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8769036 | 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 | import OAI.NumberTheory.TwoPoint.Bounds.ActiveStateVectors
import OAI.NumberTheory.TwoPoint.Bounds.EventualResidueComparison
namespace OAI
/-! Arbitrary functions of the bounded active sets at the trace vertices
are compared at every finite interval, using their exact state expansion. -/
namespace TwoPointCorrelations
open Finset Filter
theorem BravermanDepth22Input.eventually_active_state_comparison (hBr : BravermanDepth22Input) :
∃ A : ℕ, 1000 ≤ A ∧ ∀ᶠ L : ℝ in atTop,
∀ (m : ℕ) (s : Fin m → ℕ) [∀ i, NeZero (s i)],
0 < m → (m : ℝ) ≤ Real.exp L + 1 →
Pairwise (fun i j => (s i).Coprime (s j)) →
(∀ i, (s i : ℝ) ≤ Real.exp L) →
∀ (R n Ninput M : ℕ), (n : ℝ) ≤ Real.exp L →
(M : ℝ) ≤ 400 * Real.log L → (R : ℝ) ≤ 4 * L →
∀ (coord : Fin Ninput → Fin m)
(test : ∀ i, ZMod (s (coord i)) → Bool)
(index : Fin R → Fin n → Fin Ninput)
(f : (Fin R → Finset (Fin n)) → ℝ),
(∀ b : Fin R → boundedActiveStates n M,
|f (fun r => (b r).val)| ≤ Real.exp (L ^ 4)) →
∀ a N : ℕ, Real.exp (L ^ A / 2) ≤ (N : ℝ) →
let F := fun z : ∀ j, ZMod (s j) =>
let bits := residueCircuitInputs s coord test z
if ∀ r, (activeState (fun i => bits (index r i))).card ≤ M
then f (fun r => activeState (fun i => bits (index r i))) else 0
|uniformAverage (fun x : Fin N => F (fun j => (a + x.val : ZMod (s j)))) -
uniformAverage F| ≤ Real.exp (-(L ^ 9)) := by
obtain ⟨A, hA, hscalar⟩ := hBr.eventually_scalar_residue_comparison 1 (by norm_num)
refine ⟨A, hA, ?_⟩
filter_upwards [hscalar, eventually_ge_atTop (4800 : ℝ)] with L hscalar hL
intro m s _ hm hmexp hcop hs R n Ninput M hn hM hR coord test index f hf a N hN
let c := fun b : Fin R → boundedActiveStates n M =>
activeStateVectorCircuit index (fun r => (b r).val)
let coeff := fun b : Fin R → boundedActiveStates n M => f (fun r => (b r).val)
have hb := hscalar m s hm hmexp hcop hs _ Ninput coord test c coeff
(fun b => (activeStateVectorCircuit_depth index _).trans (by norm_num))
(fun b => activeStateVector_size_budget L index _ (by linarith) hn hR)
(by simpa only [one_mul] using activeStateVector_coefficient_budget L f hL hn hM hR hf)
a N hN
have hexpand (z : ∀ j, ZMod (s j)) :
(if ∀ r, (activeState (fun i => residueCircuitInputs s coord test z (index r i))).card ≤ M
then f (fun r => activeState (fun i => residueCircuitInputs s coord test z (index r i))) else 0) =
∑ b : Fin R → boundedActiveStates n M, coeff b *
eventIndicator (residueCircuitEvent s coord test (c b)) z :=
activeStateVector_scalar_expansion M index f (residueCircuitInputs s coord test z)
dsimp only
simpa only [← hexpand] using hb
end TwoPointCorrelations
end OAI
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