| import OAI.NumberTheory.TwoPoint.Walks.CanonicalComplexTesting |
| import OAI.NumberTheory.TwoPoint.Bounds.ComplexPrefixBound |
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|
| namespace OAI |
|
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| /-! Canonical directed complex prefix estimate with all boundary terms explicit. -/ |
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| namespace TwoPointCorrelations |
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|
| open Finset Filter |
| open scoped Classical |
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|
| theorem ModFiveThetaInput.eventually_canonical_complex_prefix |
| (hprime : ModFiveThetaInput) (hBr : BravermanDepth22Input) : |
| ∃ A : ℕ, 1000 ≤ A ∧ |
| ∀ (h : ℕ) (_hh : 0 < h) (E : Finset ℕ) |
| (hE : ∀ p, p.Prime → p ∣ h → p ∈ E) (W : ℝ) (hW : 1 ≤ W), |
| ∀ᶠ L : ℝ in atTop, |
| ∀ (hL : 1 ≤ L) (η : ℝ), 0 < η → η ≤ 1 → |
| ∀ eligible : ℕ → ℕ → Prop, |
| (∀ d q, eligible d q → PaddingPairEligible L η d q) → |
| ∀ (D : ℤ), 0 < D → |
| (∀ d q, eligible d q → D ≤ (h * q * d : ℕ) ∧ (h * q * d : ℕ) < 2 * D) → |
| let J := primeSupplyCount W L |
| let P := centeredPrimeBands E (L ^ (199 / 200 : ℝ)) W J |
| let Qp := paddingPrimeSupply E L |
| let Q := boundedPaddingDivisors Qp ⌊100 * Real.log L⌋₊ |
| let data := canonicalTraceFamily h E W L eligible hL hW hE |
| let active := fun d q => (d, q) ∈ data.pairs |
| let keep := fun z => ¬ProhibitedSite h ⌊L ^ (1 / 10 : ℝ)⌋₊ active z |
| let M := ⌈Real.exp (103 * L)⌉₊ |
| let K := Real.exp (4 * J) |
| let R := Real.exp 1 * (2 * (K * (2 * Real.exp 150 * Real.sqrt W) ^ J)) |
| ∀ (F G : ℤ → ℂ), (∀ n, ‖F n‖ ≤ 1) → (∀ n, ‖G n‖ ≤ 1) → |
| ∀ N : ℕ, Real.exp (L ^ A / 2) ≤ (N : ℝ) → |
| ‖retainedComplexPrefix P Q Qp actualPaddingCoefficient active L K W |
| (fun _ => actualPaddingDegreeCut Qp L) h (fun _ _ _ => True) keep F G N‖ ≤ |
| (40 * ((3 * R) * (2 * M * paddingTiltNormalizer Qp) + |
| ((M : ℝ) * (2 * K * (8 * W) ^ J) * (5 : ℝ) ^ (400 * Real.log L)) * |
| Real.exp (-(2 * ⌊L⌋₊ : ℕ)))) / (L * M) + |
| ((h * ⌊Real.exp (100 * L + 1)⌋₊ : ℕ) : ℝ) / M * |
| (K / L * (5 : ℝ) ^ (400 * Real.log L) * (8 * W) ^ J) + |
| 2 * (M : ℝ) / N * (K / L * (5 : ℝ) ^ (400 * Real.log L) * (8 * W) ^ J) := by |
| obtain ⟨A, hA, hb⟩ := hprime.eventually_canonical_complex_blocks hBr |
| refine ⟨A, hA, ?_⟩ |
| intro h hh E hE W hW |
| filter_upwards [hb h hh E hE W hW, hprime.eventually_actual_pool_masses E W hW] |
| with L hb hmass |
| intro hL η hη hηone eligible he D hD hs |
| dsimp only |
| let J := primeSupplyCount W L |
| let P := centeredPrimeBands E (L ^ (199 / 200 : ℝ)) W J |
| let Qp := paddingPrimeSupply E L |
| let Q := boundedPaddingDivisors Qp ⌊100 * Real.log L⌋₊ |
| let data := canonicalTraceFamily h E W L eligible hL hW hE |
| let active := fun d q => (d, q) ∈ data.pairs |
| let keep := fun z => ¬ProhibitedSite h ⌊L ^ (1 / 10 : ℝ)⌋₊ active z |
| let M := ⌈Real.exp (103 * L)⌉₊ |
| let K := Real.exp (4 * J) |
| let R := Real.exp 1 * (2 * (K * (2 * Real.exp 150 * Real.sqrt W) ^ J)) |
| intro F G hF hG N hN |
| have hNp : 0 < N := by exact_mod_cast (Real.exp_pos _).trans_le hN |
| have hMp : 0 < M := Nat.one_le_ceil_iff.mpr (Real.exp_pos _) |
| have hp := centeredPrimeBands_prime E (L ^ (199 / 200 : ℝ)) W J |
| have hd := centeredPrimeBands_disjoint E (L ^ (199 / 200 : ℝ)) W J |
| (Real.rpow_nonneg (by linarith) _) (by linarith) |
| have hstep (e : ((j : Fin J) → P j) × Q) : |
| retainedPrimeStep active h e ≤ h * ⌊Real.exp (100 * L + 1)⌋₊ := by |
| apply retainedPrimeStep_le h hη hηone |
| intro d q hmem |
| exact he d q (canonicalTraceFamily_pair_eligible h E W L eligible hL hW hE d q hmem) |
| have ht := hb hL η hη hηone eligible he D hD hs F G hF hG N hN |
| exact retainedComplexPrefix_bound P hp hd M (h * ⌊Real.exp (100 * L + 1)⌋₊) N |
| Q Qp actualPaddingCoefficient active L K W (fun _ => actualPaddingDegreeCut Qp L) |
| h (fun _ _ _ => True) keep F G hF hG hMp hNp (by linarith) (Real.exp_pos _).le |
| (by linarith) (fun q _ => actualPaddingCoefficient_nonneg q) |
| (fun j => (hmass.2.1 j).2.1) hstep _ ht |
|
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| end TwoPointCorrelations |
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| end OAI |
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