| import Mathlib |
| import OAI.Computability.MaxCut.Encoding.ActualAdviceSampling |
| import OAI.Computability.MaxCut.Games.StageData |
|
|
| namespace OAI |
|
|
| /-! The actual matrix reduction assembled from the gadget and decoder. |
| The dimension-uniform shortcode inverse is the sole explicit mathematical |
| input here. This intermediate theorem is not an unconditional endpoint until |
| that input is supplied by the independently proved KMS foundation. -/ |
|
|
| namespace MaxCutGames.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 MaxCutGames.Decoder.MatrixGap |
|
|
| end OAI |
|
|
