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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.NumberTheory.TwoPoint.Bounds.ComplexBlockTesting
namespace OAI
/-! The canonical spectral estimate applied to two arbitrary complex sequences.
The short-range hypothesis only selects the ten color tests. -/
namespace TwoPointCorrelations
open Finset Filter
open scoped Classical
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
end TwoPointCorrelations
end OAI
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