simpa [ht_next] using hupdate.rebased end TaoSection7Case3OneStepRebasedSourceUpdateWithGap /-- Existential one-step packet producer from the current `WithGap` state. This chooses the first hit and split witnesses instead of taking indexed `hsplit`/successor-state equalities as premises. -/ theorem taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : Nonempty (TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) := by rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAt) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) (old := old) hfull hscale hlow hroom hactive hcover hroom_next hfacts haccept with ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hstep, hrebase⟩ exact ⟨{ candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase }⟩ /-- One-step packet producer with selected-offset membership from the full local first-hit envelope. The proof keeps the later-search candidate window membership long enough to combine it with `hsrc.first_le_candidate`. This avoids replacing the selected offset by the later-search block or assuming the broad range premise `∀ first, q < first → first < P → first ∈ allowed`. -/ theorem taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace allowed (q + 1) (taoSection7Case3LaterSearchBound gapBound threshold q - (q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, hupdate.first ∈ allowed := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, hpivot, first, Γ, hsrc⟩ have hallowed : first ∈ allowed := TaoSection7Case3AllowedBlockTrace.first_mem_of_le_laterSearch_candidate (hblock := hblock) (hcandidate := hpivot.mem_later_window) (hqfirst := hsrc.first_hit.1) (hfirst_le_candidate := hsrc.first_le_candidate) have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have haccept_step : Accepts q old first Γ := haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_step) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨{ candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase }, hallowed⟩ /-- Direct-room selected-offset version of the one-step packet producer. This keeps the selected first-hit membership proof and uses it to request room only for that selected offset, rather than a universal callback for every future point below `P`. -/ theorem taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ allowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace allowed (q + 1) (taoSection7Case3LaterSearchBound gapBound threshold q - (q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, hupdate.first ∈ allowed := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, hpivot, first, Γ, hsrc⟩ have hallowed : first ∈ allowed := TaoSection7Case3AllowedBlockTrace.first_mem_of_le_laterSearch_candidate (hblock := hblock) (hcandidate := hpivot.mem_later_window) (hqfirst := hsrc.first_hit.1) (hfirst_le_candidate := hsrc.first_le_candidate) have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_selected first hallowed have haccept_step : Accepts q old first Γ := haccept first Γ hsrc.first_ltP hsrc.first_black hsrc.first_hit hsrc.new_mem_family hsrc.new_mem hsrc.new_ne_old rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_step) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨{ candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase }, hallowed⟩ /-- Direct-room one-step packet producer with transition-shaped acceptance. The acceptance premise is tied to the selected stopping transition produced by the first-hit source packet, rather than a broad callback over arbitrary future `first, Γ` data. -/ theorem taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room_accept (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ allowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace allowed (q + 1) (taoSection7Case3LaterSearchBound gapBound threshold q - (q + 1))) (haccept_step : ∀ {first : ℕ} {Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition pointAt family q old first Γ → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, hupdate.first ∈ allowed := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, hpivot, first, Γ, hsrc⟩ have hallowed : first ∈ allowed := TaoSection7Case3AllowedBlockTrace.first_mem_of_le_laterSearch_candidate (hblock := hblock) (hcandidate := hpivot.mem_later_window) (hqfirst := hsrc.first_hit.1) (hfirst_le_candidate := hsrc.first_le_candidate) have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_selected first hallowed have haccept_selected : Accepts q old first Γ := haccept_step (first := first) (Γ := Γ) hsrc.first_ltP hsrc.to_stoppingTransition hsrc.first_black rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_selected) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨{ candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase }, hallowed⟩ /-- Direct-room one-step packet producer with source-provenance-shaped acceptance. The acceptance premise consumes the selected first-hit source packet itself, before it is collapsed to a stopping transition or encounter step. -/ theorem taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room_acceptSource (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ allowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace allowed (q + 1) (taoSection7Case3LaterSearchBound gapBound threshold q - (q + 1))) (haccept_src : ∀ {candidate first : ℕ} {Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt j s pre pres P q old Γ candidate first → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, hupdate.first ∈ allowed := by have hdata : TaoSection7Case3SourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres := hfull.to_sourcePathDataWithGap hroom rcases taoSection7Case3_exists_firstHitSourceProvenance_of_sourcePathData_scale_coverage (W := W) (pointAt := pointAt) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (family := family) (old := old) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) hdata hscale hlow hroom hactive hcover hfull.point_window with ⟨candidate, _pivot, hpivot, first, Γ, hsrc⟩ have hallowed : first ∈ allowed := TaoSection7Case3AllowedBlockTrace.first_mem_of_le_laterSearch_candidate (hblock := hblock) (hcandidate := hpivot.mem_later_window) (hqfirst := hsrc.first_hit.1) (hfirst_le_candidate := hsrc.first_le_candidate) have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_selected first hallowed have haccept_selected : Accepts q old first Γ := haccept_src hsrc rcases taoSection7Case3_exists_encounterStep_and_rebasedFullWindowSourcePathDataWithGap_of_firstHitSourceProvenance (hfull := hfull) (hsrc := hsrc) (hroom_first := hroom_first) (hfacts := hfacts) (haccept := haccept_selected) with ⟨before, next, after, hsplit, hidx, hstep, hrebase⟩ exact ⟨{ candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase }, hallowed⟩ /-- No-budget active-state version of the existential one-step packet producer. This is the recursive source-state shape: it carries the successor room and active-triangle facts without requiring the fixed-start budget premise `q ∈ Icc start (start + threshold)`. -/ theorem taoSection7Case3_exists_oneStepActiveRebasedSourceUpdateWithGap_noBudget_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hallowed_first : ∀ first : ℕ, q < first → first < P → first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first ∧ TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound hupdate.first hupdate.triangle := by rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAt) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (P := P) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) (old := old) hfull hscale hlow hroom hactive hcover hroom_next hfacts haccept with ⟨hupdate⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first := hroom_next hupdate.first hupdate.q_lt_first hupdate.first_ltP have hallowed : hupdate.first ∈ allowed := hallowed_first hupdate.first hupdate.q_lt_first hupdate.first_ltP have hsmall : hupdate.triangle.size < bound hupdate.first := taoSection7Case3_triangle_size_lt_of_not_large_union (pointAt := pointAt) (family := family) (allowed := allowed) (bound := bound) hnotLarge hallowed hupdate.source.new_mem_family hupdate.source.new_mem refine ⟨hupdate, hroom_first, ?_⟩ exact { old_mem_family := hupdate.source.new_mem_family pivot_mem := hupdate.source.new_mem small_size := hsmall } /-- Selected-update active successor facts for the no-budget Case 3 recurrence. Unlike the broader existential producer above, this consumes membership of the selected update's own first-hit offset in the used/allowed set. This is the source-faithful socket for the later `E_*` provenance step: callers should prove `hupdate.first ∈ allowed` from the selected pivot/used-event data, not from an unrelated later-search block. -/ theorem taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hallowed : hupdate.first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first ∧ TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound hupdate.first hupdate.triangle := by have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first := hroom_next hupdate.first hupdate.q_lt_first hupdate.first_ltP have hsmall : hupdate.triangle.size < bound hupdate.first := taoSection7Case3_triangle_size_lt_of_not_large_union (pointAt := pointAt) (family := family) (allowed := allowed) (bound := bound) hnotLarge hallowed hupdate.source.new_mem_family hupdate.source.new_mem exact ⟨hroom_first, { old_mem_family := hupdate.source.new_mem_family pivot_mem := hupdate.source.new_mem small_size := hsmall }⟩ /-- Direct-room variant of the selected-update active successor facts. This is the narrower socket used once the selected first-hit offset is known: callers only need room at `hupdate.first`, not a universal successor-room callback for every future first. -/ theorem taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed_room {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {P threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first) (hallowed : hupdate.first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first ∧ TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound hupdate.first hupdate.triangle := by have hsmall : hupdate.triangle.size < bound hupdate.first := taoSection7Case3_triangle_size_lt_of_not_large_union (pointAt := pointAt) (family := family) (allowed := allowed) (bound := bound) hnotLarge hallowed hupdate.source.new_mem_family hupdate.source.new_mem exact ⟨hroom_first, { old_mem_family := hupdate.source.new_mem_family pivot_mem := hupdate.source.new_mem small_size := hsmall }⟩ /-- Current source-state payload for the no-budget Case 3 active recurrence. The data fields are exactly the projections that will become the indexed arrays `t`, `tri`, `j`, `s`, `pre`, and `pres` in the finite trace. -/ structure TaoSection7Case3ActiveSourceStateWithGap (W : ℕ → Prop) (family : Set TaoSection7Triangle) (pointAt : ℕ → TaoSection7Point) (bound : ℕ → ℝ) (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (P threshold : ℕ) (gapBound : ℕ → ℕ) : Type where q : ℕ old : TaoSection7Triangle j : ℕ+ s : ℕ pre : List ℕ pres : List (List ℕ) full : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres room : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q active : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old /-- A chosen no-budget active step out of one source state. This records the single packet selected by the local producer, together with the successor room and active-triangle facts that become the next state's proof fields. -/ structure TaoSection7Case3ChosenActiveStepWithGap {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : Type where update : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := st.q) (gapBound := gapBound) st.j st.s st.pre st.pres st.old room_next : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P update.first active_next : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound update.first update.triangle namespace TaoSection7Case3ChosenActiveStepWithGap def nextState {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound} (hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound := { q := hstep.update.first old := hstep.update.triangle j := taoSection7SourceBlockStartJFrom st.j (st.pre :: hstep.update.before) s := taoSection7SourceBlockStartSFrom st.s (st.pre :: hstep.update.before) pre := hstep.update.next pres := hstep.update.after full := hstep.update.rebased room := hstep.room_next active := hstep.active_next } end TaoSection7Case3ChosenActiveStepWithGap /-- Build a chosen active step from a concrete one-step update whose selected first-hit offset is already known to lie in the active used/allowed set. This is the trace-facing selected-pivot socket: it deliberately avoids the stronger range premise `∀ first, st.q < first → first < P → first ∈ allowed`. -/ def taoSection7Case3_chosenActiveStepWithGap_of_selectedAllowed_oneStep {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := st.q) (gapBound := gapBound) st.j st.s st.pre st.pres st.old) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hallowed : hupdate.first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st := by rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed (hupdate := hupdate) (hroom_next := hroom_next) (hallowed := hallowed) (hnotLarge := hnotLarge) with ⟨hroom_first, hactive_first⟩ exact { update := hupdate room_next := hroom_first active_next := hactive_first } /-- Direct-room selected-pivot constructor for a chosen active step. This is the same selected-update socket as `taoSection7Case3_chosenActiveStepWithGap_of_selectedAllowed_oneStep`, but with the successor room at the selected first-hit offset supplied directly. -/ def taoSection7Case3_chosenActiveStepWithGap_of_selectedAllowed_oneStep_room {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} {st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound} (hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := st.q) (gapBound := gapBound) st.j st.s st.pre st.pres st.old) (hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first) (hallowed : hupdate.first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st := by rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed_room (hupdate := hupdate) (hroom_first := hroom_first) (hallowed := hallowed) (hnotLarge := hnotLarge) with ⟨hroom_first', hactive_first⟩ exact { update := hupdate room_next := hroom_first' active_next := hactive_first } /-- One-state selected-envelope producer for the active source trace. It constructs the concrete one-step update from the current source state, derives the selected first-hit membership from the full `q+1` to later-search-bound envelope, and then applies the selected-allowed active-step socket. The remaining event-side obligation is the not-large-union fact for the same allowed set. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_selectedEnvelope_activeSourceState (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {allowed : Set ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace allowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family st.old st.q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAt) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := allowed) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_next hfacts hblock haccept with ⟨hupdate, hallowed⟩ exact ⟨taoSection7Case3_chosenActiveStepWithGap_of_selectedAllowed_oneStep (st := st) (hupdate := hupdate) (hroom_next := hroom_next) (hallowed := hallowed) (hnotLarge := hnotLarge)⟩ /-- One-state selected-envelope producer with the not-large input supplied by the same `E*_used` sandwich that controls the probability side. The theorem still takes the full-envelope block trace as an input; it removes the raw not-large-union premise from the selected-envelope active-step path. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_selectedUsed_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace used (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : ∃ hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st, hstep.update.first ∈ used ∧ hstep.update.first ∈ allowedFin := by have hnotLarge : ¬ taoSection7Case3LargeTriangleUnion (pointAtΩ ω) family used bound := sandwich.not_large_union_of_not_mem hω rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAtΩ ω) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := used) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_next hfacts hblock haccept with ⟨hupdate, hused⟩ rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed (hupdate := hupdate) (hroom_next := hroom_next) (hallowed := hused) (hnotLarge := hnotLarge) with ⟨hroom_first, hactive_first⟩ refine ⟨{ update := hupdate room_next := hroom_first active_next := hactive_first }, hused, ?_⟩ exact sandwich.used_mem_allowed hupdate.first hused /-- Local-envelope variant of the selected-used sandwich bridge. This keeps the deterministic selected-first envelope `localAllowed` separate from the larger finite `used` set paid by the probability-side support. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : ∃ hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st, hstep.update.first ∈ localAllowed ∧ hstep.update.first ∈ used ∧ hstep.update.first ∈ allowedFin := by have hnotLargeLocal : ¬ taoSection7Case3LargeTriangleUnion (pointAtΩ ω) family localAllowed bound := sandwich.not_large_union_of_not_mem_subset hlocal_subset_used hω rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAtΩ ω) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := localAllowed) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_next hfacts hblock haccept with ⟨hupdate, hlocal⟩ rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed (hupdate := hupdate) (hroom_next := hroom_next) (hallowed := hlocal) (hnotLarge := hnotLargeLocal) with ⟨hroom_first, hactive_first⟩ have hused : hupdate.first ∈ used := hlocal_subset_used hlocal refine ⟨{ update := hupdate room_next := hroom_first active_next := hactive_first }, hlocal, hused, ?_⟩ exact sandwich.used_mem_allowed hupdate.first hused /-- Direct-room local-envelope variant of the selected sandwich bridge. The source packet still proves that the selected first hit lies in `localAllowed`; room is requested only from that membership, not from a universal successor-room callback. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : ∃ hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st, hstep.update.first ∈ localAllowed ∧ hstep.update.first ∈ used ∧ hstep.update.first ∈ allowedFin := by have hnotLargeLocal : ¬ taoSection7Case3LargeTriangleUnion (pointAtΩ ω) family localAllowed bound := sandwich.not_large_union_of_not_mem_subset hlocal_subset_used hω rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room (W := W) (pointAt := pointAtΩ ω) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := localAllowed) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_selected hfacts hblock haccept with ⟨hupdate, hlocal⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first := hroom_selected hupdate.first hlocal rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed_room (hupdate := hupdate) (hroom_first := hroom_first) (hallowed := hlocal) (hnotLarge := hnotLargeLocal) with ⟨hroom_first', hactive_first⟩ have hused : hupdate.first ∈ used := hlocal_subset_used hlocal refine ⟨{ update := hupdate room_next := hroom_first' active_next := hactive_first }, hlocal, hused, ?_⟩ exact sandwich.used_mem_allowed hupdate.first hused /-- Direct-room local-envelope bridge with transition-shaped acceptance. This is the selected-envelope counterpart of the low-level `_room_accept` packet producer. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room_accept {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept_step : ∀ {first : ℕ} {Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAtΩ ω) family st.q st.old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → Accepts st.q st.old first Γ) : ∃ hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st, hstep.update.first ∈ localAllowed ∧ hstep.update.first ∈ used ∧ hstep.update.first ∈ allowedFin := by have hnotLargeLocal : ¬ taoSection7Case3LargeTriangleUnion (pointAtΩ ω) family localAllowed bound := sandwich.not_large_union_of_not_mem_subset hlocal_subset_used hω rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room_accept (W := W) (pointAt := pointAtΩ ω) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := localAllowed) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_selected hfacts hblock haccept_step with ⟨hupdate, hlocal⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first := hroom_selected hupdate.first hlocal rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed_room (hupdate := hupdate) (hroom_first := hroom_first) (hallowed := hlocal) (hnotLarge := hnotLargeLocal) with ⟨hroom_first', hactive_first⟩ have hused : hupdate.first ∈ used := hlocal_subset_used hlocal refine ⟨{ update := hupdate room_next := hroom_first' active_next := hactive_first }, hlocal, hused, ?_⟩ exact sandwich.used_mem_allowed hupdate.first hused /-- Direct-room local-envelope bridge with source-provenance-shaped acceptance. This keeps the acceptance callback attached to the selected first-hit source packet emitted by the current active source state. -/ theorem taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room_acceptSource {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept_src : ∀ {candidate first : ℕ} {Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (pointAtΩ ω) st.j st.s st.pre st.pres P st.q st.old Γ candidate first → Accepts st.q st.old first Γ) : ∃ hstep : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st, hstep.update.first ∈ localAllowed ∧ hstep.update.first ∈ used ∧ hstep.update.first ∈ allowedFin := by have hnotLargeLocal : ¬ taoSection7Case3LargeTriangleUnion (pointAtΩ ω) family localAllowed bound := sandwich.not_large_union_of_not_mem_subset hlocal_subset_used hω rcases taoSection7Case3_exists_oneStepRebasedSourceUpdateWithGap_and_selectedAllowed_of_fullWindowSourcePathDataWithGap_room_acceptSource (W := W) (pointAt := pointAtΩ ω) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (allowed := localAllowed) (P := P) (threshold := threshold) (q := st.q) (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (old := st.old) st.full hscale hlow st.room st.active hcover hroom_selected hfacts hblock haccept_src with ⟨hupdate, hlocal⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first := hroom_selected hupdate.first hlocal rcases taoSection7Case3_oneStepActiveFactsWithGap_of_selectedAllowed_room (hupdate := hupdate) (hroom_first := hroom_first) (hallowed := hlocal) (hnotLarge := hnotLargeLocal) with ⟨hroom_first', hactive_first⟩ have hused : hupdate.first ∈ used := hlocal_subset_used hlocal refine ⟨{ update := hupdate room_next := hroom_first' active_next := hactive_first }, hlocal, hused, ?_⟩ exact sandwich.used_mem_allowed hupdate.first hused /-- Chosen active step decorated with the selected-offset support facts that the plain trace forgets. The underlying `step` still drives the source-state recurrence; the extra fields retain the probability-side membership evidence for trace-level finite carriers. -/ structure TaoSection7Case3ChosenActiveUsedStepWithGap {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : Type where step : TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st first_mem_used : step.update.first ∈ used first_mem_allowedFin : step.update.first ∈ allowedFin /-- Package the selected-used sandwich bridge as a support-decorated chosen step. -/ theorem taoSection7Case3_exists_chosenActiveUsedStepWithGap_of_selectedUsed_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace used (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_selectedUsed_notEStar_sandwich (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hscale hlow hcover hroom_next hfacts hblock haccept with ⟨hstep, hused, hallowedFin⟩ exact ⟨{ step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin }⟩ /-- Package the local-envelope sandwich bridge as a support-decorated chosen step. -/ theorem taoSection7Case3_exists_chosenActiveUsedStepWithGap_of_localEnvelope_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_next hfacts hblock haccept with ⟨hstep, _hlocal, hused, hallowedFin⟩ exact ⟨{ step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin }⟩ /-- Chosen active step retaining both the local envelope membership used for the one-state construction and the larger probability-side used/finite support. -/ structure TaoSection7Case3ChosenActiveLocalUsedStepWithGap {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {localAllowed used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : Type where usedStep : TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st first_mem_local : usedStep.step.update.first ∈ localAllowed /-- Package the local-envelope sandwich bridge while retaining the local-envelope membership needed for bounded recursion. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_next hfacts hblock haccept with ⟨hstep, hlocal, hused, hallowedFin⟩ exact ⟨{ usedStep := { step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin } first_mem_local := hlocal }⟩ /-- Direct-room version of the local-used sandwich package. This retains the local-envelope membership while using only selected-offset room, preparing the finite bounded trace for a separate room-depth hypothesis. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept with ⟨hstep, hlocal, hused, hallowedFin⟩ exact ⟨{ usedStep := { step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin } first_mem_local := hlocal }⟩ /-- Direct-room local-used package with transition-shaped acceptance. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room_accept {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept_step : ∀ {first : ℕ} {Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAtΩ ω) family st.q st.old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room_accept (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept_step with ⟨hstep, hlocal, hused, hallowedFin⟩ exact ⟨{ usedStep := { step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin } first_mem_local := hlocal }⟩ /-- Direct-room local-used package with source-provenance-shaped acceptance. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room_acceptSource {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {used localAllowed : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold gapBound) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed used allowedFin bound) (hω : ω ∉ EStarUsed) (hlocal_subset_used : localAllowed ⊆ used) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound gapBound threshold st.q - (st.q + 1))) (haccept_src : ∀ {candidate first : ℕ} {Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (pointAtΩ ω) st.j st.s st.pre st.pres P st.q st.old Γ candidate first → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) := by rcases taoSection7Case3_exists_chosenActiveStepWithGap_of_localEnvelope_notEStar_sandwich_room_acceptSource (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept_src with ⟨hstep, hlocal, hused, hallowedFin⟩ exact ⟨{ usedStep := { step := hstep first_mem_used := hused first_mem_allowedFin := hallowedFin } first_mem_local := hlocal }⟩ /-- Pre-trace base-Kcut range instantiation of the local-used one-step bridge. At recursive step `i`, the current iterate bound places the local selected envelope inside the fixed carrier chosen before the trace is built. This keeps the `E*_used` support noncircular while producing the local-used step data needed by `seqLocalUsed`. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut R start i : ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)) bound) (hω : ω ∉ EStarUsed) (htrace_le : st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start) (hi : i + 1 ≤ R) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) := by let allowedFin : Finset ℕ := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1) let used : Set ℕ := {p : ℕ | p ∈ allowedFin} let localAllowed : Set ℕ := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q have hlocal_subset_used : localAllowed ⊆ used := by intro p hp exact taoSection7Case3_localSelectedEnvelope_subset_baseKcutIterateRange (base := base) (Kcut := Kcut) (threshold := threshold) (R := R) (start := start) (q := st.q) (i := i) htrace_le hi (by simpa [localAllowed] using hp) have hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q - (st.q + 1)) := by simpa [localAllowed] using (TaoSection7Case3AllowedBlockTrace.localSelectedEnvelope (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (threshold := threshold) (q := st.q)) change Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) exact taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_next hfacts hblock haccept /-- Room-depth split version of the base-Kcut local-used one-step bridge. The trace/support carrier uses `Rtrace`, while deterministic successor room is proved from the independent room depth `Rroom`. The hypothesis `Rtrace + 1 ≤ Rroom` is the explicit last-step margin needed because selected room consumes two base-Kcut steps from the current state. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start i : ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (htrace_le : st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start) (hi : i < Rtrace) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) := by let allowedFin : Finset ℕ := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1) let used : Set ℕ := {p : ℕ | p ∈ allowedFin} let localAllowed : Set ℕ := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q have hlocal_subset_used : localAllowed ⊆ used := by intro p hp exact taoSection7Case3_localSelectedEnvelope_subset_baseKcutIterateRange (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rtrace) (start := start) (q := st.q) (i := i) htrace_le (Nat.succ_le_of_lt hi) (by simpa [localAllowed] using hp) have hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q - (st.q + 1)) := by simpa [localAllowed] using (TaoSection7Case3AllowedBlockTrace.localSelectedEnvelope (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (threshold := threshold) (q := st.q)) have hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold) P first := by intro first hfirst have hi_room : i + 2 ≤ Rroom := by have hi_trace : i + 2 ≤ Rtrace + 1 := by omega exact hi_trace.trans hroom_depth exact taoSection7Case3_exitRoom_of_baseKcutIterateRoom_of_mem_localSelectedEnvelope (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rroom) (start := start) (P := P) (q := st.q) (i := i) (p := first) hroom_iter htrace_le hi_room (by simpa [localAllowed] using hfirst) change Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) exact taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept /-- Room-depth split base-Kcut step with transition-shaped acceptance. This keeps the room-depth count convention from `_roomDepth` while replacing the broad acceptance callback by a premise on the selected stopping transition. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth_accept {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start i : ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (htrace_le : st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start) (hi : i < Rtrace) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept_step : ∀ {first : ℕ} {Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAtΩ ω) family st.q st.old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) := by let allowedFin : Finset ℕ := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1) let used : Set ℕ := {p : ℕ | p ∈ allowedFin} let localAllowed : Set ℕ := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q have hlocal_subset_used : localAllowed ⊆ used := by intro p hp exact taoSection7Case3_localSelectedEnvelope_subset_baseKcutIterateRange (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rtrace) (start := start) (q := st.q) (i := i) htrace_le (Nat.succ_le_of_lt hi) (by simpa [localAllowed] using hp) have hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q - (st.q + 1)) := by simpa [localAllowed] using (TaoSection7Case3AllowedBlockTrace.localSelectedEnvelope (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (threshold := threshold) (q := st.q)) have hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold) P first := by intro first hfirst have hi_room : i + 2 ≤ Rroom := by have hi_trace : i + 2 ≤ Rtrace + 1 := by omega exact hi_trace.trans hroom_depth exact taoSection7Case3_exitRoom_of_baseKcutIterateRoom_of_mem_localSelectedEnvelope (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rroom) (start := start) (P := P) (q := st.q) (i := i) (p := first) hroom_iter htrace_le hi_room (by simpa [localAllowed] using hfirst) change Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) exact taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room_accept (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept_step /-- Room-depth split base-Kcut step with source-provenance-shaped acceptance. The acceptance callback is applied to the selected first-hit source packet for the current trace state. -/ theorem taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth_acceptSource {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start i : ℕ} (st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (htrace_le : st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start) (hi : i < Rtrace) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept_src : ∀ {candidate first : ℕ} {Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (pointAtΩ ω) st.j st.s st.pre st.pres P st.q st.old Γ candidate first → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) := by let allowedFin : Finset ℕ := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1) let used : Set ℕ := {p : ℕ | p ∈ allowedFin} let localAllowed : Set ℕ := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q have hlocal_subset_used : localAllowed ⊆ used := by intro p hp exact taoSection7Case3_localSelectedEnvelope_subset_baseKcutIterateRange (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rtrace) (start := start) (q := st.q) (i := i) htrace_le (Nat.succ_le_of_lt hi) (by simpa [localAllowed] using hp) have hblock : TaoSection7Case3AllowedBlockTrace localAllowed (st.q + 1) (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q - (st.q + 1)) := by simpa [localAllowed] using (TaoSection7Case3AllowedBlockTrace.localSelectedEnvelope (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (threshold := threshold) (q := st.q)) have hroom_selected : ∀ first : ℕ, first ∈ localAllowed → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold) P first := by intro first hfirst have hi_room : i + 2 ≤ Rroom := by have hi_trace : i + 2 ≤ Rtrace + 1 := by omega exact hi_trace.trans hroom_depth exact taoSection7Case3_exitRoom_of_baseKcutIterateRoom_of_mem_localSelectedEnvelope (base := base) (Kcut := Kcut) (threshold := threshold) (R := Rroom) (start := start) (P := P) (q := st.q) (i := i) (p := first) hroom_iter htrace_le hi_room (by simpa [localAllowed] using hfirst) change Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (localAllowed := localAllowed) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) exact taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_localEnvelope_notEStar_sandwich_room_acceptSource (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (used := used) (localAllowed := localAllowed) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st sandwich hω hlocal_subset_used hscale hlow hcover hroom_selected hfacts hblock haccept_src namespace TaoSection7Case3ChosenActiveLocalUsedStepWithGap theorem first_le_baseKcutNextBound {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut : ℕ} {st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hstep : TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) : hstep.usedStep.step.update.first ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold st.q := taoSection7Case3_le_baseKcutNextBound_of_mem_localSelectedEnvelope hstep.first_mem_local end TaoSection7Case3ChosenActiveLocalUsedStepWithGap /-- Finite on-trace local-used driver for the bounded Case 3 recurrence. Unlike `seqLocalUsed`, this object does not require a global advance oracle for arbitrary active states. Its steps are only required on the states already constructed, together with the propagated base-Kcut iterate bound. -/ structure TaoSection7Case3BoundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut R start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) : Type where state : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut) state_zero : state 0 = st0 transition : ∀ i : ℕ, (hi : i < R) → { step : TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold (state i).q) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (state i) // state (i + 1) = TaoSection7Case3ChosenActiveStepWithGap.nextState (step.usedStep.step) } q_bound : ∀ i : ℕ, (hi : i ≤ R) → (state i).q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start namespace TaoSection7Case3BoundedLocalUsedTrace /-- The bounded local-used trace's explicit first-`Rtail + 1` stopping prefix. This is a prefix object only. It is not the full terminal source stopping run unless separate terminal/no-later-black evidence extends it to Tao's maximal run. -/ def prefixSteps {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : List (ℕ × TaoSection7Triangle) := ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail theorem prefixSteps_length {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : htrace.prefixSteps.length = Rtail + 1 := by simp [prefixSteps, taoSection7Case3IndexedEncounterTail_length] theorem prefixSteps_hasAtLeast {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : TaoSection7Case3SourceStoppingRun.HasAtLeast htrace.prefixSteps (Rtail + 1) := by simpa [prefixSteps] using taoSection7Case3_hasAtLeast_cons_indexedTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail theorem prefixSteps_tR? {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : TaoSection7Case3SourceStoppingRun.tR? htrace.prefixSteps (Rtail + 1) = some ((htrace.state Rtail).q) := by simpa [prefixSteps] using taoSection7Case3_tR?_cons_indexedTail_last (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail theorem prefixSteps_last? {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : TaoSection7Case3StoppingLast? htrace.prefixSteps = some ((htrace.state Rtail).q, (htrace.state Rtail).old) := by simpa [prefixSteps] using taoSection7Case3StoppingLast?_cons_indexedTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail theorem prefixSteps_length_and_tR? {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : htrace.prefixSteps.length = Rtail + 1 ∧ TaoSection7Case3SourceStoppingRun.tR? htrace.prefixSteps (Rtail + 1) = some ((htrace.state Rtail).q) := ⟨htrace.prefixSteps_length, htrace.prefixSteps_tR?⟩ theorem q_ltP {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : ∀ i : ℕ, i ≤ Rtrace → (htrace.state i).q < P := by intro i hi cases i with | zero => simpa [htrace.state_zero] using hfirst0_ltP | succ k => have hk : k < Rtrace := Nat.succ_le_iff.mp hi let htrans := htrace.transition k hk let hstep := htrans.1.usedStep.step have hq : (htrace.state (k + 1)).q = hstep.update.first := by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.q) htrans.2 simpa [hq] using hstep.update.first_ltP theorem endpoint_q_ltP {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : (htrace.state Rtrace).q < P := htrace.q_ltP hfirst0_ltP Rtrace (Nat.le_refl Rtrace) end TaoSection7Case3BoundedLocalUsedTrace /-- Project one bounded local-used trace transition to the indexed rebased-update packet consumed by the finite `PrefixRInputs` layer. -/ def taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_boundedLocalUsedTrace_step {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) (i : ℕ) (hi : i < Rtail) : TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) (fun k => (htrace.state k).j) (fun k => (htrace.state k).s) (fun k => (htrace.state k).pre) (fun k => (htrace.state k).pres) i := by let htrans := htrace.transition i hi let hstep := htrans.1.usedStep.step exact TaoSection7Case3OneStepRebasedSourceUpdateWithGap.to_indexedRebasedSourceUpdateWithGap (hupdate := hstep.update) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.q) htrans.2) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.old) htrans.2) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.j) htrans.2) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.s) htrans.2) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.pre) htrans.2) (by simpa [hstep, TaoSection7Case3ChosenActiveStepWithGap.nextState] using congrArg (fun st => st.pres) htrans.2) /-- Bounded local-used traces feed the existing indexed rebased-update consumer. The trace contributes the state arrays and per-step update packets. The initial stopping step and initial finite-window proof remain explicit inputs, because a bounded active trace by itself does not identify the source run's chosen starting stop. -/ theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_boundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : TaoSection7Case3PrefixRInputs (taoSection7SourceBlackInDomain n xi epsilon J) family pointAt Accepts P (Rtail + 1) ((htrace.state 0).q) ((htrace.state 0).old) (taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold ((htrace.state i).q) (taoSection7Case3ExitGapBoundWithBase base Kcut) ((htrace.state i).j) ((htrace.state i).s) ((htrace.state i).pre) ((htrace.state i).pres) := by have hinit0 : TaoSection7Case3InitialStoppingStep pointAt family ((htrace.state 0).q) ((htrace.state 0).old) := by simpa [htrace.state_zero] using hinit have hfirst0 : (htrace.state 0).q < P := by simpa [htrace.state_zero] using hfirst0_ltP exact taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedSourceStateUpdates (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) (fun k => (htrace.state k).j) (fun k => (htrace.state k).s) (fun k => (htrace.state k).pre) (fun k => (htrace.state k).pres) hinit0 hfirst0 (htrace.state 0).full (by intro i hi exact taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_boundedLocalUsedTrace_step (htrace := htrace) i hi |>.to_sourceStateUpdate) namespace TaoSection7Case3ActualRunSourceIdentity /-- Endpoint-window extractor for an actual run whose first stops are identified with a bounded local-used trace. This is still only a take-prefix adapter: the equality identifying `steps.take (Rtail + 1)` with the constructed trace prefix is an explicit premise. -/ theorem tR_ltP_of_boundedLocalUsedTrace_take_eq {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start tR lemma79Count : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps (Rtail + 1) tR lemma79Count) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) (htake : steps.take (Rtail + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail) : tR < P := h.tR_ltP_of_prefixRInputs_take_eq htake (taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_boundedLocalUsedTrace (hinit := hinit) (hfirst0_ltP := hfirst0_ltP) (htrace := htrace)).1 end TaoSection7Case3ActualRunSourceIdentity namespace TaoSection7Case3StoppingTail /-- Compare an actual stopping tail with the tail carried by a bounded local-used trace. The equalities `hq` and `hold` are the predecessor transport needed at each induction step: after comparing one pair, the recursive call uses that pair equality to align the next actual predecessor with the next bounded-trace state. -/ theorem take_eq_indexedEncounterTail_of_boundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) {q : ℕ} {old : TaoSection7Triangle} {steps : List (ℕ × TaoSection7Triangle)} (htail : TaoSection7Case3StoppingTail pointAt family q old steps) : ∀ (i Rtail : ℕ), i + Rtail ≤ Rtrace → q = (htrace.state i).q → old = (htrace.state i).old → Rtail ≤ steps.length → steps.take Rtail = taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) i Rtail := by intro i Rtail induction Rtail generalizing i q old steps with | zero => intro _htrace_room _hq _hold _hlen simp [taoSection7Case3IndexedEncounterTail] | succ Rtail ih => intro htrace_room hq hold hlen cases steps with | nil => simp at hlen | cons head rest => cases head with | mk first Γ => have hstep : TaoSection7Case3StoppingTransition pointAt family q old first Γ := TaoSection7Case3StoppingTail.head_step htail have htail_rest : TaoSection7Case3StoppingTail pointAt family first Γ rest := TaoSection7Case3StoppingTail.tail htail have hi : i < Rtrace := by omega let htrans := htrace.transition i hi let hchosen := htrans.1.usedStep.step let hupdate := hchosen.update subst q subst old have hactual_eq_update : (first, Γ) = (hupdate.first, hupdate.triangle) := hupdate.pair_eq_of_pairwiseDisjoint_stoppingTransition hpair hstep have hupdate_eq_next : (hupdate.first, hupdate.triangle) = ((htrace.state (i + 1)).q, (htrace.state (i + 1)).old) := by apply Prod.ext · simpa [hchosen, hupdate, TaoSection7Case3ChosenActiveStepWithGap.nextState] using (congrArg (fun st => st.q) htrans.2).symm · simpa [hchosen, hupdate, TaoSection7Case3ChosenActiveStepWithGap.nextState] using (congrArg (fun st => st.old) htrans.2).symm have hactual_eq_next : (first, Γ) = ((htrace.state (i + 1)).q, (htrace.state (i + 1)).old) := hactual_eq_update.trans hupdate_eq_next have hlen_tail : Rtail ≤ rest.length := by simpa using Nat.succ_le_succ_iff.mp hlen have hrec : rest.take Rtail = taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) (i + 1) Rtail := ih htail_rest (i + 1) (by omega) (congrArg Prod.fst hactual_eq_next) (congrArg Prod.snd hactual_eq_next) hlen_tail simp [taoSection7Case3IndexedEncounterTail, hactual_eq_next, hrec] end TaoSection7Case3StoppingTail namespace TaoSection7Case3StoppingPrefix /-- The first `Rtail + 1` stops of an actual stopping prefix agree with a bounded local-used trace when the initial pair agrees. The proof uses the initial pairwise-disjoint comparison for the head and `TaoSection7Case3StoppingTail.take_eq_indexedEncounterTail_of_boundedLocalUsedTrace` for the predecessor-transport tail induction. -/ theorem take_eq_boundedLocalUsedTrace_prefix {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (hlen : Rtail + 1 ≤ steps.length) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : steps.take (Rtail + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail := by cases steps with | nil => simp at hlen | cons head rest => cases head with | mk first Γ => have hinit_actual : TaoSection7Case3InitialStoppingStep pointAt family first Γ := TaoSection7Case3StoppingPrefix.head_initial hprefix have htail_actual : TaoSection7Case3StoppingTail pointAt family first Γ rest := TaoSection7Case3StoppingPrefix.tail hprefix have hinit_trace : TaoSection7Case3InitialStoppingStep pointAt family ((htrace.state 0).q) ((htrace.state 0).old) := by simpa [htrace.state_zero] using hinit have hhead_eq : (first, Γ) = ((htrace.state 0).q, (htrace.state 0).old) := hinit_actual.pair_eq_of_pairwiseDisjoint hpair hinit_trace have hlen_tail : Rtail ≤ rest.length := by simpa using Nat.succ_le_succ_iff.mp hlen have htail_eq : rest.take Rtail = taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail := TaoSection7Case3StoppingTail.take_eq_indexedEncounterTail_of_boundedLocalUsedTrace (hpair := hpair) (htrace := htrace) htail_actual 0 Rtail (by omega) (congrArg Prod.fst hhead_eq) (congrArg Prod.snd hhead_eq) hlen_tail simp [hhead_eq, htail_eq] theorem take_eq_boundedLocalUsedTrace_prefix_of_le {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rtail start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hprefix : TaoSection7Case3StoppingPrefix pointAt family steps) (htrace_len : Rtail ≤ Rtrace) (hlen : Rtail + 1 ≤ steps.length) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : steps.take (Rtail + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail := by cases steps with | nil => simp at hlen | cons head rest => cases head with | mk first Γ => have hinit_actual : TaoSection7Case3InitialStoppingStep pointAt family first Γ := TaoSection7Case3StoppingPrefix.head_initial hprefix have htail_actual : TaoSection7Case3StoppingTail pointAt family first Γ rest := TaoSection7Case3StoppingPrefix.tail hprefix have hinit_trace : TaoSection7Case3InitialStoppingStep pointAt family ((htrace.state 0).q) ((htrace.state 0).old) := by simpa [htrace.state_zero] using hinit have hhead_eq : (first, Γ) = ((htrace.state 0).q, (htrace.state 0).old) := hinit_actual.pair_eq_of_pairwiseDisjoint hpair hinit_trace have hlen_tail : Rtail ≤ rest.length := by simpa using Nat.succ_le_succ_iff.mp hlen have htail_eq : rest.take Rtail = taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail := TaoSection7Case3StoppingTail.take_eq_indexedEncounterTail_of_boundedLocalUsedTrace (hpair := hpair) (htrace := htrace) htail_actual 0 Rtail (by simpa using htrace_len) (congrArg Prod.fst hhead_eq) (congrArg Prod.snd hhead_eq) hlen_tail simp [hhead_eq, htail_eq] end TaoSection7Case3StoppingPrefix namespace TaoSection7Case3SourceStoppingRun /-- A complete source stopping run cannot terminate before the bounded local-used trace length. If the run were empty, its terminal no-initial-hit certificate would contradict the supplied initial stopping step. If it were nonempty but too short, the last actual stop agrees with the corresponding bounded-trace state by the pairwise-disjoint first-hit comparison, and the next bounded-trace transition supplies a later post-exit source-black point, contradicting terminality of the complete run. -/ theorem hasAtLeast_of_boundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : TaoSection7Case3SourceStoppingRun.HasAtLeast steps (Rtrace + 1) := by by_contra hnot have hnot_len : ¬ Rtrace + 1 ≤ steps.length := by simpa [HasAtLeast, r] using hnot have hshort : steps.length < Rtrace + 1 := Nat.lt_of_not_ge hnot_len cases steps with | nil => have hnoInitial : TaoSection7Case3NoInitialTriangleHitAfter pointAt family 0 := TaoSection7Case3StoppingTerminal.noInitialTriangleHitAfter_of_nil hrun.terminal_condition exact hnoInitial st0.q hinit.first_hit.1 hinit.first_hit.2.1 | cons head rest => cases head with | mk first Γ => let k : ℕ := rest.length have hk_lt : k < Rtrace := by dsimp [k] at hshort ⊢ simpa using Nat.succ_lt_succ_iff.mp hshort have htake_eq : ((first, Γ) :: rest).take (k + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 k := TaoSection7Case3StoppingPrefix.take_eq_boundedLocalUsedTrace_prefix_of_le (hpair := hpair) hrun.trace_prefix (Rtail := k) (Rtrace := Rtrace) (Nat.le_of_lt hk_lt) (by simp [k]) hinit htrace have htake_self : ((first, Γ) :: rest).take (k + 1) = (first, Γ) :: rest := by simp [k] have hsteps_eq : (first, Γ) :: rest = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 k := by rw [← htake_self] exact htake_eq have hlast_trace : TaoSection7Case3StoppingLast? (((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 k) = some ((htrace.state k).q, (htrace.state k).old) := by simpa using taoSection7Case3StoppingLast?_cons_indexedTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 k have hlast_steps : TaoSection7Case3StoppingLast? ((first, Γ) :: rest) = some ((htrace.state k).q, (htrace.state k).old) := by rw [hsteps_eq] exact hlast_trace have hnoLater : TaoSection7Case3NoLaterBlackAfterExit (taoSection7SourceBlackInDomain n xi epsilon J) pointAt ((htrace.state k).old) := TaoSection7Case3StoppingTerminal.noLaterBlackAfterExit_of_last?_eq_some hrun.terminal_condition hlast_steps let htrans := htrace.transition k hk_lt let hchosen := htrans.1.usedStep.step let hupdate := hchosen.update have hk_first : (htrace.state k).q < hupdate.first := TaoSection7Case3OneStepRebasedSourceUpdateWithGap.q_lt_first hupdate have hfirst_pos : 0 < hupdate.first := lt_of_le_of_lt (Nat.zero_le _) hk_first have hexit : (htrace.state k).old.cornerL < (pointAt hupdate.first).l := hupdate.source.first_hit.2.1.1 exact hnoLater hupdate.first hfirst_pos hexit hupdate.source.first_black theorem tR?_eq_boundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : tR? steps (Rtrace + 1) = some ((htrace.state Rtrace).q) := by have hR : HasAtLeast steps (Rtrace + 1) := hrun.hasAtLeast_of_boundedLocalUsedTrace hpair hinit htrace have hlen : Rtrace + 1 ≤ steps.length := by simpa [HasAtLeast, r] using hR have htake : steps.take (Rtrace + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtrace := TaoSection7Case3StoppingPrefix.take_eq_boundedLocalUsedTrace_prefix_of_le (hpair := hpair) hrun.trace_prefix (Rtail := Rtrace) (Rtrace := Rtrace) (Nat.le_refl Rtrace) hlen hinit htrace have htR_take : tR? (steps.take (Rtrace + 1)) (Rtrace + 1) = some ((htrace.state Rtrace).q) := by rw [htake] simpa using taoSection7Case3_tR?_cons_indexedTail_last (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtrace simpa [tR?_take_self] using htR_take theorem hasTWithinWindow_of_boundedLocalUsedTrace {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) : HasTWithinWindow steps (Rtrace + 1) P := ⟨(htrace.state Rtrace).q, hrun.tR?_eq_boundedLocalUsedTrace hpair hinit htrace, htrace.endpoint_q_ltP hfirst0_ltP⟩ /-- Producer-facing Lemma 7.9/Markov lower-bound packet before the `r >= R` specialization. The source statement first gives a lower bound at `t_min(r,R)`. The `tMin_eq` field records that pre-specialization accessor directly as `tR? steps (min (r steps) R)`, while `sourceWhite_sum_gt` keeps the raw positive-offset source-white sum visible. This is an input contract only: it does not prove Lemma 7.9, the Markov estimate, or the outside-`FSlack` probability bound. -/ structure Lemma79MarkovSourceWhiteMinBound (pointAt : ℕ → TaoSection7Point) (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] (steps : List (ℕ × TaoSection7Triangle)) (R threshold tMin : ℕ) : Prop where tMin_eq : tR? steps (min (r steps) R) = some tMin sourceWhite_sum_gt : threshold < (Finset.Icc 1 tMin).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0) namespace Lemma79MarkovSourceWhiteMinBound theorem sourceWhitePoint_sum_gt_threshold_of_tR? {pointAt : ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {steps : List (ℕ × TaoSection7Triangle)} {R threshold tMin t : ℕ} (h : Lemma79MarkovSourceWhiteMinBound pointAt n xi epsilon steps R threshold tMin) (_hRpos : 0 < R) (hR : HasAtLeast steps R) (htR : tR? steps R = some t) : threshold < (Finset.Icc 1 t).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0) := by have hminIndex : min (r steps) R = R := Nat.min_eq_right hR have htMin_at_R : tR? steps R = some tMin := by simpa [hminIndex] using h.tMin_eq have htMin_eq : tMin = t := by rw [htR] at htMin_at_R cases htMin_at_R rfl simpa [htMin_eq] using h.sourceWhite_sum_gt /-- Pointwise shifted-`FSlack` complement and scalar-room producer for the Lemma 7.9 min-bound socket. This is still a source-facing input theorem: the caller supplies the moment identity and the real scalar room. The theorem only performs the Markov complement algebra, logarithmic cutoff comparison, and Nat count transport needed to fill `Lemma79MarkovSourceWhiteMinBound`. -/ theorem of_shiftedFSlack_pointwise_room {Ω : Type*} {FSlack : Set Ω} {ω : Ω} {pointAt : Ω → ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {R threshold tMin lemma79MinCount lowerBound Amarkov : ℕ} {moment : Ω → ℝ} (hout : ω ∉ FSlack) (hFSlack : FSlack = {ω | (10 : ℝ) ^ (Amarkov + 2) * Real.exp epsilon < moment ω}) (hmoment_eq : moment ω = Real.exp (-(lemma79MinCount : ℝ) + epsilon * (min (r (steps ω)) R : ℝ))) (hscalar_room : (lowerBound : ℝ) + ((Amarkov + 2 : ℕ) : ℝ) * Real.log 10 + epsilon < epsilon * (min (r (steps ω)) R : ℝ)) (hthreshold : threshold ≤ lowerBound) (htMin_eq : tR? (steps ω) (min (r (steps ω)) R) = some tMin) (hcount_identity_min : lemma79MinCount = (Finset.Icc 1 tMin).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0)) : Lemma79MarkovSourceWhiteMinBound (pointAt ω) n xi epsilon (steps ω) R threshold tMin where tMin_eq := htMin_eq sourceWhite_sum_gt := by let k : ℕ := Amarkov + 2 let tMinIndex : ℕ := min (r (steps ω)) R have hnot_cutoff : ¬ ((10 : ℝ) ^ k * Real.exp epsilon < moment ω) := by have hout' : ω ∉ {ω | (10 : ℝ) ^ (Amarkov + 2) * Real.exp epsilon < moment ω} := by simpa [hFSlack] using hout simpa [k] using hout' have hmoment_le_cutoff : moment ω ≤ (10 : ℝ) ^ k * Real.exp epsilon := le_of_not_gt hnot_cutoff have hpow_exp : (10 : ℝ) ^ k = Real.exp ((k : ℝ) * Real.log 10) := by calc (10 : ℝ) ^ k = (Real.exp (Real.log 10)) ^ k := by rw [Real.exp_log (by norm_num : (0 : ℝ) < 10)] _ = Real.exp ((k : ℝ) * Real.log 10) := (Real.exp_nat_mul (Real.log 10) k).symm have hcutoff_eq : (10 : ℝ) ^ k * Real.exp epsilon = Real.exp ((k : ℝ) * Real.log 10 + epsilon) := by rw [hpow_exp, ← Real.exp_add] have hexp_le : Real.exp (-(lemma79MinCount : ℝ) + epsilon * (tMinIndex : ℝ)) ≤ Real.exp ((k : ℝ) * Real.log 10 + epsilon) := by simpa [hmoment_eq, hcutoff_eq, tMinIndex] using hmoment_le_cutoff have harg_le : -(lemma79MinCount : ℝ) + epsilon * (tMinIndex : ℝ) ≤ (k : ℝ) * Real.log 10 + epsilon := Real.exp_le_exp.mp hexp_le have hroom_local : (lowerBound : ℝ) + ((k : ℝ) * Real.log 10 + epsilon) < epsilon * (tMinIndex : ℝ) := by have hroom_base : (lowerBound : ℝ) + (k : ℝ) * Real.log 10 + epsilon < epsilon * (tMinIndex : ℝ) := by simpa [k, tMinIndex] using hscalar_room calc (lowerBound : ℝ) + ((k : ℝ) * Real.log 10 + epsilon) = (lowerBound : ℝ) + (k : ℝ) * Real.log 10 + epsilon := by ring _ < epsilon * (tMinIndex : ℝ) := hroom_base have hlower_real : (lowerBound : ℝ) < (lemma79MinCount : ℝ) := by linarith [harg_le, hroom_local] have hcount_lower : lowerBound < lemma79MinCount := by exact_mod_cast hlower_real have hsource_lower : lowerBound < (Finset.Icc 1 tMin).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0) := by simpa [hcount_identity_min] using hcount_lower exact lt_of_le_of_lt hthreshold hsource_lower end Lemma79MarkovSourceWhiteMinBound /-- Source-facing pin for the repaired Lemma 7.9 tail statistic. The global Markov moment is the repaired tail statistic, not the printed unconditional `t_min(r,R)` statistic. On samples with `r >= R` this statistic reduces to the local exponential moment used by the pointwise min-bound socket. -/ structure Lemma79RepairedTailOnRGeR {Ω : Type*} (steps : Ω → List (ℕ × TaoSection7Triangle)) (R : ℕ) (tailCount : Ω → ℕ) (moment : Ω → ℝ) (epsilon : ℝ) : Prop where tail_moment_eq : ∀ ω : Ω, moment ω = if R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω) then Real.exp (-(tailCount ω : ℝ) + epsilon * (R : ℝ)) else 0 namespace Lemma79RepairedTailOnRGeR theorem moment_eq_min_of_r_ge {Ω : Type*} {steps : Ω → List (ℕ × TaoSection7Triangle)} {R : ℕ} {tailCount : Ω → ℕ} {moment : Ω → ℝ} {epsilon : ℝ} (h : Lemma79RepairedTailOnRGeR steps R tailCount moment epsilon) (ω : Ω) (hrge : R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω)) : moment ω = Real.exp (-(tailCount ω : ℝ) + epsilon * (min (TaoSection7Case3SourceStoppingRun.r (steps ω)) R : ℝ)) := by rw [h.tail_moment_eq, if_pos hrge] have hrge_real : (R : ℝ) ≤ (TaoSection7Case3SourceStoppingRun.r (steps ω) : ℝ) := by exact_mod_cast hrge rw [min_eq_right hrge_real] end Lemma79RepairedTailOnRGeR /-- Shared source-facing repaired Lemma 7.9 packet for the shifted `FSlack` gate. This record ties the global PMF event and the pointwise min-bound complement to the same repaired tail moment, event definition, `steps`, and `pointAt`. It still contains source-facing hypotheses: the moment construction, expectation bound, scalar room, threshold comparison, `t_min` accessor, and source-white count identity are supplied rather than proved here. -/ structure Lemma79SourceFSlackMomentPacket {Ω : Type*} [Fintype Ω] (μ : PMF Ω) (FSlack : Set Ω) (pointAt : Ω → ℕ → TaoSection7Point) (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] (steps : Ω → List (ℕ × TaoSection7Triangle)) (R threshold Amarkov Aweight : ℕ) (tMin lemma79MinCount lowerBound : Ω → ℕ) (moment : Ω → ℝ) : Prop where Amarkov_eq : Amarkov = Aweight + 1 R_pos : 0 < R repaired_tail_on_r_ge_R : Lemma79RepairedTailOnRGeR steps R lemma79MinCount moment epsilon FSlack_eq : FSlack = {ω | (10 : ℝ) ^ (Amarkov + 2) * Real.exp epsilon < moment ω} moment_nonneg : ∀ ω : Ω, 0 ≤ moment ω repaired_tail_expectation_le : pmfExpectation μ moment ≤ Real.exp epsilon scalar_room : ∀ ω : Ω, R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω) → (lowerBound ω : ℝ) + ((Amarkov + 2 : ℕ) : ℝ) * Real.log 10 + epsilon < epsilon * (min (TaoSection7Case3SourceStoppingRun.r (steps ω)) R : ℝ) threshold_compare_min : ∀ ω : Ω, threshold ≤ lowerBound ω tMin_eq_on_r_ge : ∀ ω : Ω, R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω) → TaoSection7Case3SourceStoppingRun.tR? (steps ω) R = some (tMin ω) count_identity_on_r_ge : ∀ ω : Ω, R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω) → lemma79MinCount ω = (Finset.Icc 1 (tMin ω)).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0) namespace Lemma79SourceFSlackMomentPacket theorem fSlackBudget {Ω : Type*} [Fintype Ω] {μ : PMF Ω} {FSlack : Set Ω} {pointAt : Ω → ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {R threshold Amarkov Aweight : ℕ} {tMin lemma79MinCount lowerBound : Ω → ℕ} {moment : Ω → ℝ} (h : Lemma79SourceFSlackMomentPacket μ FSlack pointAt n xi epsilon steps R threshold Amarkov Aweight tMin lemma79MinCount lowerBound moment) : pmfProb μ FSlack ≤ 1 / ((10 : ℝ) ^ (Aweight + 3)) := taoSection7Lemma79_fSlackBudget_of_expect μ FSlack moment Amarkov Aweight epsilon h.Amarkov_eq h.FSlack_eq h.moment_nonneg h.repaired_tail_expectation_le theorem minBound {Ω : Type*} [Fintype Ω] {μ : PMF Ω} {FSlack : Set Ω} {pointAt : Ω → ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {R threshold Amarkov Aweight : ℕ} {tMin lemma79MinCount lowerBound : Ω → ℕ} {moment : Ω → ℝ} (h : Lemma79SourceFSlackMomentPacket μ FSlack pointAt n xi epsilon steps R threshold Amarkov Aweight tMin lemma79MinCount lowerBound moment) (ω : Ω) (hout : ω ∉ FSlack) (hrge : R ≤ TaoSection7Case3SourceStoppingRun.r (steps ω)) : Lemma79MarkovSourceWhiteMinBound (pointAt ω) n xi epsilon (steps ω) R threshold (tMin ω) := by have htMin_eq : TaoSection7Case3SourceStoppingRun.tR? (steps ω) (min (TaoSection7Case3SourceStoppingRun.r (steps ω)) R) = some (tMin ω) := by have hmin : min (TaoSection7Case3SourceStoppingRun.r (steps ω)) R = R := Nat.min_eq_right hrge simpa [hmin] using h.tMin_eq_on_r_ge ω hrge exact Lemma79MarkovSourceWhiteMinBound.of_shiftedFSlack_pointwise_room (FSlack := FSlack) (ω := ω) (pointAt := pointAt) (steps := steps) (R := R) (threshold := threshold) (tMin := tMin ω) (lemma79MinCount := lemma79MinCount ω) (lowerBound := lowerBound ω) (Amarkov := Amarkov) (moment := moment) hout h.FSlack_eq (h.repaired_tail_on_r_ge_R.moment_eq_min_of_r_ge ω hrge) (h.scalar_room ω hrge) (h.threshold_compare_min ω) htMin_eq (h.count_identity_on_r_ge ω hrge) end Lemma79SourceFSlackMomentPacket /-- Pointwise producer-facing inputs for the Lemma 7.9/Markov min-bound socket. This record does not prove the analytic estimates. It keeps the future source producer honest about the pieces it must supply: the outside-`FSlack` sample, the `R = floor(A^2 / epsilon^4)` floor sandwich, the Lemma 7.9 moment bound, the `FSlack` cutoff event, the outside-`FSlack` Markov/count lower bound at `t_min(r,R)`, the count identity to the raw positive-offset source-white sum, and the comparison to the Lean threshold. -/ structure Lemma79MarkovSourceWhiteMinBoundInputs {Ω : Type*} (FSlack : Set Ω) (ω : Ω) (pointAt : Ω → ℕ → TaoSection7Point) (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] (steps : Ω → List (ℕ × TaoSection7Triangle)) (R threshold tMin lemma79MinCount lowerBound Aweight : ℕ) (moment momentUpper markovCutoff : ℝ) : Prop where outside_FSlack : ω ∉ FSlack R_floor_lower : (R : ℝ) ≤ ((Aweight : ℝ) ^ 2) / (epsilon ^ 4) R_floor_upper : ((Aweight : ℝ) ^ 2) / (epsilon ^ 4) < (R : ℝ) + 1 lemma79_mgf_min : moment ≤ momentUpper FSlack_def : ω ∈ FSlack ↔ markovCutoff < moment markov_count_lower_of_not_FSlack : ω ∉ FSlack → lowerBound < lemma79MinCount threshold_compare_min : threshold ≤ lowerBound tMin_eq : tR? (steps ω) (min (r (steps ω)) R) = some tMin count_identity_min : lemma79MinCount = (Finset.Icc 1 tMin).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0) namespace Lemma79MarkovSourceWhiteMinBoundInputs theorem to_minBound {Ω : Type*} {FSlack : Set Ω} {ω : Ω} {pointAt : Ω → ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {R threshold tMin lemma79MinCount lowerBound Aweight : ℕ} {moment momentUpper markovCutoff : ℝ} (h : Lemma79MarkovSourceWhiteMinBoundInputs FSlack ω pointAt n xi epsilon steps R threshold tMin lemma79MinCount lowerBound Aweight moment momentUpper markovCutoff) : Lemma79MarkovSourceWhiteMinBound (pointAt ω) n xi epsilon (steps ω) R threshold tMin where tMin_eq := h.tMin_eq sourceWhite_sum_gt := by have hcount_lower : lowerBound < lemma79MinCount := h.markov_count_lower_of_not_FSlack h.outside_FSlack have hsource_lower : lowerBound < (Finset.Icc 1 tMin).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0) := by simpa [h.count_identity_min] using hcount_lower exact lt_of_le_of_lt h.threshold_compare_min hsource_lower end Lemma79MarkovSourceWhiteMinBoundInputs theorem sourceWhitePoint_sum_gt_threshold_of_boundedLocalUsedTrace_lemma79 {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {P threshold base Kcut Rtrace start tMin : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hlemma79 : Lemma79MarkovSourceWhiteMinBound pointAt n xi epsilon steps (Rtrace + 1) threshold tMin) : threshold < (Finset.Icc 1 ((htrace.state Rtrace).q)).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0) := by have hR : HasAtLeast steps (Rtrace + 1) := hrun.hasAtLeast_of_boundedLocalUsedTrace hpair hinit htrace have htR : tR? steps (Rtrace + 1) = some ((htrace.state Rtrace).q) := hrun.tR?_eq_boundedLocalUsedTrace hpair hinit htrace exact hlemma79.sourceWhitePoint_sum_gt_threshold_of_tR? (Nat.succ_pos Rtrace) hR htR theorem many_positive_of_boundedLocalUsedTrace_count_identity {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start lemma79Count : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hcount : lemma79Count = taoSection7Case3PositiveWhiteCount W ((htrace.state Rtrace).q)) (hmany : threshold < lemma79Count) : ∀ t : ℕ, tR? steps (Rtrace + 1) = some t → threshold < taoSection7Case3PositiveWhiteCount W t := by intro t ht have htR : tR? steps (Rtrace + 1) = some ((htrace.state Rtrace).q) := hrun.tR?_eq_boundedLocalUsedTrace hpair hinit htrace rw [htR] at ht cases ht simpa [hcount] using hmany theorem many_positive_of_boundedLocalUsedTrace_sourceWhitePoint_sum {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hsource : threshold < (Finset.Icc 1 ((htrace.state Rtrace).q)).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0)) : ∀ t : ℕ, tR? steps (Rtrace + 1) = some t → threshold < taoSection7Case3PositiveWhiteCount W t := by intro t ht have htR : tR? steps (Rtrace + 1) = some ((htrace.state Rtrace).q) := hrun.tR?_eq_boundedLocalUsedTrace hpair hinit htrace rw [htR] at ht cases ht exact hfacts.threshold_lt_positiveWhiteCount_of_sourceWhitePoint_sum (htrace.endpoint_q_ltP hfirst0_ltP) hsource theorem not_low_window_of_boundedLocalUsedTrace_many_positive {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hmany : ∀ t : ℕ, tR? steps (Rtrace + 1) = some t → threshold < taoSection7Case3PositiveWhiteCount W t) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := not_low_window_of_hasTWithinWindow_many_positive (W := W) (steps := steps) (R := Rtrace + 1) (P := P) (threshold := threshold) (hrun.hasTWithinWindow_of_boundedLocalUsedTrace hpair hinit hfirst0_ltP htrace) hmany theorem not_low_window_of_boundedLocalUsedTrace_count_identity {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace start lemma79Count : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hcount : lemma79Count = taoSection7Case3PositiveWhiteCount W ((htrace.state Rtrace).q)) (hmany : threshold < lemma79Count) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := hrun.not_low_window_of_boundedLocalUsedTrace_many_positive hpair hinit hfirst0_ltP htrace (hrun.many_positive_of_boundedLocalUsedTrace_count_identity hpair hinit htrace hcount hmany) theorem not_low_window_of_boundedLocalUsedTrace_sourceWhitePoint_sum {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {P threshold base Kcut Rtrace start : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hrun : TaoSection7Case3SourceStoppingRun (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family steps) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) (hsource : threshold < (Finset.Icc 1 ((htrace.state Rtrace).q)).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0)) : ¬ taoSection7Case3WindowWhiteCount W P ≤ threshold := hrun.not_low_window_of_boundedLocalUsedTrace_many_positive hpair hinit hfirst0_ltP htrace (hrun.many_positive_of_boundedLocalUsedTrace_sourceWhitePoint_sum hpair hinit hfacts hfirst0_ltP htrace hsource) end TaoSection7Case3SourceStoppingRun namespace TaoSection7Case3ActualRunSourceIdentity /-- Produce the take-prefix equality consumed by the bounded-trace endpoint adapter. This is the finite compatibility bridge from an actual stopping run to a bounded local-used trace. It still assumes the bounded trace and the shared initial stopping step; it does not construct either from source probability data. -/ theorem take_eq_boundedLocalUsedTrace {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start tR lemma79Count : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps (Rtail + 1) tR lemma79Count) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : steps.take (Rtail + 1) = ((htrace.state 0).q, (htrace.state 0).old) :: taoSection7Case3IndexedEncounterTail (fun k => (htrace.state k).q) (fun k => (htrace.state k).old) 0 Rtail := by have hlen : Rtail + 1 ≤ steps.length := by simpa [TaoSection7Case3SourceStoppingRun.HasAtLeast, TaoSection7Case3SourceStoppingRun.r] using h.R_le_r exact TaoSection7Case3StoppingPrefix.take_eq_boundedLocalUsedTrace_prefix (hpair := hpair) h.source_run.trace_prefix hlen hinit htrace /-- Endpoint-window extractor from an actual run and a bounded local-used trace, with the take-prefix equality supplied by `take_eq_boundedLocalUsedTrace`. -/ theorem tR_ltP_of_boundedLocalUsedTrace {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtail start tR lemma79Count : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps (Rtail + 1) tR lemma79Count) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) : tR < P := h.tR_ltP_of_boundedLocalUsedTrace_take_eq hinit hfirst0_ltP htrace (h.take_eq_boundedLocalUsedTrace hpair hinit htrace) /-- Checked bridge from the Lemma 7.9/Markov min-bound socket to the `hmany` input consumed by the low-window contradiction. This composes only already-local infrastructure: specialization of the min-bound source-white sum to `t_R`, all-window source/`W` transport, and the actual-run count identity. It does not prove the Lemma 7.9 producer or the global `FSlack` probability budget. -/ theorem manyWhite_of_lemma79MinBound {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {P R tR lemma79Count threshold tMin : ℕ} {steps : List (ℕ × TaoSection7Triangle)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W black pointAt family n J xi epsilon P steps R tR lemma79Count) (htR_ltP : tR < P) (hlemma79 : TaoSection7Case3SourceStoppingRun.Lemma79MarkovSourceWhiteMinBound pointAt n xi epsilon steps R threshold tMin) : threshold < lemma79Count := by have hsource : threshold < (Finset.Icc 1 tR).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt p) then 1 else 0) := hlemma79.sourceWhitePoint_sum_gt_threshold_of_tR? h.R_pos h.R_le_r h.tR_eq have htR_positive : threshold < taoSection7Case3PositiveWhiteCount W tR := h.sameW.threshold_lt_positiveWhiteCount_of_sourceWhitePoint_sum htR_ltP hsource simpa [h.count_identity] using htR_positive /-- Bounded-trace specialization of `manyWhite_of_lemma79MinBound`. The bounded trace supplies the endpoint fact `t_R < P`; the Lemma 7.9 socket still remains an explicit input. -/ theorem manyWhite_of_boundedLocalUsedTrace_lemma79_minBound {sourceAt : ℕ → TaoSection7Point} {restart : ℕ} {W : ℕ → Prop} [DecidablePred W] {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt p))] {P threshold base Kcut Rtail start tR lemma79Count tMin : ℕ} {steps : List (ℕ × TaoSection7Triangle)} {st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (h : TaoSection7Case3ActualRunSourceIdentity sourceAt restart W (taoSection7SourceBlackInDomain n xi epsilon J) pointAt family n J xi epsilon P steps (Rtail + 1) tR lemma79Count) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep pointAt family st0.q st0.old) (hfirst0_ltP : st0.q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) st0) (hlemma79 : TaoSection7Case3SourceStoppingRun.Lemma79MarkovSourceWhiteMinBound pointAt n xi epsilon steps (Rtail + 1) threshold tMin) : threshold < lemma79Count := h.manyWhite_of_lemma79MinBound (h.tR_ltP_of_boundedLocalUsedTrace hpair hinit hfirst0_ltP htrace) hlemma79 /-- Bounded-trace specialization of the repaired-tail Lemma 7.9 source packet. The actual-run identity supplies the `R <= r` branch needed by the packet's pointwise projection, while the bounded trace supplies the endpoint window fact `t_R < P`. The repaired expectation, source moment construction, and scalar-room fields remain inside the supplied packet. -/ theorem manyWhite_of_boundedLocalUsedTrace_lemma79_sourcePacket {Ω : Type*} [Fintype Ω] {μ : PMF Ω} {FSlack : Set Ω} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {family : Set TaoSection7Triangle} {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Ω → Set ℕ} {allowedFin : Ω → Finset ℕ} {bound : Ω → ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {P threshold base Kcut Rtail start Amarkov Aweight : ℕ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count tMin : Ω → ℕ} {lemma79MinCount lowerBound : Ω → ℕ} {moment : Ω → ℝ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) (bound ω) n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} (hpacket : TaoSection7Case3SourceStoppingRun.Lemma79SourceFSlackMomentPacket μ FSlack pointAt n xi epsilon steps (Rtail + 1) threshold Amarkov Aweight tMin lemma79MinCount lowerBound moment) (ω : Ω) (h : TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtail + 1) (tR ω) (lemma79Count ω)) (hnotF : ω ∉ FSlack) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : (st0 ω).q < P) (htrace : TaoSection7Case3BoundedLocalUsedTrace (W := W ω) (family := family) (pointAt := pointAt ω) (Accepts := Accepts ω) (used := used ω) (allowedFin := allowedFin ω) (bound := bound ω) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) (st0 ω)) : threshold < lemma79Count ω := h.manyWhite_of_boundedLocalUsedTrace_lemma79_minBound hpair hinit hfirst0_ltP htrace (hpacket.minBound ω hnotF h.R_le_r_steps) end TaoSection7Case3ActualRunSourceIdentity namespace TaoSection7Case3SourceStoppingRun namespace Lemma79SourceFSlackMomentPacket /-- Source-white count through the supplied `t_min` accessor. -/ def sourceWhiteCountAtTMin {Ω : Type*} (pointAt : Ω → ℕ → TaoSection7Point) (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] (tMin : Ω → ℕ) : Ω → ℕ := fun ω => (Finset.Icc 1 (tMin ω)).sum (fun p : ℕ => if taoSection7SourceWhitePoint n xi epsilon (pointAt ω p) then 1 else 0) /-- Repaired tail moment used by the source-facing Lemma 7.9 packet. -/ noncomputable def repairedTailMoment {Ω : Type*} (steps : Ω → List (ℕ × TaoSection7Triangle)) (R : ℕ) (tailCount : Ω → ℕ) (epsilon : ℝ) : Ω → ℝ := fun ω => if R ≤ r (steps ω) then Real.exp (-(tailCount ω : ℝ) + epsilon * (R : ℝ)) else 0 /-- Strict shifted `FSlack` event attached to the repaired tail moment. -/ noncomputable def shiftedFSlackEvent {Ω : Type*} (steps : Ω → List (ℕ × TaoSection7Triangle)) (R : ℕ) (tailCount : Ω → ℕ) (Amarkov : ℕ) (epsilon : ℝ) : Set Ω := {ω | (10 : ℝ) ^ (Amarkov + 2) * Real.exp epsilon < repairedTailMoment steps R tailCount epsilon ω} /-- Deterministic `t_min` accessor extracted from `tR?`, with default unused. -/ def tMinIndex {Ω : Type*} (steps : Ω → List (ℕ × TaoSection7Triangle)) (R : ℕ) : Ω → ℕ := fun ω => (tR? (steps ω) (min (r (steps ω)) R)).getD 0 theorem tMinIndex_min_pos_of_isSome {Ω : Type*} {steps : Ω → List (ℕ × TaoSection7Triangle)} {R : ℕ} (h : ∀ ω : Ω, (tR? (steps ω) (min (r (steps ω)) R)).isSome = true) : ∀ ω : Ω, 0 < min (r (steps ω)) R := by intro ω exact tR?_isSome_pos (h ω) /-- Reduce the packet's scalar-room field to a cleaner `R`-room premise. This is only the `min(r,R) = R` rewrite under the branch evidence `R <= r`. It does not prove the large-`Aweight`/floor inequality that must eventually produce the `R`-room bound. -/ theorem scalar_room_of_R_room {Ω : Type*} {steps : Ω → List (ℕ × TaoSection7Triangle)} {R Amarkov : ℕ} {lowerBound : Ω → ℕ} {epsilon : ℝ} (hroomR : ∀ ω : Ω, (lowerBound ω : ℝ) + ((Amarkov + 2 : ℕ) : ℝ) * Real.log 10 + epsilon < epsilon * (R : ℝ)) : ∀ ω : Ω, R ≤ r (steps ω) → (lowerBound ω : ℝ) + ((Amarkov + 2 : ℕ) : ℝ) * Real.log 10 + epsilon < epsilon * (min (r (steps ω)) R : ℝ) := by intro ω hrge have hrge_real : (R : ℝ) ≤ (r (steps ω) : ℝ) := by exact_mod_cast hrge simpa [min_eq_right hrge_real] using hroomR ω /-- Producer pilot for the repaired-tail Lemma 7.9 source packet. The repaired tail moment, strict shifted `FSlack` event, and source-white count are definitional here. The analytic expectation estimate, the large-parameter `R`-room inequality, threshold comparison, `0 < R`, and the branch-local `t_min` accessor remain explicit producer obligations. -/ theorem of_repairedTailMoment {Ω : Type*} [Fintype Ω] (μ : PMF Ω) {pointAt : Ω → ℕ → TaoSection7Point} {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {R threshold Amarkov Aweight : ℕ} {tMin lowerBound : Ω → ℕ} (hAmarkov : Amarkov = Aweight + 1) (hR_pos : 0 < R) (hexpect : pmfExpectation μ (repairedTailMoment steps R (sourceWhiteCountAtTMin pointAt n xi epsilon tMin) epsilon) ≤ Real.exp epsilon) (hroomR : ∀ ω : Ω, (lowerBound ω : ℝ) + ((Amarkov + 2 : ℕ) : ℝ) * Real.log 10 + epsilon < epsilon * (R : ℝ)) (hthreshold : ∀ ω : Ω, threshold ≤ lowerBound ω) (htMin_eq_on_r_ge : ∀ ω : Ω, R ≤ r (steps ω) → tR? (steps ω) R = some (tMin ω)) : Lemma79SourceFSlackMomentPacket μ (shiftedFSlackEvent steps R (sourceWhiteCountAtTMin pointAt n xi epsilon tMin) Amarkov epsilon) pointAt n xi epsilon steps R threshold Amarkov Aweight tMin (sourceWhiteCountAtTMin pointAt n xi epsilon tMin) lowerBound (repairedTailMoment steps R (sourceWhiteCountAtTMin pointAt n xi epsilon tMin) epsilon) where Amarkov_eq := hAmarkov R_pos := hR_pos repaired_tail_on_r_ge_R := { tail_moment_eq := by intro ω rfl } FSlack_eq := rfl moment_nonneg := by intro ω by_cases hR : R ≤ r (steps ω) · have hpos : 0 < Real.exp (-(sourceWhiteCountAtTMin pointAt n xi epsilon tMin ω : ℝ) + epsilon * (R : ℝ)) := Real.exp_pos _ simpa [repairedTailMoment, hR] using hpos.le · simp [repairedTailMoment, hR] repaired_tail_expectation_le := hexpect scalar_room := scalar_room_of_R_room hroomR threshold_compare_min := hthreshold tMin_eq_on_r_ge := htMin_eq_on_r_ge count_identity_on_r_ge := by intro ω _hrge rfl /-- Packet-namespaced actual-run consumer for the repaired-tail Lemma 7.9 source packet. For a fixed sample, the actual-run identity supplies the `R <= r` branch needed by `minBound`; the endpoint fact `t_R < P` stays explicit. This is the narrow projection that feeds the existing `manyWhite` consumer without proving the repaired expectation, source moment construction, endpoint control, or scalar-room inputs. -/ theorem manyWhite_of_actualRun {Ω : Type*} [Fintype Ω] {μ : PMF Ω} {FSlack : Set Ω} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {black : Ω → TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : Ω → ℕ → TaoSection7Point} {n J P R threshold Amarkov Aweight : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count tMin : Ω → ℕ} {lemma79MinCount lowerBound : Ω → ℕ} {moment : Ω → ℝ} (h : Lemma79SourceFSlackMomentPacket μ FSlack pointAt n xi epsilon steps R threshold Amarkov Aweight tMin lemma79MinCount lowerBound moment) (ω : Ω) (hidentity : TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (black ω) (pointAt ω) family n J xi epsilon P (steps ω) R (tR ω) (lemma79Count ω)) (htR_ltP : tR ω < P) (hnotF : ω ∉ FSlack) : threshold < lemma79Count ω := hidentity.manyWhite_of_lemma79MinBound htR_ltP (h.minBound ω hnotF hidentity.R_le_r_steps) end Lemma79SourceFSlackMomentPacket end TaoSection7Case3SourceStoppingRun /-- Low-white subset bridge whose endpoint half is derived directly from bounded local-used traces. This removes the separate `PrefixRInputs` and `steps.take` premises from the outside-sample endpoint path. It still assumes the actual-run identities, bounded traces, initial stopping data, and Lemma 7.9-style many-white lower bound. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {used : Ω → Set ℕ} {allowedFin : Ω → Finset ℕ} {bound : Ω → ℕ → ℝ} {n J P threshold base Kcut Rtail start : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) (bound ω) n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtail + 1) (tR ω) (lemma79Count ω)) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q < P) (htrace : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3BoundedLocalUsedTrace (W := W ω) (family := family) (pointAt := pointAt ω) (Accepts := Accepts ω) (used := used ω) (allowedFin := allowedFin ω) (bound := bound ω) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) (st0 ω)) (hmany : ∀ ω : Ω, ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity (hidentity := hidentity) ?_ intro ω hnotE hnotF exact ⟨(hidentity ω).tR_ltP_of_boundedLocalUsedTrace hpair (hinit ω hnotE hnotF) (hfirst0_ltP ω hnotE hnotF) (htrace ω hnotE hnotF), hmany ω hnotE hnotF⟩ /-- Branch-local bounded-trace low-white subset bridge. Unlike `taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace`, the endpoint and many-white inputs may use the low-white membership currently being contradicted. This matches the source Case 3 branch shape: trace production is only required on samples that are low-white and outside the two exceptional events. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace_onLowWhite {Ω : Type*} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {used : Ω → Set ℕ} {allowedFin : Ω → Finset ℕ} {bound : Ω → ℕ → ℝ} {n J P threshold base Kcut Rtail start : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) (bound ω) n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtail + 1) (tR ω) (lemma79Count ω)) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q < P) (htrace : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3BoundedLocalUsedTrace (W := W ω) (family := family) (pointAt := pointAt ω) (Accepts := Accepts ω) (used := used ω) (allowedFin := allowedFin ω) (bound := bound ω) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) (st0 ω)) (hmany : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhite_subset_EStarUsed_union_FSlack ?_ intro ω hlow hnotE hnotF have htR_ltP : tR ω < P := (hidentity ω).tR_ltP_of_boundedLocalUsedTrace hpair (hinit ω hlow hnotE hnotF) (hfirst0_ltP ω hlow hnotE hnotF) (htrace ω hlow hnotE hnotF) have hnotLow : ¬ taoSection7Case3WindowWhiteCount (W ω) P ≤ threshold := (hidentity ω).not_low_window_of_tR_ltP_many_positive htR_ltP (hmany ω hlow hnotE hnotF) exact hnotLow (by simpa [taoSection7Case3LowWhiteEvent, taoSection7Case3LowWhite] using hlow) /-- Low-white subset bridge from the repaired-tail Lemma 7.9 source packet on the bounded-trace branch. This composes the checked repaired-tail packet with the bounded-trace actual-run route. It supplies the packet's pointwise `R <= r` requirement from the actual-run identity and leaves only the packet inputs, bounded-trace construction, and endpoint initialization as external producers. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace_sourcePacket_onLowWhite {Ω : Type*} [Fintype Ω] {μ : PMF Ω} {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {used : Ω → Set ℕ} {allowedFin : Ω → Finset ℕ} {bound : Ω → ℕ → ℝ} {n J P threshold base Kcut Rtail start Amarkov Aweight : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} [∀ ω : Ω, DecidablePred (fun p : ℕ => taoSection7SourceWhitePoint n xi epsilon (pointAt ω p))] {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count tMin : Ω → ℕ} {lemma79MinCount lowerBound : Ω → ℕ} {moment : Ω → ℝ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) (bound ω) n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} {EStarUsed FSlack : Set Ω} (hpacket : TaoSection7Case3SourceStoppingRun.Lemma79SourceFSlackMomentPacket μ FSlack pointAt n xi epsilon steps (Rtail + 1) threshold Amarkov Aweight tMin lemma79MinCount lowerBound moment) (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtail + 1) (tR ω) (lemma79Count ω)) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q < P) (htrace : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3BoundedLocalUsedTrace (W := W ω) (family := family) (pointAt := pointAt ω) (Accepts := Accepts ω) (used := used ω) (allowedFin := allowedFin ω) (bound := bound ω) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtail) (start := start) (st0 ω)) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace_onLowWhite (hidentity := hidentity) hpair hinit hfirst0_ltP htrace ?_ intro ω hlow hnotE hnotF exact (hidentity ω).manyWhite_of_boundedLocalUsedTrace_lemma79_sourcePacket (hpacket := hpacket) ω hnotF hpair (hinit ω hlow hnotE hnotF) (hfirst0_ltP ω hlow hnotE hnotF) (htrace ω hlow hnotE hnotF) /-- Construct a finite bounded local-used trace from an on-trace step producer. The producer receives the current state only after the recursion has built it, and also receives the propagated proof that the state lies under the base-Kcut iterate bound. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_statefulStep {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut R start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (hstep : ∀ i : ℕ, i < R → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → Nonempty (TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st)) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := R) (start := start) st0) := by classical induction R generalizing st0 with | zero => refine ⟨?_⟩ exact { state := fun _i => st0 state_zero := rfl transition := by intro i hi omega q_bound := by intro i hi have hi0 : i = 0 := by omega subst i simpa using h0 } | succ Rtail ih => let hrec := Classical.choice (ih st0 h0 (fun i hi st hbound => hstep i (by omega) st hbound)) let lastState := hrec.state Rtail let lastStep := Classical.choice (hstep Rtail (by omega) lastState (by simpa [lastState] using hrec.q_bound Rtail (Nat.le_refl Rtail))) let nextState := TaoSection7Case3ChosenActiveStepWithGap.nextState ((lastStep).usedStep.step) let state : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut) := fun i => if hle : i ≤ Rtail then hrec.state i else nextState refine ⟨?_⟩ exact { state := state state_zero := by have hle0 : 0 ≤ Rtail := Nat.zero_le Rtail simpa [state, hle0] using hrec.state_zero transition := by intro i hi by_cases hlt : i < Rtail · have hle_succ : i + 1 ≤ Rtail := Nat.succ_le_of_lt hlt have hstate_i : state i = hrec.state i := by simp [state, Nat.le_of_lt hlt] have hstate_succ : state (i + 1) = hrec.state (i + 1) := by simp [state, hle_succ] rw [hstate_i, hstate_succ] exact hrec.transition i hlt · have hi_eq : i = Rtail := by omega subst i have hnot : ¬ Rtail + 1 ≤ Rtail := by omega have hstate_i : state Rtail = hrec.state Rtail := by simp [state] have hstate_succ : state (Rtail + 1) = nextState := by simp [state, hnot] rw [hstate_i, hstate_succ] refine ⟨lastStep, ?_⟩ rfl q_bound := by intro i hi by_cases hle : i ≤ Rtail · simpa [state, hle] using hrec.q_bound i hle · have hi_eq : i = Rtail + 1 := by omega subst i have hnot : ¬ Rtail + 1 ≤ Rtail := by omega have hfirst : ((lastStep).usedStep.step.update.first) ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold lastState.q := TaoSection7Case3ChosenActiveLocalUsedStepWithGap.first_le_baseKcutNextBound (base := base) (Kcut := Kcut) (hstep := lastStep) have hmono : taoSection7Case3BaseKcutNextBound base Kcut threshold lastState.q ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtail]) start) := taoSection7Case3BaseKcutNextBound_mono base Kcut threshold (by simpa [lastState] using hrec.q_bound Rtail (Nat.le_refl Rtail)) have hnext : nextState.q = (lastStep).usedStep.step.update.first := by rfl rw [show state (Rtail + 1) = nextState by simp [state, hnot]] rw [hnext] exact le_trans hfirst (by simpa [Function.iterate_succ_apply'] using hmono) } /-- Source-facing fixed-carrier instantiation of the bounded local-used trace recursor. This removes the abstract bounded `hstep` premise by calling the checked base-Kcut range one-step constructor at each reached state. The room and acceptance inputs are still explicit per-state sockets; producing them from the source process remains the next gate. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut R start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)) bound) (hω : ω ∉ EStarUsed) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ i : ℕ, i < R → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → ∀ first : ℕ, st.q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept : ∀ i : ℕ, i < R → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := R) (start := start) st0) := by exact taoSection7Case3_boundedLocalUsedTrace_of_statefulStep (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[R]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := R) (start := start) st0 h0 (fun i hi st htrace_le => taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := R) (start := start) (i := i) st sandwich hω htrace_le (Nat.succ_le_of_lt hi) hscale hlow hcover (hroom_next i hi st htrace_le) hfacts (haccept i hi st htrace_le)) /-- Room-depth split fixed-carrier bounded trace wrapper. The support carrier and trace length are governed by `Rtrace`, while the deterministic room hypothesis is allowed to run to `Rroom`. The explicit `Rtrace + 1 ≤ Rroom` margin is what lets the final selected transition obtain successor room from the local selected-envelope room lemma without replacing Tao's `Rtrace` selected stops by `Rtrace - 1`. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtrace → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → TaoSection7Case3FirstAfterTriangleHitFrom (pointAtΩ ω) family st.old st.q first → Γ ∈ family → Γ.Mem ((pointAtΩ ω) first) → Γ ≠ st.old → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) := by exact taoSection7Case3_boundedLocalUsedTrace_of_statefulStep (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0 h0 (fun i hi st htrace_le => taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) (i := i) st sandwich hω htrace_le hi hroom_iter hroom_depth hscale hlow hcover hfacts (haccept i hi st htrace_le)) /-- Room-depth split fixed-carrier bounded trace wrapper with transition-shaped acceptance. Compared with `_roomDepth`, the remaining acceptance premise is narrowed to the exact stopping transition selected by the source first-hit packet at each reached state. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_accept {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept_step : ∀ i : ℕ, i < Rtrace → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → ∀ {first : ℕ} {Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAtΩ ω) family st.q st.old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) := by exact taoSection7Case3_boundedLocalUsedTrace_of_statefulStep (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0 h0 (fun i hi st htrace_le => taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth_accept (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) (i := i) st sandwich hω htrace_le hi hroom_iter hroom_depth hscale hlow hcover hfacts (haccept_step i hi st htrace_le)) /-- Global transition-acceptance wrapper for the room-depth bounded trace. This is a transport layer over `_roomDepth_accept`: callers may provide one finite-window, transition-shaped acceptance compatibility premise for the whole sample, rather than an indexed per-state callback. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_transitionAccept {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept_transition : ∀ {q first : ℕ} {old Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAtΩ ω) family q old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAtΩ ω) first) → Accepts q old first Γ) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) := taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_accept (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) st0 h0 sandwich hω hroom_iter hroom_depth hscale hlow hcover hfacts (fun _i _hi st _htrace_le {first} {Γ} hfirst_lt hstep hblack => haccept_transition (q := st.q) (first := first) (old := st.old) (Γ := Γ) hfirst_lt hstep hblack) /-- Room-depth split fixed-carrier bounded trace wrapper with source-provenance acceptance. At each reached state, the acceptance callback consumes the selected first-hit source packet before any transition-shaped erasure is needed by downstream encounter data. -/ theorem taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_acceptSource {Ω : Type*} [Fintype Ω] (W : ℕ → Prop) [DecidablePred W] (pointAtΩ : Ω → ℕ → TaoSection7Point) {ω : Ω} {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {EStarUsed : Set Ω} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut Rtrace Rroom start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (h0 : st0.q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAtΩ family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hω : ω ∉ EStarUsed) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : TaoSection7Case3AllWindowSourceFacts W (pointAtΩ ω) n J xi epsilon P) (haccept_src : ∀ i : ℕ, i < Rtrace → ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family (pointAtΩ ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), st.q ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start → ∀ {candidate first : ℕ} {Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (pointAtΩ ω) st.j st.s st.pre st.pres P st.q st.old Γ candidate first → Accepts st.q st.old first Γ) : Nonempty (TaoSection7Case3BoundedLocalUsedTrace (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0) := by exact taoSection7Case3_boundedLocalUsedTrace_of_statefulStep (W := W) (family := family) (pointAt := pointAtΩ ω) (Accepts := Accepts) (used := {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (R := Rtrace) (start := start) st0 h0 (fun i hi st htrace_le => taoSection7Case3_exists_chosenActiveLocalUsedStepWithGap_of_baseKcutRange_notEStar_sandwich_roomDepth_acceptSource (W := W) (pointAtΩ := pointAtΩ) (ω := ω) (family := family) (Accepts := Accepts) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) (i := i) st sandwich hω htrace_le hi hroom_iter hroom_depth hscale hlow hcover hfacts (haccept_src i hi st htrace_le)) /-- Branch-local low-white subset bridge whose bounded trace is produced from the fixed base-Kcut source carrier. This composes the room-depth transition-acceptance trace producer with the branch-local bounded-trace low-white endpoint consumer. Trace production is only requested for samples that are already in the low-white event and outside the two exceptional events. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_baseKcutRange_notEStar_sandwich_roomDepth_transitionAccept_onLowWhite {Ω : Type*} [Fintype Ω] {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {bound : ℕ → ℝ} {n J P threshold base Kcut Rtrace Rroom start : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtrace + 1) (tR ω) (lemma79Count ω)) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q < P) (h0 : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAt family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3AllWindowSourceFacts (W ω) (pointAt ω) n J xi epsilon P) (haccept_transition : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → ∀ {q first : ℕ} {old Γ : TaoSection7Triangle}, first < P → TaoSection7Case3StoppingTransition (pointAt ω) family q old first Γ → taoSection7SourceBlackInDomain n xi epsilon J ((pointAt ω) first) → Accepts ω q old first Γ) (hmany : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace_onLowWhite (Accepts := Accepts) (used := fun _ω : Ω => {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := fun _ω : Ω => Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := fun _ω : Ω => bound) (Rtail := Rtrace) (start := start) (st0 := st0) (hidentity := hidentity) (hpair := hpair) (hinit := hinit) (hfirst0_ltP := hfirst0_ltP) ?_ (hmany := hmany) intro ω hlow hnotE hnotF have hlow_count : taoSection7Case3WindowWhiteCount (W ω) P ≤ threshold := by simpa [taoSection7Case3LowWhiteEvent, taoSection7Case3LowWhite] using hlow exact Classical.choice (taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_transitionAccept (W := W ω) (pointAtΩ := pointAt) (ω := ω) (family := family) (Accepts := Accepts ω) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) (st0 ω) (h0 ω hlow hnotE hnotF) sandwich hnotE hroom_iter hroom_depth hscale hlow_count hcover (hfacts ω hlow hnotE hnotF) (fun {q} {first} {old} {Γ} hfirst_lt hstep hblack => haccept_transition ω hlow hnotE hnotF (q := q) (first := first) (old := old) (Γ := Γ) hfirst_lt hstep hblack)) /-- Branch-local low-white subset bridge with source-provenance-shaped acceptance. This is the branch-local endpoint composer over the room-depth source acceptance trace producer. Acceptance is only requested after fixing a low-white sample outside `E*_used` and `F*`. -/ theorem taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_baseKcutRange_notEStar_sandwich_roomDepth_acceptSource_onLowWhite {Ω : Type*} [Fintype Ω] {sourceAt : Ω → ℕ → TaoSection7Point} {restart : Ω → ℕ} {W : Ω → ℕ → Prop} [∀ ω, DecidablePred (W ω)] {pointAt : Ω → ℕ → TaoSection7Point} {Accepts : Ω → ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {family : Set TaoSection7Triangle} {bound : ℕ → ℝ} {n J P threshold base Kcut Rtrace Rroom start : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {steps : Ω → List (ℕ × TaoSection7Triangle)} {tR lemma79Count : Ω → ℕ} {st0 : ∀ ω : Ω, TaoSection7Case3ActiveSourceStateWithGap (W ω) family (pointAt ω) bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)} {EStarUsed FSlack : Set Ω} (hidentity : ∀ ω : Ω, TaoSection7Case3ActualRunSourceIdentity (sourceAt ω) (restart ω) (W ω) (taoSection7SourceBlackInDomain n xi epsilon J) (pointAt ω) family n J xi epsilon P (steps ω) (Rtrace + 1) (tR ω) (lemma79Count ω)) (hpair : TaoSection7TriangleFamilyPairwiseDisjoint family) (hinit : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3InitialStoppingStep (pointAt ω) family (st0 ω).q (st0 ω).old) (hfirst0_ltP : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q < P) (h0 : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → (st0 ω).q ≤ start) (sandwich : TaoSection7Case3EStarUsedLargeTriangleSandwich pointAt family EStarUsed {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)} (Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) bound) (hroom_iter : taoSection7Case3BaseKcutNextBoundIterateRoom base Kcut threshold Rroom start P) (hroom_depth : Rtrace + 1 ≤ Rroom) (hscale : TaoSection7Case3RecurrenceScale bound (taoSection7Case3ExitGapBoundWithBase base Kcut)) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hfacts : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → TaoSection7Case3AllWindowSourceFacts (W ω) (pointAt ω) n J xi epsilon P) (haccept_src : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → ∀ {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {q candidate first : ℕ} {old Γ : TaoSection7Triangle}, TaoSection7Case3FirstHitSourceProvenance (taoSection7SourceBlackInDomain n xi epsilon J) family (pointAt ω) j s pre pres P q old Γ candidate first → Accepts ω q old first Γ) (hmany : ∀ ω : Ω, ω ∈ taoSection7Case3LowWhiteEvent W P threshold → ω ∉ EStarUsed → ω ∉ FSlack → threshold < lemma79Count ω) : taoSection7Case3LowWhiteEvent W P threshold ⊆ EStarUsed ∪ FSlack := by refine taoSection7Case3_lowWhiteEvent_subset_EStarUsed_union_FSlack_of_actualRunSourceIdentity_boundedLocalUsedTrace_onLowWhite (Accepts := Accepts) (used := fun _ω : Ω => {p : ℕ | p ∈ Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)}) (allowedFin := fun _ω : Ω => Finset.range (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[Rtrace]) start + 1)) (bound := fun _ω : Ω => bound) (Rtail := Rtrace) (start := start) (st0 := st0) (hidentity := hidentity) (hpair := hpair) (hinit := hinit) (hfirst0_ltP := hfirst0_ltP) ?_ (hmany := hmany) intro ω hlow hnotE hnotF have hlow_count : taoSection7Case3WindowWhiteCount (W ω) P ≤ threshold := by simpa [taoSection7Case3LowWhiteEvent, taoSection7Case3LowWhite] using hlow exact Classical.choice (taoSection7Case3_boundedLocalUsedTrace_of_baseKcutRange_notEStar_sandwich_roomDepth_acceptSource (W := W ω) (pointAtΩ := pointAt) (ω := ω) (family := family) (Accepts := Accepts ω) (EStarUsed := EStarUsed) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) (Rtrace := Rtrace) (Rroom := Rroom) (start := start) (st0 ω) (h0 ω hlow hnotE hnotF) sandwich hnotE hroom_iter hroom_depth hscale hlow_count hcover (hfacts ω hlow hnotE hnotF) (fun _i _hi st _htrace_le {candidate} {first} {Γ} hsrc => haccept_src ω hlow hnotE hnotF (j := st.j) (s := st.s) (pre := st.pre) (pres := st.pres) (q := st.q) (candidate := candidate) (first := first) (old := st.old) (Γ := Γ) hsrc)) namespace TaoSection7Case3ActiveSourceStateWithGap def seq {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound := fun i => Nat.rec st0 (fun i st => TaoSection7Case3ChosenActiveStepWithGap.nextState (advance i st)) i def seqUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound := TaoSection7Case3ActiveSourceStateWithGap.seq (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 (fun i st => (advance i st).step) def seqLocalUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (localAllowed : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound → Set ℕ) (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := localAllowed st) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound := TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 (fun i st => (advance i st).usedStep) end TaoSection7Case3ActiveSourceStateWithGap namespace TaoSection7Case3ActiveSourceTraceWithGap def t {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → ℕ := fun i => (trace i).q def tri {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → TaoSection7Triangle := fun i => (trace i).old def j {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → ℕ+ := fun i => (trace i).j def s {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → ℕ := fun i => (trace i).s def pre {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → List ℕ := fun i => (trace i).pre def pres {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) : ℕ → List (List ℕ) := fun i => (trace i).pres def selectedStopsFinset {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (trace : ℕ → TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (R : ℕ) : Finset ℕ := (Finset.range R).image (fun i => TaoSection7Case3ActiveSourceTraceWithGap.t trace (i + 1)) end TaoSection7Case3ActiveSourceTraceWithGap theorem taoSection7Case3_activeSourceTraceWithGap_t_succ_mem_used_of_seqUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) (i : ℕ) : let trace := TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance TaoSection7Case3ActiveSourceTraceWithGap.t trace (i + 1) ∈ used := by simpa [TaoSection7Case3ActiveSourceTraceWithGap.t, TaoSection7Case3ActiveSourceStateWithGap.seqUsed, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState] using (advance i (TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance i)).first_mem_used theorem taoSection7Case3_activeSourceTraceWithGap_t_succ_mem_allowedFin_of_seqUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) (i : ℕ) : let trace := TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance TaoSection7Case3ActiveSourceTraceWithGap.t trace (i + 1) ∈ allowedFin := by simpa [TaoSection7Case3ActiveSourceTraceWithGap.t, TaoSection7Case3ActiveSourceStateWithGap.seqUsed, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState] using (advance i (TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance i)).first_mem_allowedFin theorem taoSection7Case3_selectedStopsFinset_mem_used_of_seqUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) (R p : ℕ) : p ∈ TaoSection7Case3ActiveSourceTraceWithGap.selectedStopsFinset (TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance) R → p ∈ used := by intro hp rcases Finset.mem_image.mp hp with ⟨i, _hi, rfl⟩ exact taoSection7Case3_activeSourceTraceWithGap_t_succ_mem_used_of_seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance i theorem taoSection7Case3_selectedStopsFinset_subset_allowedFin_of_seqUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) (R : ℕ) : TaoSection7Case3ActiveSourceTraceWithGap.selectedStopsFinset (TaoSection7Case3ActiveSourceStateWithGap.seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance) R ⊆ allowedFin := by intro p hp rcases Finset.mem_image.mp hp with ⟨i, _hi, rfl⟩ exact taoSection7Case3_activeSourceTraceWithGap_t_succ_mem_allowedFin_of_seqUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance i theorem taoSection7Case3_activeSourceTraceWithGap_t_succ_le_baseKcutNextBound_of_seqLocalUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) (i : ℕ) : let trace := TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun st => taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) st0 advance TaoSection7Case3ActiveSourceTraceWithGap.t trace (i + 1) ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold (TaoSection7Case3ActiveSourceTraceWithGap.t trace i) := by simpa [TaoSection7Case3ActiveSourceTraceWithGap.t, TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed, TaoSection7Case3ActiveSourceStateWithGap.seqUsed, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState] using (TaoSection7Case3ChosenActiveLocalUsedStepWithGap.first_le_baseKcutNextBound (base := base) (Kcut := Kcut) (hstep := advance i (TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun st => taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) st0 advance i))) theorem taoSection7Case3_activeSourceTraceWithGap_t_le_baseKcut_iterate_of_seqLocalUsed {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {used : Set ℕ} {allowedFin : Finset ℕ} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold base Kcut start : ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut)) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold (taoSection7Case3ExitGapBoundWithBase base Kcut), TaoSection7Case3ChosenActiveLocalUsedStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (localAllowed := taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) st) (h0 : st0.q ≤ start) (i : ℕ) : let trace := TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun st => taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) st0 advance TaoSection7Case3ActiveSourceTraceWithGap.t trace i ≤ ((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start := by induction i with | zero => simpa [TaoSection7Case3ActiveSourceTraceWithGap.t, TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed, TaoSection7Case3ActiveSourceStateWithGap.seqUsed, TaoSection7Case3ActiveSourceStateWithGap.seq] using h0 | succ i ih => have hstep : let trace := TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun st => taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) st0 advance TaoSection7Case3ActiveSourceTraceWithGap.t trace (i + 1) ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold (TaoSection7Case3ActiveSourceTraceWithGap.t trace i) := taoSection7Case3_activeSourceTraceWithGap_t_succ_le_baseKcutNextBound_of_seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (base := base) (Kcut := Kcut) st0 advance i have hmono : taoSection7Case3BaseKcutNextBound base Kcut threshold (TaoSection7Case3ActiveSourceTraceWithGap.t (TaoSection7Case3ActiveSourceStateWithGap.seqLocalUsed (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (used := used) (allowedFin := allowedFin) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := taoSection7Case3ExitGapBoundWithBase base Kcut) (fun st => taoSection7Case3LocalSelectedEnvelope (taoSection7Case3ExitGapBoundWithBase base Kcut) threshold st.q) st0 advance) i) ≤ taoSection7Case3BaseKcutNextBound base Kcut threshold (((taoSection7Case3BaseKcutNextBound base Kcut threshold)^[i]) start) := taoSection7Case3BaseKcutNextBound_mono base Kcut threshold ih exact le_trans hstep (by simpa [Function.iterate_succ_apply'] using hmono) def taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_activeSourceStateTrace_step {W : ℕ → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {bound : ℕ → ℝ} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} (st0 : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound) (advance : ∀ _i : ℕ, ∀ st : TaoSection7Case3ActiveSourceStateWithGap W family pointAt bound n J xi epsilon P threshold gapBound, TaoSection7Case3ChosenActiveStepWithGap (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st) (i : ℕ) : let trace := TaoSection7Case3ActiveSourceStateWithGap.seq (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) (TaoSection7Case3ActiveSourceTraceWithGap.t trace) (TaoSection7Case3ActiveSourceTraceWithGap.tri trace) (TaoSection7Case3ActiveSourceTraceWithGap.j trace) (TaoSection7Case3ActiveSourceTraceWithGap.s trace) (TaoSection7Case3ActiveSourceTraceWithGap.pre trace) (TaoSection7Case3ActiveSourceTraceWithGap.pres trace) i := by dsimp only exact TaoSection7Case3OneStepRebasedSourceUpdateWithGap.to_indexedRebasedSourceUpdateWithGap (hupdate := (advance i (TaoSection7Case3ActiveSourceStateWithGap.seq (W := W) (family := family) (pointAt := pointAt) (Accepts := Accepts) (bound := bound) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) st0 advance i)).update) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.t, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.tri, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.j, TaoSection7Case3ActiveSourceTraceWithGap.pre, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.s, TaoSection7Case3ActiveSourceTraceWithGap.pre, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.pre, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) (by simp [TaoSection7Case3ActiveSourceTraceWithGap.pres, TaoSection7Case3ActiveSourceStateWithGap.seq, TaoSection7Case3ChosenActiveStepWithGap.nextState]) /-- Active-state version of the existential one-step packet producer. It keeps the chosen split/source packet while adding the successor active triangle and start-budget facts needed by the iterative route. -/ theorem taoSection7Case3_exists_oneStepActiveRebasedSourceUpdateWithGap_of_fullWindowSourcePathDataWithGap (W : ℕ → Prop) [DecidablePred W] (pointAt : ℕ → TaoSection7Point) {family : Set TaoSection7Triangle} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {bound : ℕ → ℝ} {gapBound B : ℕ → ℕ} {allowed : Set ℕ} {P start threshold q : ℕ} {j : ℕ+} {s : ℕ} {pre : List ℕ} {pres : List (List ℕ)} {old : TaoSection7Triangle} (hq : q ∈ Finset.Icc start (start + threshold)) (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold q gapBound j s pre pres) (hscale : TaoSection7Case3RecurrenceScale bound gapBound) (hlow : taoSection7Case3WindowWhiteCount W P ≤ threshold) (hroom : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P q) (hactive : TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound q old) (hcover : TaoSection7TriangleFamilyCoverBlack (taoSection7SourceBlackInDomain n xi epsilon J) family) (hroom_next : ∀ first : ℕ, q < first → first < P → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (hright : taoSection7Case3LaterSearchBound gapBound threshold (start + threshold) ≤ B start) (hallowed_first : ∀ first : ℕ, q < first → first < P → first ∈ allowed) (hnotLarge : ¬ taoSection7Case3LargeTriangleUnion pointAt family allowed bound) (haccept : ∀ first Γ, first < P → taoSection7SourceBlackInDomain n xi epsilon J (pointAt first) → TaoSection7Case3FirstAfterTriangleHitFrom pointAt family old q first → Γ ∈ family → Γ.Mem (pointAt first) → Γ ≠ old → Accepts q old first Γ) : ∃ hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old, hupdate.first ≤ B start ∧ taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P hupdate.first ∧ TaoSection7Case3ActiveTriangleAtWithBound family pointAt bound hupdate.first hupdate.triangle := by rcases taoSection7Case3_exists_encounterStep_rebasedWithGap_and_budget_of_fullWindowSourcePathDataWithGap (W := W) (pointAt := pointAt) (family := family) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (bound := bound) (gapBound := gapBound) (B := B) (P := P) (start := start) (threshold := threshold) (q := q) (j := j) (s := s) (pre := pre) (pres := pres) (old := old) hq hfull hscale hlow hroom hactive hcover hroom_next hfacts hright haccept with ⟨candidate, first, Γ, before, next, after, hsplit, hidx, hsrc, hstep, hrebase, hbudget⟩ have hroom_first : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P first := hroom_next first hsrc.first_hit.1 hsrc.first_ltP have hallowed : first ∈ allowed := hallowed_first first hsrc.first_hit.1 hsrc.first_ltP have hsmall : Γ.size < bound first := taoSection7Case3_triangle_size_lt_of_not_large_union (pointAt := pointAt) (family := family) (allowed := allowed) (bound := bound) hnotLarge hallowed hsrc.new_mem_family hsrc.new_mem let hupdate : TaoSection7Case3OneStepRebasedSourceUpdateWithGap (W := W) (black := taoSection7SourceBlackInDomain n xi epsilon J) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (q := q) (gapBound := gapBound) j s pre pres old := { candidate := candidate first := first triangle := Γ before := before next := next after := after split := hsplit index := hidx source := hsrc encounter := hstep rebased := hrebase } refine ⟨hupdate, hbudget, hroom_first, ?_⟩ exact { old_mem_family := hsrc.new_mem_family pivot_mem := hsrc.new_mem small_size := hsmall } /-- Stateful recursor for indexed rebased source updates. This packages the induction that propagates the `WithGap` full-window state while constructing each indexed update from a local step. The local step sees the current state, so callers do not need to supply a separate all-index `hfull` oracle. -/ def taoSection7Case3_indexedRebasedSourceUpdatesWithGap_and_stateInvariant_of_statefulStep {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hstep : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) → TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i) : { _updates : ∀ i : ℕ, i < Rtail → TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i // ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) } := by induction Rtail with | zero => refine ⟨?_, ?_⟩ · intro i hi omega · intro i hi have hi0 : i = 0 := by omega subst i exact hstate0 | succ Rtail ih => let hrec := ih (fun i hi hstate => hstep i (by omega) hstate) let hupdateRtail := hstep Rtail (by omega) (hrec.2 Rtail (by omega)) refine ⟨?_, ?_⟩ · intro i hi by_cases hlt : i < Rtail · exact hrec.1 i hlt · have hi_eq : i = Rtail := by omega subst i exact hupdateRtail · intro i hi by_cases hle : i ≤ Rtail · exact hrec.2 i hle · have hi_eq : i = Rtail + 1 := by omega subst i exact hupdateRtail.to_sourceStateUpdate.2 theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_statefulRebasedSourceUpdateStep {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hstep : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) → TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) := by let hrec := taoSection7Case3_indexedRebasedSourceUpdatesWithGap_and_stateInvariant_of_statefulStep (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (Rtail := Rtail) (gapBound := gapBound) t tri j s pre pres hstate0 hstep constructor · exact taoSection7Case3_prefixRInputs_of_indexedEncounterTail (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (P := P) t tri (Rtail := Rtail) hinit hfirst_ltP (by intro i hi exact (hrec.1 i hi).encounter) · exact hrec.2 def taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_firstHitSourceProvenance_split {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold : ℕ} {gapBound : ℕ → ℕ} {t : ℕ → ℕ} {tri : ℕ → TaoSection7Triangle} {j : ℕ → ℕ+} {s : ℕ → ℕ} {pre : ℕ → List ℕ} {pres : ℕ → List (List ℕ)} {i candidate : ℕ} {before : List (List ℕ)} {next : List ℕ} {after : List (List ℕ)} (hfull : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i)) (hsrc : TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) candidate (t (i + 1))) (hsplit : pres i = before ++ next :: after) (hidx : before.length + 1 = t (i + 1) - t i) (hroom_next : taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) (hj_next : j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: before)) (hs_next : s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: before)) (hpre_next : pre (i + 1) = next) (hpres_next : pres (i + 1) = after) : TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i := by refine { before := before next := next after := after split := hsplit j_next := hj_next s_next := hs_next pre_next := hpre_next pres_next := hpres_next encounter := hsrc.to_encounterStep haccept rebased := ?_ } exact taoSection7Case3_fullWindowSourcePathDataWithGap_rebase_of_split (hfull := hfull) hsplit hidx hsrc.first_hit.1 hsrc.first_ltP (TaoSection7Case3AllWindowSourceFacts.white_iff_later_withGap hfacts hroom_next) (TaoSection7Case3AllWindowSourceFacts.domain_later_withGap hfacts hroom_next) def taoSection7Case3_indexedRebasedSourceUpdatesWithGap_of_indexedFirstHitSourceProvenance_splits {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (candidate : ℕ → ℕ) (before : ℕ → List (List ℕ)) (next : ℕ → List ℕ) (after : ℕ → List (List ℕ)) (hfull : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i)) (hsrc : ∀ i : ℕ, i < Rtail → TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) (candidate i) (t (i + 1))) (hsplit : ∀ i : ℕ, i < Rtail → pres i = before i ++ next i :: after i) (hidx : ∀ i : ℕ, i < Rtail → (before i).length + 1 = t (i + 1) - t i) (hroom_next : ∀ i : ℕ, i < Rtail → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtail → Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) (hj_next : ∀ i : ℕ, i < Rtail → j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: before i)) (hs_next : ∀ i : ℕ, i < Rtail → s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: before i)) (hpre_next : ∀ i : ℕ, i < Rtail → pre (i + 1) = next i) (hpres_next : ∀ i : ℕ, i < Rtail → pres (i + 1) = after i) : ∀ i : ℕ, i < Rtail → TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i := by intro i hi exact taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_firstHitSourceProvenance_split (hfull := hfull i hi) (hsrc := hsrc i hi) (hsplit := hsplit i hi) (hidx := hidx i hi) (hroom_next := hroom_next i hi) (hfacts := hfacts) (haccept := haccept i hi) (hj_next := hj_next i hi) (hs_next := hs_next i hi) (hpre_next := hpre_next i hi) (hpres_next := hpres_next i hi) theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_statefulIndexedFirstHitSourceProvenance_splits {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (candidate : ℕ → ℕ) (before : ℕ → List (List ℕ)) (next : ℕ → List ℕ) (after : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hsrc : ∀ i : ℕ, i < Rtail → TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) (candidate i) (t (i + 1))) (hsplit : ∀ i : ℕ, i < Rtail → pres i = before i ++ next i :: after i) (hidx : ∀ i : ℕ, i < Rtail → (before i).length + 1 = t (i + 1) - t i) (hroom_next : ∀ i : ℕ, i < Rtail → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtail → Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) (hj_next : ∀ i : ℕ, i < Rtail → j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: before i)) (hs_next : ∀ i : ℕ, i < Rtail → s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: before i)) (hpre_next : ∀ i : ℕ, i < Rtail → pre (i + 1) = next i) (hpres_next : ∀ i : ℕ, i < Rtail → pres (i + 1) = after i) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) := by exact taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_statefulRebasedSourceUpdateStep (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (Rtail := Rtail) (gapBound := gapBound) t tri j s pre pres hinit hfirst_ltP hstate0 (by intro i hi hstate exact taoSection7Case3_indexedRebasedSourceUpdateWithGap_of_firstHitSourceProvenance_split (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) (t := t) (tri := tri) (j := j) (s := s) (pre := pre) (pres := pres) (i := i) (candidate := candidate i) (before := before i) (next := next i) (after := after i) (hfull := hstate) (hsrc := hsrc i hi) (hsplit := hsplit i hi) (hidx := hidx i hi) (hroom_next := hroom_next i hi) (hfacts := hfacts) (haccept := haccept i hi) (hj_next := hj_next i hi) (hs_next := hs_next i hi) (hpre_next := hpre_next i hi) (hpres_next := hpres_next i hi)) theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedFirstHitSourceProvenance_splits {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (candidate : ℕ → ℕ) (before : ℕ → List (List ℕ)) (next : ℕ → List ℕ) (after : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hfull : ∀ i : ℕ, i < Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i)) (hsrc : ∀ i : ℕ, i < Rtail → TaoSection7Case3FirstHitSourceProvenance black family pointAt (j i) (s i) (pre i) (pres i) P (t i) (tri i) (tri (i + 1)) (candidate i) (t (i + 1))) (hsplit : ∀ i : ℕ, i < Rtail → pres i = before i ++ next i :: after i) (hidx : ∀ i : ℕ, i < Rtail → (before i).length + 1 = t (i + 1) - t i) (hroom_next : ∀ i : ℕ, i < Rtail → taoSection7Case3ExitRoom (taoSection7Case3LaterSearchBound gapBound threshold) P (t (i + 1))) (hfacts : TaoSection7Case3AllWindowSourceFacts W pointAt n J xi epsilon P) (haccept : ∀ i : ℕ, i < Rtail → Accepts (t i) (tri i) (t (i + 1)) (tri (i + 1))) (hj_next : ∀ i : ℕ, i < Rtail → j (i + 1) = taoSection7SourceBlockStartJFrom (j i) (pre i :: before i)) (hs_next : ∀ i : ℕ, i < Rtail → s (i + 1) = taoSection7SourceBlockStartSFrom (s i) (pre i :: before i)) (hpre_next : ∀ i : ℕ, i < Rtail → pre (i + 1) = next i) (hpres_next : ∀ i : ℕ, i < Rtail → pres (i + 1) = after i) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) := by exact taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedSourceStateUpdates (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres hinit hfirst_ltP hstate0 (by intro i hi exact (taoSection7Case3_indexedRebasedSourceUpdatesWithGap_of_indexedFirstHitSourceProvenance_splits (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (Rtail := Rtail) (gapBound := gapBound) t tri j s pre pres candidate before next after hfull hsrc hsplit hidx hroom_next hfacts haccept hj_next hs_next hpre_next hpres_next i hi).to_sourceStateUpdate) theorem taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedRebasedSourceUpdates {W : ℕ → Prop} {black : TaoSection7Point → Prop} {family : Set TaoSection7Triangle} {pointAt : ℕ → TaoSection7Point} {Accepts : ℕ → TaoSection7Triangle → ℕ → TaoSection7Triangle → Prop} {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {P threshold Rtail : ℕ} {gapBound : ℕ → ℕ} (t : ℕ → ℕ) (tri : ℕ → TaoSection7Triangle) (j : ℕ → ℕ+) (s : ℕ → ℕ) (pre : ℕ → List ℕ) (pres : ℕ → List (List ℕ)) (hinit : TaoSection7Case3InitialStoppingStep pointAt family (t 0) (tri 0)) (hfirst_ltP : t 0 < P) (hstate0 : TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t 0) gapBound (j 0) (s 0) (pre 0) (pres 0)) (hupdates : ∀ i : ℕ, i < Rtail → TaoSection7Case3IndexedRebasedSourceUpdateWithGap (W := W) (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres i) : TaoSection7Case3PrefixRInputs black family pointAt Accepts P (Rtail + 1) (t 0) (tri 0) (taoSection7Case3IndexedEncounterTail t tri 0 Rtail) ∧ ∀ i : ℕ, i ≤ Rtail → TaoSection7Case3FullWindowSourcePathDataWithGap W pointAt n J xi epsilon P threshold (t i) gapBound (j i) (s i) (pre i) (pres i) := by exact taoSection7Case3_prefixRInputs_and_stateInvariantWithGap_of_indexedSourceStateUpdates (black := black) (family := family) (pointAt := pointAt) (Accepts := Accepts) (n := n) (J := J) (xi := xi) (epsilon := epsilon) (P := P) (threshold := threshold) (gapBound := gapBound) t tri j s pre pres hinit hfirst_ltP hstate0 (by intro i hi exact (hupdates i hi).to_sourceStateUpdate) end Tao end Erdos1135