import Erdos1135.Tao.Section3.AmbientScalePilot /-! # Seed-Forward Ambient Section 3 Trace This leaf reindexes the descending ambient windows forward from the selected seed. A bounded telescope requires only the steps in the requested prefix. A later ambient-cut consumer chooses the prefix ending at `N_(X^(1 / alpha))`; its proof therefore never inserts the unused next call to `N_X`. -/ namespace Erdos1135 namespace Tao open Filter open scoped Topology BigOperators noncomputable section /-- Scale obtained after `m` forward `alpha`-power steps from seed `y`. -/ def taoSection3ForwardScale (y : ℝ) (m : ℕ) : ℝ := y ^ (taoAlpha ^ m) /-- Proposition 1.11 threshold for the `m`-th seed-forward step. -/ def taoSection3ForwardThreshold (y : ℝ) (m : ℕ) : ℝ := (taoSection3ForwardScale y m) ^ taoSection3AmbientRatio /-- Source window at the `m`-th seed-forward scale. -/ def taoSection3ForwardWindow (y : ℝ) (m : ℕ) : Finset ℕ := taoNyOddWindow (taoSection3ForwardScale y m) taoAlpha /-- Fixed-target good probability in the `m`-th seed-forward window. -/ def taoSection3ForwardGoodProb (B : ℕ) (y : ℝ) (m : ℕ) : ℝ := logFinsetProb (taoSection3ForwardWindow y m) (syracuseThresholdGood B) /-- Error paid by the `m`-th seed-forward passage step. -/ def taoSection3ForwardStepError (errNoHit errTV : ℕ → ℝ) (y : ℝ) (m : ℕ) : ℝ := taoSection3AmbientStepError errNoHit errTV (taoSection3ForwardThreshold y m) theorem taoSection3ForwardScale_succ {y : ℝ} (hy : 0 ≤ y) (m : ℕ) : taoSection3ForwardScale y (m + 1) = (taoSection3ForwardScale y m) ^ taoAlpha := by simpa only [taoSection3ForwardScale, pow_succ] using Real.rpow_mul hy (taoAlpha ^ m) taoAlpha theorem taoSection3ForwardThreshold_rpow_alpha {y : ℝ} (hy : 0 ≤ y) (m : ℕ) : (taoSection3ForwardThreshold y m) ^ taoAlpha = taoSection3ForwardScale y m := by unfold taoSection3ForwardThreshold simpa only [taoSection3AmbientRatio] using Real.rpow_inv_rpow (Real.rpow_nonneg hy (taoAlpha ^ m)) taoAlpha_pos.ne' theorem taoSection3ForwardThreshold_rpow_alpha_sq {y : ℝ} (hy : 0 ≤ y) (m : ℕ) : (taoSection3ForwardThreshold y m) ^ (taoAlpha ^ 2) = taoSection3ForwardScale y (m + 1) := by calc (taoSection3ForwardThreshold y m) ^ (taoAlpha ^ 2) = ((taoSection3ForwardThreshold y m) ^ taoAlpha) ^ taoAlpha := by simpa only [pow_two] using Real.rpow_mul (x := taoSection3ForwardThreshold y m) (by unfold taoSection3ForwardThreshold exact Real.rpow_nonneg (Real.rpow_nonneg hy (taoAlpha ^ m)) taoSection3AmbientRatio) taoAlpha taoAlpha _ = (taoSection3ForwardScale y m) ^ taoAlpha := by rw [taoSection3ForwardThreshold_rpow_alpha hy] _ = taoSection3ForwardScale y (m + 1) := by rw [taoSection3ForwardScale_succ hy] /-- Each seed-forward threshold compares the current window with the literal next window. -/ theorem taoSection3Forward_prop111_pair {y : ℝ} (hy : 0 ≤ y) (m : ℕ) : TaoProp111RealWindowPair (taoSection3ForwardThreshold y m) (taoNyLo (taoSection3ForwardScale y m)) (taoNyHi (taoSection3ForwardScale y m) taoAlpha) (taoNyLo (taoSection3ForwardScale y (m + 1))) (taoNyHi (taoSection3ForwardScale y (m + 1)) taoAlpha) := by simp [TaoProp111RealWindowPair, taoProp111RealY1, taoProp111RealY2, taoSection3ForwardThreshold_rpow_alpha hy, taoSection3ForwardThreshold_rpow_alpha_sq hy] /-- Starting `m` forward steps below ambient index `i` returns exactly the ambient boundary at `i`. -/ theorem taoSection3ForwardScale_ambient_add {X : ℝ} (hX : 0 ≤ X) (i m : ℕ) : taoSection3ForwardScale (taoSection3AmbientBoundary X (i + m)) m = taoSection3AmbientBoundary X i := by have hexponent : taoSection3AmbientRatio ^ (i + m) * taoAlpha ^ m = taoSection3AmbientRatio ^ i := by calc taoSection3AmbientRatio ^ (i + m) * taoAlpha ^ m = taoSection3AmbientRatio ^ i * (taoSection3AmbientRatio ^ m * taoAlpha ^ m) := by rw [pow_add] ring _ = taoSection3AmbientRatio ^ i * (taoSection3AmbientRatio * taoAlpha) ^ m := by rw [mul_pow] _ = taoSection3AmbientRatio ^ i := by simp [taoSection3AmbientRatio, taoAlpha_pos.ne'] unfold taoSection3ForwardScale taoSection3AmbientBoundary rw [← Real.rpow_mul hX, hexponent] /-- The corresponding forward threshold is the next lower ambient boundary. -/ theorem taoSection3ForwardThreshold_ambient_add {X : ℝ} (hX : 0 ≤ X) (i m : ℕ) : taoSection3ForwardThreshold (taoSection3AmbientBoundary X (i + m)) m = taoSection3AmbientBoundary X (i + 1) := by unfold taoSection3ForwardThreshold rw [taoSection3ForwardScale_ambient_add hX] symm simpa only [taoSection3AmbientRatio] using taoSection3AmbientBoundary_succ_eq_rpow_inv hX i theorem taoSection3ForwardWindow_ambient_add {X : ℝ} (hX : 0 ≤ X) (i m : ℕ) : taoSection3ForwardWindow (taoSection3AmbientBoundary X (i + m)) m = taoSection3AmbientWindow X i := by unfold taoSection3ForwardWindow taoSection3AmbientWindow rw [taoSection3ForwardScale_ambient_add hX] /-- A bounded seed-forward call has an ambient remainder index, with source, target, and threshold all identified without natural subtraction. -/ theorem taoSection3ForwardCall_ambient_of_le {X : ℝ} (hX : 0 ≤ X) {j m : ℕ} (hm : m ≤ j) : ∃ d : ℕ, j = m + d ∧ taoSection3ForwardThreshold (taoSection3AmbientBoundary X (j + 2)) m = taoSection3AmbientBoundary X (d + 3) ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X (j + 2)) m = taoSection3AmbientWindow X (d + 2) ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X (j + 2)) (m + 1) = taoSection3AmbientWindow X (d + 1) := by obtain ⟨d, rfl⟩ := Nat.exists_eq_add_of_le hm refine ⟨d, rfl, ?_, ?_, ?_⟩ · simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.reduceAdd] using taoSection3ForwardThreshold_ambient_add hX (d + 2) m · simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using taoSection3ForwardWindow_ambient_add hX (d + 2) m · simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.reduceAdd] using taoSection3ForwardWindow_ambient_add hX (d + 1) (m + 1) /-- Ambient `K=2` is one forward call at `b₃`, from `N_(b₂)` to `N_(b₁)`. -/ theorem taoSection3Forward_ambientK2_identification {X : ℝ} (hX : 0 ≤ X) : taoSection3ForwardThreshold (taoSection3AmbientBoundary X 2) 0 = taoSection3AmbientBoundary X 3 ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X 2) 0 = taoSection3AmbientWindow X 2 ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X 2) 1 = taoSection3AmbientWindow X 1 := by constructor · simpa using taoSection3ForwardThreshold_ambient_add hX 2 0 constructor · simpa using taoSection3ForwardWindow_ambient_add hX 2 0 · simpa using taoSection3ForwardWindow_ambient_add hX 1 1 /-- Ambient `K=3` has calls at `b₄,b₃` and one literal shared middle forward window. -/ theorem taoSection3Forward_ambientK3_identification {X : ℝ} (hX : 0 ≤ X) : taoSection3ForwardThreshold (taoSection3AmbientBoundary X 3) 0 = taoSection3AmbientBoundary X 4 ∧ taoSection3ForwardThreshold (taoSection3AmbientBoundary X 3) 1 = taoSection3AmbientBoundary X 3 ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X 3) 0 = taoSection3AmbientWindow X 3 ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X 3) 1 = taoSection3AmbientWindow X 2 ∧ taoSection3ForwardWindow (taoSection3AmbientBoundary X 3) 2 = taoSection3AmbientWindow X 1 := by constructor · simpa using taoSection3ForwardThreshold_ambient_add hX 3 0 constructor · simpa using taoSection3ForwardThreshold_ambient_add hX 2 1 constructor · simpa using taoSection3ForwardWindow_ambient_add hX 3 0 constructor · simpa using taoSection3ForwardWindow_ambient_add hX 2 1 · simpa using taoSection3ForwardWindow_ambient_add hX 1 2 theorem taoSection3ForwardScale_monotone {y : ℝ} (hy : 1 ≤ y) : Monotone (taoSection3ForwardScale y) := by simpa only [taoSection3ForwardScale, Function.comp_apply] using (Real.monotone_rpow_of_base_ge_one hy).comp (pow_right_mono₀ taoAlpha_one_lt.le) theorem taoSection3ForwardScale_seed_le {y : ℝ} (hy : 1 ≤ y) (m : ℕ) : y ≤ taoSection3ForwardScale y m := by have hmono := taoSection3ForwardScale_monotone hy (Nat.zero_le m) simpa only [taoSection3ForwardScale, pow_zero, Real.rpow_one] using hmono theorem taoSection3ForwardThreshold_one_le {y : ℝ} (hy : 1 ≤ y) (m : ℕ) : 1 ≤ taoSection3ForwardThreshold y m := by unfold taoSection3ForwardThreshold exact Real.one_le_rpow (hy.trans (taoSection3ForwardScale_seed_le hy m)) taoSection3AmbientRatio_pos.le /-- One real cutoff supplies every seed-forward threshold and normalizer. The mass predicate itself is not claimed to be monotone. -/ theorem exists_taoSection3ForwardFamilyCutoff : ∃ Ymass : ℝ, 1 ≤ Ymass ∧ ∀ {y : ℝ}, Ymass ≤ y → ∀ m : ℕ, 1 ≤ taoSection3ForwardThreshold y m ∧ 0 < logFinsetMass (taoSection3ForwardWindow y m) := by obtain ⟨Y₀, hY₀⟩ := eventually_taoNyOddWindow_mass_pos.exists_forall_of_atTop refine ⟨max 1 Y₀, le_max_left _ _, ?_⟩ intro y hy m have hyOne : 1 ≤ y := (le_max_left 1 Y₀).trans hy have hseed := taoSection3ForwardScale_seed_le hyOne m refine ⟨taoSection3ForwardThreshold_one_le hyOne m, ?_⟩ unfold taoSection3ForwardWindow exact hY₀ _ ((le_max_right 1 Y₀).trans (hy.trans hseed)) /-- Fixed-target no-hit is the exact complement of forward-window goodness. -/ theorem taoSection3ForwardNoHitProb_eq_one_sub_goodProb (B : ℕ) (y : ℝ) (m : ℕ) (hmass : 0 < logFinsetMass (taoSection3ForwardWindow y m)) : syracuseNoHitWindowProb B (taoNyLo (taoSection3ForwardScale y m)) (taoNyHi (taoSection3ForwardScale y m) taoAlpha) hmass = 1 - taoSection3ForwardGoodProb B y m := by simpa only [taoSection3ForwardGoodProb, taoSection3ForwardWindow, taoNyOddWindow] using syracuseNoHitWindowProb_eq_one_sub_thresholdGoodProb B (taoNyLo (taoSection3ForwardScale y m)) (taoNyHi (taoSection3ForwardScale y m) taoAlpha) hmass /-- A partial good-probability lower bound is equivalently a fixed-target no-hit upper bound on that forward window. -/ theorem taoSection3ForwardNoHitProb_le_sum_of_partial {B : ℕ} {y : ℝ} {J : ℕ} {errNoHit errTV : ℕ → ℝ} (hmass : 0 < logFinsetMass (taoSection3ForwardWindow y J)) (hpartial : 1 - (∑ m ∈ Finset.range J, taoSection3ForwardStepError errNoHit errTV y m) ≤ taoSection3ForwardGoodProb B y J) : syracuseNoHitWindowProb B (taoNyLo (taoSection3ForwardScale y J)) (taoNyHi (taoSection3ForwardScale y J) taoAlpha) hmass ≤ ∑ m ∈ Finset.range J, taoSection3ForwardStepError errNoHit errTV y m := by rw [taoSection3ForwardNoHitProb_eq_one_sub_goodProb] linarith /-- Bounded all-partial trace consuming one fixed actual-real Proposition 1.11 error packet. This is the uniform core used after the packet's witnesses have been chosen outside all trace parameters. -/ theorem taoSection3Forward_allPartial_of_realRateBounds {errNoHit errTV : ℕ → ℝ} (hbounds : ∀ x lo₁ hi₁ lo₂ hi₂, (hx : 1 ≤ x) → TaoProp111RealWindowPair x lo₁ hi₁ lo₂ hi₂ → ∀ (hmass₁ : 0 < logFinsetMass (oddLogWindow lo₁ hi₁)) (hmass₂ : 0 < logFinsetMass (oddLogWindow lo₂ hi₂)), syracuseNoHitRealWindowProb x lo₁ hi₁ hmass₁ ≤ errNoHit (Nat.floor x) ∧ syracuseNoHitRealWindowProb x lo₂ hi₂ hmass₂ ≤ errNoHit (Nat.floor x) ∧ syracusePassRealFloorWindowTV x lo₁ hi₁ lo₂ hi₂ hx hmass₁ hmass₂ ≤ errTV (Nat.floor x)) {B Jmax : ℕ} {y : ℝ} (hB : 1 ≤ B) (hy : 1 ≤ y) (hmass : ∀ m : ℕ, m ≤ Jmax → 0 < logFinsetMass (taoSection3ForwardWindow y m)) (hseed : taoSection3ForwardScale y 1 ≤ (B : ℝ)) : ∀ J : ℕ, J ≤ Jmax → 1 - (∑ m ∈ Finset.range J, taoSection3ForwardStepError errNoHit errTV y m) ≤ taoSection3ForwardGoodProb B y J := by have hy0 : 0 ≤ y := zero_le_one.trans hy have hbase : taoSection3ForwardGoodProb B y 0 = 1 := by apply taoSection3Ambient_seed_prob_eq_one (y := taoSection3ForwardScale y 0) · exact Real.rpow_nonneg hy0 _ · exact hmass 0 (Nat.zero_le Jmax) · rw [show (taoSection3ForwardScale y 0) ^ taoAlpha = taoSection3ForwardScale y 1 by symm simpa using taoSection3ForwardScale_succ hy0 0] exact hseed have hstep : ∀ m : ℕ, m < Jmax → taoSection3ForwardGoodProb B y m - taoSection3ForwardStepError errNoHit errTV y m ≤ taoSection3ForwardGoodProb B y (m + 1) := by intro m hm have hmle : m ≤ Jmax := hm.le have hmsucc : m + 1 ≤ Jmax := hm have hpair := taoSection3Forward_prop111_pair hy0 m have hpacket := hbounds (taoSection3ForwardThreshold y m) (taoNyLo (taoSection3ForwardScale y m)) (taoNyHi (taoSection3ForwardScale y m) taoAlpha) (taoNyLo (taoSection3ForwardScale y (m + 1))) (taoNyHi (taoSection3ForwardScale y (m + 1)) taoAlpha) (taoSection3ForwardThreshold_one_le hy m) hpair (hmass m hmle) (hmass (m + 1) hmsucc) have hone := taoSection3_fixedTargetGood_real_oneStep (b := B) (x := taoSection3ForwardThreshold y m) (lo₁ := taoNyLo (taoSection3ForwardScale y m)) (hi₁ := taoNyHi (taoSection3ForwardScale y m) taoAlpha) (lo₂ := taoNyLo (taoSection3ForwardScale y (m + 1))) (hi₂ := taoNyHi (taoSection3ForwardScale y (m + 1)) taoAlpha) hB (taoSection3ForwardThreshold_one_le hy m) (hmass m hmle) (hmass (m + 1) hmsucc) have hone' : taoSection3ForwardGoodProb B y m - syracusePassRealFloorWindowTV (taoSection3ForwardThreshold y m) (taoNyLo (taoSection3ForwardScale y m)) (taoNyHi (taoSection3ForwardScale y m) taoAlpha) (taoNyLo (taoSection3ForwardScale y (m + 1))) (taoNyHi (taoSection3ForwardScale y (m + 1)) taoAlpha) (taoSection3ForwardThreshold_one_le hy m) (hmass m hmle) (hmass (m + 1) hmsucc) - syracuseNoHitRealWindowProb (taoSection3ForwardThreshold y m) (taoNyLo (taoSection3ForwardScale y (m + 1))) (taoNyHi (taoSection3ForwardScale y (m + 1)) taoAlpha) (hmass (m + 1) hmsucc) ≤ taoSection3ForwardGoodProb B y (m + 1) := by simpa only [taoSection3ForwardGoodProb, taoSection3ForwardWindow, taoNyOddWindow] using hone unfold taoSection3ForwardStepError taoSection3AmbientStepError linarith [hone', hpacket.2.2, hpacket.2.1] intro J hJ have htelescope := finite_telescope_lower_bound_bounded (taoSection3ForwardGoodProb B y) (taoSection3ForwardStepError errNoHit errTV y) 0 Jmax (by simp [hbase]) hstep J hJ simpa using htelescope /-- Local checked bounded trace with rate witnesses selected for this call. -/ theorem taoSection3Forward_allPartial_checked {B Jmax : ℕ} {y : ℝ} (hB : 1 ≤ B) (hy : 1 ≤ y) (hmass : ∀ m : ℕ, m ≤ Jmax → 0 < logFinsetMass (taoSection3ForwardWindow y m)) (hseed : taoSection3ForwardScale y 1 ≤ (B : ℝ)) : ∃ errNoHit errTV : ℕ → ℝ, ∃ C c : ℝ, 0 ≤ C ∧ 0 < c ∧ (∀ᶠ x : ℝ in atTop, 0 ≤ errNoHit (Nat.floor x) ∧ 0 ≤ errTV (Nat.floor x) ∧ errNoHit (Nat.floor x) ≤ C * x ^ (-c) ∧ errTV (Nat.floor x) ≤ C * (Real.log x) ^ (-c)) ∧ ∀ J : ℕ, J ≤ Jmax → 1 - (∑ m ∈ Finset.range J, taoSection3ForwardStepError errNoHit errTV y m) ≤ taoSection3ForwardGoodProb B y J := by rcases taoProp111RealFirstPassageStabilizationRate_checked with ⟨errNoHit, errTV, C, c, hC, hc, hrate, hbounds⟩ refine ⟨errNoHit, errTV, C, c, hC, hc, hrate, ?_⟩ exact taoSection3Forward_allPartial_of_realRateBounds hbounds hB hy hmass hseed end end Tao end Erdos1135