import Erdos1135.Tao.Section5.PassageSum import Erdos1135.Tao.Section5.PassNormalizer /-! # Section 5 Ideal Passage Center This leaf combines the denominator-preserving ideal passage sum with the exact passage normalizer. It stops before comparing the ideal affine atoms with the genuine passage event. -/ open scoped BigOperators namespace Erdos1135 namespace Tao noncomputable section /-- The complete ideal passage sum is centered at Tao's limiting passage constant times the branch-independent Common-Z term. -/ theorem abs_taoSection5_idealAffineAtomSum_sub_two_div_log_mul_commonZ_le_of_mixing {A B : ℕ} {Cmix : ℝ} (facts : TaoSection5PowerInteriorBoundaryMassFacts B) (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (branch : TaoSection5SourceBranch) (E : Set ℕ) (hmix : syracFineScaleMixingAt A Cmix) : let D := logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)) let I := {n // n ∈ taoSection5PassTimes B branch} let r := (Fintype.card I : ℝ) / D let Z := taoSection5PayloadFreeCommonZ (taoSection5M0 B) (taoSection5EPrime B E) |(∑ i : TaoSection5ClosedAffineAtomIndex B branch E, taoSection5IdealAffineAtomMass (facts.residue.mass_pos branch) i) - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ r * (5 * taoSection5PowerInteriorDelta B + 5 * (Cmix / (taoSection5M0 B : ℝ) ^ A)) + 5 * (184000 * Real.rpow (Real.log B) (-1 / 5 : ℝ)) := by classical dsimp only let D := logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)) let r := (Fintype.card {n // n ∈ taoSection5PassTimes B branch} : ℝ) / D let Z := taoSection5PayloadFreeCommonZ (taoSection5M0 B) (taoSection5EPrime B E) let S := ∑ i : TaoSection5ClosedAffineAtomIndex B branch E, taoSection5IdealAffineAtomMass (facts.residue.mass_pos branch) i let e := 5 * taoSection5PowerInteriorDelta B + 5 * (Cmix / (taoSection5M0 B : ℝ) ^ A) let eta := 184000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) change |S - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ r * e + 5 * eta have hpass : |S - r * Z| ≤ r * e := by simpa only [S, r, D, Z, e] using (abs_taoSection5_idealAffineAtomSum_sub_commonZ_le_of_mixing branch E (facts.residue.mass_pos branch) facts.geometry.schedule lost failure hmix) have hnormalizer : |r - 2 / Real.log (4 / 3 : ℝ)| ≤ eta := by simpa only [r, D, eta] using (abs_taoSection5_passTimeRatio_sub_two_div_log_four_thirds_le facts branch) have hm0Guard : taoSection5M0 B + taoSection5M0 B ≤ taoSection5N0 B := by obtain ⟨n, hn⟩ := facts.geometry.schedule.nonempty branch have hrange := facts.geometry.schedule.range branch n hn omega have hZ : Z ∈ Set.Icc (0 : ℝ) 5 := by simpa only [Z] using (taoSection5PayloadFreeCommonZ_ePrime_mem_Icc_five (E := E) lost hm0Guard) have heta : 0 ≤ eta := by dsimp only [eta] exact mul_nonneg (by norm_num) (Real.rpow_nonneg (by nlinarith [facts.log_large]) _) have hscaledNormalizer : |(r - 2 / Real.log (4 / 3 : ℝ)) * Z| ≤ 5 * eta := by rw [abs_mul, abs_of_nonneg hZ.1] calc |r - 2 / Real.log (4 / 3 : ℝ)| * Z ≤ eta * Z := mul_le_mul_of_nonneg_right hnormalizer hZ.1 _ ≤ eta * 5 := mul_le_mul_of_nonneg_left hZ.2 heta _ = 5 * eta := by ring calc |S - (2 / Real.log (4 / 3 : ℝ)) * Z| = |(S - r * Z) + (r - 2 / Real.log (4 / 3 : ℝ)) * Z| := by congr 1 ring _ ≤ |S - r * Z| + |(r - 2 / Real.log (4 / 3 : ℝ)) * Z| := abs_add_le _ _ _ ≤ r * e + 5 * eta := add_le_add hpass hscaledNormalizer /-- Given the public passage-ratio cap, the ideal passage-center error has a branch-uniform scalar bound. -/ theorem abs_taoSection5_idealAffineAtomSum_sub_two_div_log_mul_commonZ_le_capped_of_mixing {A B : ℕ} {Cmix : ℝ} (facts : TaoSection5PowerInteriorBoundaryMassFacts B) (lost : TaoSection5PassLostWindowFacts B) (failure : TaoSection5PassTypicalFailureFacts B) (branch : TaoSection5SourceBranch) (E : Set ℕ) (hmix : syracFineScaleMixingAt A Cmix) (hCmix : 0 ≤ Cmix) (hratio : (Fintype.card {n // n ∈ taoSection5PassTimes B branch} : ℝ) / logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)) ≤ 9) : let Z := taoSection5PayloadFreeCommonZ (taoSection5M0 B) (taoSection5EPrime B E) |(∑ i : TaoSection5ClosedAffineAtomIndex B branch E, taoSection5IdealAffineAtomMass (facts.residue.mass_pos branch) i) - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ 45 * taoSection5PowerInteriorDelta B + 45 * (Cmix / (taoSection5M0 B : ℝ) ^ A) + 920000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) := by classical dsimp only let D := logFinsetMass (oddLogWindow (taoSection5SourceLo B branch) (taoSection5SourceHi B branch)) let r := (Fintype.card {n // n ∈ taoSection5PassTimes B branch} : ℝ) / D let Z := taoSection5PayloadFreeCommonZ (taoSection5M0 B) (taoSection5EPrime B E) let S := ∑ i : TaoSection5ClosedAffineAtomIndex B branch E, taoSection5IdealAffineAtomMass (facts.residue.mass_pos branch) i let mix := Cmix / (taoSection5M0 B : ℝ) ^ A let e := 5 * taoSection5PowerInteriorDelta B + 5 * mix let eta := 184000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) change |S - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ 45 * taoSection5PowerInteriorDelta B + 45 * mix + 920000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) have hcenter : |S - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ r * e + 5 * eta := by simpa only [S, r, D, Z, mix, e, eta] using (abs_taoSection5_idealAffineAtomSum_sub_two_div_log_mul_commonZ_le_of_mixing facts lost failure branch E hmix) have hr : r ≤ 9 := by simpa only [r, D] using hratio have hmixNonneg : 0 ≤ mix := by dsimp only [mix] exact div_nonneg hCmix (by positivity) have he : 0 ≤ e := by dsimp only [e] exact add_nonneg (mul_nonneg (by norm_num) facts.geometry.delta_pos.le) (mul_nonneg (by norm_num) hmixNonneg) calc |S - (2 / Real.log (4 / 3 : ℝ)) * Z| ≤ r * e + 5 * eta := hcenter _ ≤ 9 * e + 5 * eta := add_le_add (mul_le_mul_of_nonneg_right hr he) (le_refl _) _ = 45 * taoSection5PowerInteriorDelta B + 45 * mix + 920000 * Real.rpow (Real.log B) (-1 / 5 : ℝ) := by dsimp only [e, eta] ring end end Tao end Erdos1135