| import OAI.NumberTheory.TwoPoint.Bounds.ComplexBlockTesting |
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| namespace OAI |
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| /-! The canonical spectral estimate applied to two arbitrary complex sequences. |
| The short-range hypothesis only selects the ten color tests. -/ |
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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_blocks |
| (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) → |
| uniformAverage (fun t : Fin N => |
| ‖retainedComplexBlock P M Q Qp actualPaddingCoefficient active L K W |
| (fun _ => actualPaddingDegreeCut Qp L) h (fun _ _ _ => True) keep F G t.val‖) ≤ |
| 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))) := by |
| obtain ⟨A, hA, hb⟩ := hprime.eventually_actual_affine_bilinear_testing hBr |
| refine ⟨A, hA, ?_⟩ |
| intro h hh E hE W hW |
| filter_upwards [hb h 1 hh (by norm_num) E hE |
| (fun p hp hdiv => False.elim (hp.not_dvd_one hdiv)) W hW] with L hb |
| 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 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 hshift : ∀ d : (j : Fin J) → P j, ∀ q ∈ Q, active (∏ j, (d j).val) q → |
| D ≤ (h * q * ∏ j, (d j).val : ℕ) ∧ (h * q * ∏ j, (d j).val : ℕ) < 2 * D := by |
| intro d q _hq hact |
| exact hs _ _ (canonicalTraceFamily_pair_eligible h E W L eligible hL hW hE _ _ hact) |
| have ht (c : ForwardColorCode) : uniformAverage (fun t : Fin N => |
| ‖ambientComplexBlockTest P M Q Qp actualPaddingCoefficient active L K W |
| (fun _ => actualPaddingDegreeCut Qp L) h (fun _ _ _ => True) keep |
| (fun n => if sourceColor D c n then F n else 0) |
| (fun n => if targetColor D c n then G n else 0) t.val‖) ≤ |
| 4 * ((3 * R) * (2 * M * paddingTiltNormalizer Qp) + |
| ((M : ℝ) * (2 * K * (8 * W) ^ J) * (5 : ℝ) ^ (400 * Real.log L)) * |
| Real.exp (-(2 * ⌊L⌋₊ : ℕ))) := by |
| have hf (n : ℤ) : ‖star (if sourceColor D c n then F n else 0)‖ ≤ 1 := by |
| rw [norm_star] |
| exact sourceColorFunction_norm_le D id c F hF n |
| have hg (n : ℤ) : ‖if targetColor D c n then G n else 0‖ ≤ 1 := |
| targetColorFunction_norm_le D id c G hG n |
| have htest := hb hL η hη hηone eligible he (fun _ _ _ => True) |
| (fun _ _ _ => Iff.rfl) (fun n => star (if sourceColor D c n then F n else 0)) |
| (fun n => if targetColor D c n then G n else 0) hf hg 2 N hN |
| simpa only [ambientComplexBlockTest, Nat.cast_one, one_mul, Nat.one_mul, Nat.add_comm] using htest |
| have hr := retainedComplexBlock_average_le P hp hd M N Q Qp actualPaddingCoefficient |
| active L K W (fun _ => actualPaddingDegreeCut Qp L) h (fun _ _ _ => True) |
| (fun _ _ _ => Iff.rfl) keep D hD hshift F G _ ht |
| linarith |
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| end TwoPointCorrelations |
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| end OAI |
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