| 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 |
|
|
