import Erdos1135.Tao.Section5.AffineSourceLaw import Erdos1135.Tao.Section5.AffineTypicalityRemoval import Erdos1135.Tao.Section5.CommonZ import Erdos1135.Tao.Section5.WeightedMixing /-! # Section 5 Fixed-Time Mixing And Common Z This leaf consumes the unrestricted affine source law at one primitive time. It compares the full affine coefficient expectation with Tao's branch-free Common-Z term, then restores the already-paid prefix-typicality error. Passage-time summation and source-window normalization remain downstream. -/ namespace Erdos1135 namespace Tao noncomputable section /-- The unrestricted affine expectation is exactly the fine Syracuse weighted term at the same modulus. -/ theorem taoSection5FullAffineCoefficientExpectation_eq_fineSyracWeightedTerm (B q : ℕ) (E : Set ℕ) : taoSection5FullAffineCoefficientExpectation B q E = taoSection5FineSyracWeightedTerm q (taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E)) := by unfold taoSection5FullAffineCoefficientExpectation taoSection5FineSyracWeightedTerm syracPMFMassVector rw [← syracPMF_eq_geom2PNatListPMF_map_taoAffineOffsetZMod] /-- The repaired endpoint coefficient has the exact pointwise bound `5`. -/ theorem taoSection5PayloadFreeCoefficient_ePrime_supBound_five {B q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) : TaoSection5CoefficientSupBound q (taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E)) 5 := by refine ⟨by norm_num, ?_⟩ intro X have hcoeff := taoSection5PayloadFreeCoefficient_ePrime_mem_Icc_five lost (E := E) (X := X) hindex rw [abs_of_nonneg hcoeff.1] exact hcoeff.2 /-- Tao's branch-independent Common-Z term is a nonnegative expectation bounded by the repaired coefficient constant `5`. -/ theorem taoSection5PayloadFreeCommonZ_ePrime_mem_Icc_five {B m : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + m ≤ taoSection5N0 B) : taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E) ∈ Set.Icc (0 : ℝ) 5 := by rw [taoSection5PayloadFreeCommonZ_eq_fineSyracWeightedTerm] unfold taoSection5FineSyracWeightedTerm constructor · apply Finset.sum_nonneg intro X _hX exact mul_nonneg (taoSection5PayloadFreeCoefficient_ePrime_mem_Icc_five lost (E := E) (X := X) hindex).1 ENNReal.toReal_nonneg · calc (∑ X : ZMod (3 ^ m), taoSection5PayloadFreeCoefficient m (taoSection5EPrime B E) X * syracPMFMassVector m X) ≤ ∑ X : ZMod (3 ^ m), 5 * syracPMFMassVector m X := by apply Finset.sum_le_sum intro X _hX exact mul_le_mul_of_nonneg_right (taoSection5PayloadFreeCoefficient_ePrime_mem_Icc_five lost (E := E) (X := X) hindex).2 ENNReal.toReal_nonneg _ = 5 * ∑ X : ZMod (3 ^ m), syracPMFMassVector m X := by rw [Finset.mul_sum] _ = 5 := by rw [show (∑ X : ZMod (3 ^ m), syracPMFMassVector m X) = 1 by exact pmf_sum_toReal (syracPMF m)] norm_num /-- A later fixed-time index guard implies the smaller same-level guard needed to bound Common Z. -/ theorem taoSection5PayloadFreeCommonZ_ePrime_mem_Icc_five_of_le {B m q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) (hmq : m ≤ q) : taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E) ∈ Set.Icc (0 : ℝ) 5 := by apply taoSection5PayloadFreeCommonZ_ePrime_mem_Icc_five lost omega /-- At one primitive time, the unrestricted affine expectation differs from the common lower-level term by at most five times the full-L1 oscillation. -/ theorem abs_taoSection5FullAffineCoefficientExpectation_sub_commonZ_le {B m q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) (hmq : m ≤ q) : |taoSection5FullAffineCoefficientExpectation B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| ≤ 5 * syracFineScaleOscillation m q := by let c : ZMod (3 ^ q) → ℝ := taoSection5PayloadFreeCoefficient q (taoSection5EPrime B E) have hfine : taoSection5FullAffineCoefficientExpectation B q E = taoSection5FineSyracWeightedTerm q c := by exact taoSection5FullAffineCoefficientExpectation_eq_fineSyracWeightedTerm B q E have hcoarse := taoSection5PayloadFreeCoarseSyracWeightedTerm_coefficient_eq_commonZ hmq (taoSection5EPrime B E) have hcoarse' : taoSection5CoarseSyracWeightedTerm m q c = taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E) := by simpa [c, taoSection5CoarseSyracWeightedTerm, taoSection5PayloadFreeCoarseSyracWeightedTerm] using hcoarse calc |taoSection5FullAffineCoefficientExpectation B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| = taoSection5WeightedMixingError m q c := by rw [hfine, ← hcoarse'] rfl _ ≤ 5 * syracFineScaleOscillation m q := taoSection5WeightedMixingError_le_sup_mul_oscillation (taoSection5PayloadFreeCoefficient_ePrime_supBound_five lost hindex) /-- A supplied Proposition 1.14 bound gives the fixed-time full-affine/Common-Z comparison with no extra factor two or power of three. -/ theorem abs_taoSection5FullAffineCoefficientExpectation_sub_commonZ_le_of_mixing {A B m q : ℕ} {Cmix : ℝ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) (hm : 1 ≤ m) (hmq : m ≤ q) (hmix : syracFineScaleMixingAt A Cmix) : |taoSection5FullAffineCoefficientExpectation B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| ≤ 5 * (Cmix / (m : ℝ) ^ A) := by exact (abs_taoSection5FullAffineCoefficientExpectation_sub_commonZ_le lost hindex hmq).trans (mul_le_mul_of_nonneg_left (syracFineScaleMixingAt.apply hmix hm hmq) (by norm_num)) /-- Restoring chronological prefix typicality adds exactly the previously paid `5 * delta` error. -/ theorem abs_taoSection5TypicalAffineCoefficientSum_sub_commonZ_le {B m q : ℕ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) (hmq : m ≤ q) : |taoSection5TypicalAffineCoefficientSum B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| ≤ 5 * taoSection5PowerInteriorDelta B + 5 * syracFineScaleOscillation m q := by have htriangle : |taoSection5TypicalAffineCoefficientSum B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| ≤ |taoSection5TypicalAffineCoefficientSum B q E - taoSection5FullAffineCoefficientExpectation B q E| + |taoSection5FullAffineCoefficientExpectation B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| := by calc |taoSection5TypicalAffineCoefficientSum B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| = |(taoSection5TypicalAffineCoefficientSum B q E - taoSection5FullAffineCoefficientExpectation B q E) + (taoSection5FullAffineCoefficientExpectation B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E))| := by congr 1 ring _ ≤ _ := abs_add_le _ _ exact htriangle.trans (add_le_add (abs_taoSection5TypicalAffineCoefficientSum_sub_full_le_five_mul_delta lost failure hindex) (abs_taoSection5FullAffineCoefficientExpectation_sub_commonZ_le lost hindex hmq)) /-- Fixed-time typical/Common-Z comparison after applying a supplied mixing bound. -/ theorem abs_taoSection5TypicalAffineCoefficientSum_sub_commonZ_le_of_mixing {A B m q : ℕ} {Cmix : ℝ} {E : Set ℕ} (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (hindex : taoSection5M0 B + q ≤ taoSection5N0 B) (hm : 1 ≤ m) (hmq : m ≤ q) (hmix : syracFineScaleMixingAt A Cmix) : |taoSection5TypicalAffineCoefficientSum B q E - taoSection5PayloadFreeCommonZ m (taoSection5EPrime B E)| ≤ 5 * taoSection5PowerInteriorDelta B + 5 * (Cmix / (m : ℝ) ^ A) := by have hosc : 5 * syracFineScaleOscillation m q ≤ 5 * (Cmix / (m : ℝ) ^ A) := mul_le_mul_of_nonneg_left (syracFineScaleMixingAt.apply hmix hm hmq) (by norm_num) exact (abs_taoSection5TypicalAffineCoefficientSum_sub_commonZ_le lost failure hindex hmq).trans (add_le_add_right hosc (5 * taoSection5PowerInteriorDelta B)) end end Tao end Erdos1135