import Erdos1135.ND.Band.A5TerminalAtomNominalization /-! # A5 Nominal Terminal-Total Cap This leaf turns the checked harmonic and flat nominalization errors into one common upper bound for the two nominal terminal sums in a fixed band. -/ namespace Erdos1135 namespace ND noncomputable section /-- The real mass of a PMF event is at most one. -/ private theorem ndPMFEventMass_toReal_le_one {Ω : Type*} (μ : PMF Ω) (S : Set Ω) : (μ.toOuterMeasure S).toReal ≤ 1 := by have huniv : μ.toOuterMeasure Set.univ = 1 := (μ.toOuterMeasure_apply_eq_one_iff Set.univ).2 (Set.subset_univ _) have hmono : (μ.toOuterMeasure S).toReal ≤ (μ.toOuterMeasure Set.univ).toReal := ENNReal.toReal_mono (by simp [huniv]) (μ.toOuterMeasure.mono (Set.subset_univ S)) simpa [huniv] using hmono /-- One harmonic nominal terminal atom is nonnegative. -/ theorem ndA5HarmonicTerminalAtomNominalIdealMass_nonneg {B : ℕ} (facts : Tao.TaoSection5AffineSourceScaleFacts B) {branch : Tao.TaoSection5SourceBranch} {C : ℝ} {j : ℕ} {E : Set ℕ} (hcount : 0 < ndA5BandCount B branch) (i : NDA5TerminalAtomIndex B branch C j E) (ht2 : NDA5T2ScaleFacts B i.1.1 C) : 0 ≤ ndA5HarmonicTerminalAtomNominalIdealMass i := by unfold ndA5HarmonicTerminalAtomNominalIdealMass by_cases hnom : ndA5TerminalAtomNominalSupported i · rw [if_pos hnom] exact div_nonneg (ndA5HarmonicTerminalAtomMWeight_nonneg_le_one_div_candidate facts i ht2 hnom.1).1 (logFinsetMass_ndA5OddBand_pos facts.schedule.one_le_B j hcount).le · simp [hnom] /-- One flat nominal terminal atom is nonnegative. -/ theorem ndA5FlatTerminalAtomNominalIdealMass_nonneg {B : ℕ} {branch : Tao.TaoSection5SourceBranch} {C : ℝ} {j : ℕ} {E : Set ℕ} (hB : 1 ≤ B) (i : NDA5TerminalAtomIndex B branch C j E) : 0 ≤ ndA5FlatTerminalAtomNominalIdealMass i := by rw [ndA5FlatTerminalAtomNominalIdealMass_eq_ite hB i] positivity /-- The two literal nominal sums in one retained `(n,L)` cell are nonnegative. This is the sign adapter consumed before true absorption; it retains the terminal endpoint rather than reindexing to physical residues. -/ theorem ndA5NominalFixedCell_sums_nonneg {B : ℕ} (facts : Tao.TaoSection5AffineSourceScaleFacts B) {branch : Tao.TaoSection5SourceBranch} {C : ℝ} {j : ℕ} {E : Set ℕ} (hcount : 0 < ndA5BandCount B branch) (n : {n // n ∈ ndA5PaddedBandWindow B branch C j}) (L : {L // L ∈ ndA5TerminalTotals B C n.1}) (ht2 : NDA5T2ScaleFacts B n.1 C) : let nu := n.1 - Tao.taoSection5M0 B 0 ≤ ∑ M : {M // M ∈ Tao.taoSection5EPrime B E}, ∑ u : Sym (Fin nu) (L.1 - nu), ndA5HarmonicTerminalAtomNominalIdealMass (⟨n, ⟨L, (u, M)⟩⟩ : NDA5TerminalAtomIndex B branch C j E) ∧ 0 ≤ ∑ M : {M // M ∈ Tao.taoSection5EPrime B E}, ∑ u : Sym (Fin nu) (L.1 - nu), ndA5FlatTerminalAtomNominalIdealMass (⟨n, ⟨L, (u, M)⟩⟩ : NDA5TerminalAtomIndex B branch C j E) := by classical dsimp only constructor · apply Finset.sum_nonneg intro M _hM apply Finset.sum_nonneg intro u _hu exact ndA5HarmonicTerminalAtomNominalIdealMass_nonneg facts hcount _ ht2 · apply Finset.sum_nonneg intro M _hM apply Finset.sum_nonneg intro u _hu exact ndA5FlatTerminalAtomNominalIdealMass_nonneg facts.schedule.one_le_B _ /-- Both fixed-band nominal terminal sums obey the same cap consumed by true absorption. The harmonic estimate is internally sharper (`1 + 6 eps`), but the common `1 + 11 eps` interface avoids a law-dependent downstream bound. -/ theorem ndA5NominalTerminalIdealMass_sums_le_commonCap {B : ℕ} (facts : Tao.TaoSection5AffineSourceScaleFacts B) {branch : Tao.TaoSection5SourceBranch} {C : ℝ} {j : ℕ} {E : Set ℕ} (hlogB : (300000 : ℝ) ≤ Real.log B) (hcount : 0 < ndA5BandCount B branch) (ht2 : ∀ i : NDA5TerminalAtomIndex B branch C j E, NDA5T2ScaleFacts B i.1.1 C) (htime : ∀ n ∈ ndA5PaddedBandWindow B branch C j, Tao.taoSection5M0 B < n) (hdiam : ∀ n ∈ ndA5PaddedBandWindow B branch C j, ∀ n' ∈ ndA5PaddedBandWindow B branch C j, n ≤ n' → n' - n ≤ Tao.taoSection5M0 B) : (∑ i : NDA5TerminalAtomIndex B branch C j E, ndA5HarmonicTerminalAtomNominalIdealMass i) ≤ 1 + 11 * (B : ℝ) ^ (-(9 / 10 : ℝ)) ∧ (∑ i : NDA5TerminalAtomIndex B branch C j E, ndA5FlatTerminalAtomNominalIdealMass i) ≤ 1 + 11 * (B : ℝ) ^ (-(9 / 10 : ℝ)) := by let eps : ℝ := (B : ℝ) ^ (-(9 / 10 : ℝ)) let strong : ℝ := (B : ℝ) ^ (-(19749 / 20000 : ℝ)) let μh := ndA5HarmonicBandPMF B branch j facts.schedule.one_le_B hcount let μf := ndA5FlatBandPMF B branch j facts.schedule.one_le_B hcount let U_h := (μh.toOuterMeasure (ndA5TerminalAtomUnion B branch C j E)).toReal let U_f := (μf.toOuterMeasure (ndA5TerminalAtomUnion B branch C j E)).toReal let I_h := ∑ i : NDA5TerminalAtomIndex B branch C j E, ndA5HarmonicTerminalAtomNominalIdealMass i let I_f := ∑ i : NDA5TerminalAtomIndex B branch C j E, ndA5FlatTerminalAtomNominalIdealMass i have hUh : U_h ≤ 1 := by exact ndPMFEventMass_toReal_le_one μh _ have hUf : U_f ≤ 1 := by exact ndPMFEventMass_toReal_le_one μf _ have hUh0 : 0 ≤ U_h := ENNReal.toReal_nonneg have heps0 : 0 ≤ eps := Real.rpow_nonneg (by positivity) _ have hstrong0 : 0 ≤ strong := Real.rpow_nonneg (by positivity) _ have hbase : (1 : ℝ) ≤ (B : ℝ) := by exact_mod_cast facts.schedule.one_le_B have hstrongEps : strong ≤ eps := by dsimp [strong, eps] apply Real.rpow_le_rpow_of_exponent_le hbase norm_num have hstrongMass : strong * U_h ≤ eps := by calc strong * U_h ≤ strong * 1 := mul_le_mul_of_nonneg_left hUh hstrong0 _ = strong := mul_one strong _ ≤ eps := hstrongEps have hharm := ndA5HarmonicTerminalAtomUnionMass_sub_nominalIdealSum_abs_le facts hlogB hcount ht2 htime hdiam have hflat := abs_ndA5FlatTerminalAtomUnionMass_sub_nominalIdealSum_lt facts hlogB hcount ht2 htime hdiam change |U_h - I_h| ≤ strong * U_h + 5 * eps at hharm change |U_f - I_f| < 11 * eps at hflat have hIhSub : I_h - U_h ≤ |U_h - I_h| := by rw [abs_sub_comm] exact le_abs_self (I_h - U_h) have hIfSub : I_f - U_f ≤ |U_f - I_f| := by rw [abs_sub_comm] exact le_abs_self (I_f - U_f) have hIhSix : I_h ≤ 1 + 6 * eps := by linarith have hIh : I_h ≤ 1 + 11 * eps := by linarith have hIf : I_f ≤ 1 + 11 * eps := by linarith simpa [I_h, I_f, eps] using And.intro hIh hIf end end ND end Erdos1135