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760f90e | 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 | import OAI.Computability.UniqueGames.Decoding.ActualMatrixContradiction
import OAI.Computability.UniqueGames.Decoding.ActualSeedSamplingLemmas
import OAI.Computability.UniqueGames.Decoding.ActualSpectralLemmas
import OAI.Computability.UniqueGames.Decoding.TableKeysAddressGameLemmas
import OAI.Computability.UniqueGames.Gadgets.Final
import OAI.Computability.UniqueGames.Inverse.ShortcodeTheorem
namespace OAI
section
/-! The matrix reduction assembled from the gadget and decoder, conditional
on the dimension-uniform shortcode inverse. -/
namespace UniqueGamesTheorem.Decoder.MatrixGap
open Integration.BinaryLinear Reduction ActualSource Foundations.Games
open Integration
noncomputable section
theorem parameters_of_inverse (hinverse : Inverse.Shortcode.InversePrinciple)
(p : ℚ) (hp : 0 < p) : Nonempty (Parameters p) := by
obtain ⟨rstar, hrstar, hgadget⟩ := Gadget.exists_split_gadget p hp
obtain ⟨P⟩ := MatrixGapChoice.exists_inverse_parameters hinverse rstar
obtain ⟨d, g, hstable, hkernel⟩ := hgadget P.s P.s_large
let en := NoiseEnumeration.ofFintype g
let T := NoiseTables.Table.ofEnumeration g en
let gT := T.gadget g.f g.equivariant
have hstableT : gT.stabilityError ≤ p := by
exact (NoiseTables.Table.stabilityError_ofEnumeration g en).trans_le hstable
have hkernelT : ∀ (Q : Type) [AddCommGroup Q] [Module F2 Q]
(L : Ambient P.s d →ₗ[F2] Q),
rstar ≤ Module.finrank F2 (L.comp (alphabetEmbedding P.s d)).range →
Integration.SplitGadget.kernelProbability gT L ≤ 7 / 8 := by
intro Q _ _ L hL
exact (NoiseTables.Table.kernelProbability_ofEnumeration g en L).trans_le (hkernel Q L hL)
obtain ⟨m, hm, hlarge, hβ, hβ', htv, hrate⟩ := MatrixGapChoice.exists_cube P d
refine ⟨{
k := m ^ 3
s := P.s
d := d
k_pos := Nat.succ_le_iff.mpr (pow_pos hm _)
s_pos := hrstar.trans P.s_large
T := T
f := g.f
equivariant := g.equivariant
stability := hstableT
sound := ?_ }⟩
intro S hdistinct hopt
apply address_value_le_of_semantic_acceptance S (m ^ 3) T g.f g.equivariant
intro labeling
apply le_of_lt
by_contra hnot
have haccept : (99 : ℚ) / 100 ≤ TableKeysGame.acceptanceProbability S (m ^ 3) gT labeling :=
le_of_not_gt hnot
have hgood := P.advice (m ^ 3) hlarge S d gT labeling
(fun L hL => hkernelT (ActualHomogeneous.E (m ^ 3)) L hL) haccept
exact ActualMatrixContradiction.contradiction S labeling P.α
(ActualGoodRows.goodRowMass rstar P.s P.r) (1 / (m : ℝ) ^ 2)
P.positive P.trivial_small hβ hβ' hgood htv hrate hdistinct hopt
end
end UniqueGamesTheorem.Decoder.MatrixGap
end
section
/-! The fixed finite matrix reduction obtained from the latent
gadget, dimension-uniform shortcode inverse, sparse full-table law and private
local decoder. The construction precedes the parity input and its accuracy.
No inverse, noise, spectral, or decoder certificate is supplied as a premise. -/
namespace UniqueGamesTheorem.Decoder.MatrixGap
theorem exists_parameters (p : ℚ) (hp : 0 < p) : Nonempty (Parameters p) :=
parameters_of_inverse Inverse.ShortcodeTheorem.inversePrinciple p hp
end UniqueGamesTheorem.Decoder.MatrixGap
end
end OAI
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