import Mathlib import OAI.Computability.MaxCut.Encoding.MatrixCoordinates import OAI.Computability.MaxCut.Games.RowErasureConcentration namespace OAI namespace MaxCutGames.Decoder.ActualGoodRows open Integration.BinaryLinear Reduction ActualSource open Inverse Inverse.RowErasure open Inverse.Shortcode (Mat HasAffineSlice) open scoped BigOperators Classical noncomputable section local instance homFintype {D F : Type*} [AddCommGroup D] [Module F2 D] [AddCommGroup F] [Module F2 F] [Fintype D] [Fintype F] : Fintype (D →ₗ[F2] F) := Fintype.ofInjective (fun M : D →ₗ[F2] F => (M : D → F)) DFunLike.coe_injective def table (S : Source) (k s d : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (U : ActualGame.Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) (M : Mat s (1 + 2 * k)) : Vector s := TableKeysGame.unfolded S k s d labeling (U, ((MatrixCoordinates.mapEquiv k s).symm M).prod T) theorem table_acceptance (S : Source) (k s d : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (U : ActualGame.Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) : Shortcode.equalityAcceptance (table S k s d labeling U T) = ActualSpectral.fiberAcceptance S k s d labeling U T := by unfold table rw [← MatrixCoordinates.shortcodeAcceptance_eq k s (fun M : ActualHomogeneous.E k →ₗ[F2] Vector s => TableKeysGame.unfolded S k s d labeling (U, M.prod T))] unfold ActualSpectral.fiberAcceptance SurrogateFibers.shortcodeAcceptance apply Finset.expect_congr (by ext M; simp only [Finset.mem_univ]) intro M _ apply Finset.expect_congr (by ext a; simp only [Finset.mem_univ]) intro a _ apply Finset.expect_congr (by ext l; simp only [Finset.mem_univ]) intro l _ rfl /-- The row-advice probability in actual matrix coordinates. In particular, dependent rows of `A` are sampled with their full multiplicity. -/ def adviceMass (S : Source) (k s d r : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α : ℝ) : ℝ := 𝔼 U : ActualGame.Question S k, 𝔼 T : ActualHomogeneous.E k →ₗ[F2] Vector d, RowErasure.adviceMass ((RowErasureMatrix.family s (1 + 2 * k) r).goodAt (table S k s d labeling U T) α) def goodRowMass (rstar s r : ℕ) : ℝ := 10 * ActualSpectral.eta rstar ^ 2 / (2 : ℝ) ^ (s * r) theorem goodRowMass_pos (rstar s r : ℕ) : 0 < goodRowMass rstar s r := by unfold goodRowMass have h := ActualSpectral.eta_pos rstar positivity theorem exists_actual_advice_threshold (rstar : ℕ) (α : ℝ) (hα : 0 < α) (hαone : α ≤ 1) (s r : ℕ) (hr : r < s) (halphabet : 1 / (2 : ℝ) ^ s ≤ α / 8) : ∃ k₀ : ℕ, ∀ k : ℕ, k₀ ≤ k → (∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ Shortcode.equalityAcceptance f → HasAffineSlice f α r) → ∀ (S : Source) (d : ℕ) (g : SplitGadget s d) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)), (∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k, rstar ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range → Integration.SplitGadget.kernelProbability g L ≤ 7 / 8) → (99 : ℚ) / 100 ≤ TableKeysGame.acceptanceProbability S k g labeling → goodRowMass rstar s r ≤ adviceMass S k s d r labeling α := by obtain ⟨m₀, hm₀⟩ := RowErasureMatrix.exists_contextual_advice_threshold (ActualSpectral.eta rstar) α (ActualSpectral.eta_pos rstar) hα hαone s r hr halphabet refine ⟨m₀, ?_⟩ intro k hk hinverse S d g labeling hkernel haccept let Q := ActualGame.Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d) let f : Q → Mat s (1 + 2 * k) → Vector s := fun q => table S k s d labeling q.1 q.2 have hcontext : 10 * ActualSpectral.eta rstar ≤ uniformMass (fun q : Q => 4 * ActualSpectral.eta rstar ≤ Shortcode.equalityAcceptance (f q)) := by have h := ActualSpectral.actual_good_fiber_mass S k g labeling rstar hkernel haccept dsimp only [Q, f] unfold uniformMass rw [ActualSpectral.expect_prod] simpa only [indicator, table_acceptance] using h have h := hm₀ (1 + 2 * k) (by omega) hinverse Q f hcontext dsimp only [Q, f] at h rw [ActualSpectral.expect_prod] at h simpa only [goodRowMass, adviceMass] using h end end MaxCutGames.Decoder.ActualGoodRows namespace MaxCutGames.Decoder.ConstantSelection open MaxCutGames.Foundations.Information open SparseLaw noncomputable section /-- Probability of actual visible data carrying a transferred witness. -/ def visibleMass (α g₀ : ℝ) (ell r : ℕ) : ℝ := α * g₀ / (8 * (2 : ℝ) ^ (ell * r)) /-- Independent private decoding success on a transferred witness. -/ def conditionalMass (α : ℝ) (ell r : ℕ) : ℝ := (α / 8) ^ 2 / (2 : ℝ) ^ (ell * r) / (2 : ℝ) ^ r def decodingMass (α g₀ : ℝ) (ell r : ℕ) : ℝ := visibleMass α g₀ ell r * conditionalMass α ell r /-- The whole-table TV budget includes the coefficient of both compared events in the witness-margin inequality. -/ def variationBudget (α g₀ : ℝ) (ell r : ℕ) : ℝ := visibleMass α g₀ ell r / (1 + α / 4) theorem visibleMass_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < visibleMass α g₀ ell r := by unfold visibleMass positivity theorem conditionalMass_pos {α : ℝ} (hα : 0 < α) (ell r : ℕ) : 0 < conditionalMass α ell r := by unfold conditionalMass positivity theorem decodingMass_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < decodingMass α g₀ ell r := mul_pos (visibleMass_pos hα hg₀ ell r) (conditionalMass_pos hα ell r) theorem variationBudget_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < variationBudget α g₀ ell r := by exact div_pos (visibleMass_pos hα hg₀ ell r) (by positivity) theorem variationBudget_suffices {α g₀ ε : ℝ} (hα : 0 < α) (ell r : ℕ) (hε : ε ≤ variationBudget α g₀ ell r) : (1 + α / 4) * ε ≤ α * g₀ * (1 / (2 : ℝ) ^ (ell * r)) / 8 := by have hd : 0 < 1 + α / 4 := by positivity have h := (le_div_iff₀ hd).mp hε calc (1 + α / 4) * ε = ε * (1 + α / 4) := mul_comm _ _ _ ≤ visibleMass α g₀ ell r := h _ = α * g₀ * (1 / (2 : ℝ) ^ (ell * r)) / 8 := by unfold visibleMass ring /-- Exponential clean-mask decay persists after increasing the cube root. -/ theorem cube_exponential_antitone {m n : ℕ} (hmn : m ≤ n) {N : ℝ} (hN : 0 < N) : Real.exp (-(n : ℝ) / (3600 * N)) ≤ Real.exp (-(m : ℝ) / (3600 * N)) := by apply Real.exp_le_exp.mpr apply div_le_div_of_nonneg_right _ (by positivity) exact neg_le_neg (by exact_mod_cast hmn) /-- One tuple length meets the sparse-law and repetition requirements, while exceeding any independent inverse/erasure dimension threshold. -/ theorem exists_simultaneous_cube (V : Type*) [Fintype V] [Zero V] [DecidableEq V] {N ε γ : ℝ} (hN : 1 ≤ N) (hε : 0 < ε) (hγ : 0 < γ) (lower : ℕ) : ∃ m : ℕ, lower ≤ m ∧ 0 < m ∧ totalVariation (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3)) (uniformWeights (Fin (m ^ 3) → V × V)) ≤ ε ∧ (1 - ((1 / (m : ℝ) ^ 2) / N) / 3600) ^ (m ^ 3) < γ := by obtain ⟨m₀, hm₀, hexp⟩ := Clean.exists_cube_exponential_lt (show 0 < N by linarith) hγ obtain ⟨m, hlarge, hm, htv⟩ := exists_cubeSlope_totalVariation_le V ε hε (max lower m₀) have hbase : m₀ ≤ m := (le_max_right _ _).trans hlarge refine ⟨m, (le_max_left _ _).trans hlarge, hm, htv, ?_⟩ have hdecay := (Clean.cube_mask_rate_le_exp (Nat.succ_le_iff.mpr hm) hN).trans (cube_exponential_antitone hbase (by linarith : 0 < N)) have hbeta : (1 / (m : ℝ)) ^ 2 = 1 / (m : ℝ) ^ 2 := by ring rw [hbeta] at hdecay exact hdecay.trans_lt hexp /-- All scalar requirements for the final decoding contradiction can be met after the fixed alphabet and inverse parameters have been chosen. -/ theorem exists_matrix_cube (V : Type*) [Fintype V] [Zero V] [DecidableEq V] {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r dimW lower : ℕ) : ∃ m : ℕ, lower ≤ m ∧ 0 < m ∧ totalVariation (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3)) (uniformWeights (Fin (m ^ 3) → V × V)) ≤ variationBudget α g₀ ell r ∧ (1 - ((1 / (m : ℝ) ^ 2) / (2 : ℝ) ^ (dimW + r)) / 3600) ^ (m ^ 3) < decodingMass α g₀ ell r := by exact exists_simultaneous_cube V (one_le_pow₀ (by norm_num)) (variationBudget_pos hα hg₀ ell r) (decodingMass_pos hα hg₀ ell r) lower end end MaxCutGames.Decoder.ConstantSelection namespace MaxCutGames.Decoder.MatrixGapChoice open Integration.BinaryLinear Reduction ActualSource open Inverse.Shortcode (Mat HasAffineSlice InversePrinciple equalityAcceptance) noncomputable section /-- Fixed dimensions, density and sparse mask for one logical alphabet. The remaining gadget codimension may be arbitrary when choosing the cube. -/ structure InverseParameters (rstar : ℕ) where α : ℝ positive : 0 < α bounded : α ≤ 1 r : ℕ s : ℕ s_large : rstar ≤ s r_small : r < s trivial_small : 1 / (2 : ℝ) ^ (s - r) < α / 8 tupleThreshold : ℕ inverse : ∀ k : ℕ, tupleThreshold ≤ k → ∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ equalityAcceptance f → HasAffineSlice f α r advice : ∀ k : ℕ, tupleThreshold ≤ k → ∀ (S : Source) (d : ℕ) (g : SplitGadget s d) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)), (∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k, rstar ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range → Integration.SplitGadget.kernelProbability g L ≤ 7 / 8) → (99 : ℚ) / 100 ≤ TableKeysGame.acceptanceProbability S k g labeling → ActualGoodRows.goodRowMass rstar s r ≤ ActualGoodRows.adviceMass S k s d r labeling α theorem eta_lt_one (rstar : ℕ) : ActualSpectral.eta rstar < 1 := by have hp : (1 : ℝ) ≤ 2 ^ rstar := one_le_pow₀ (by norm_num) have hi : 1 / (2 : ℝ) ^ rstar ≤ 1 := by exact (div_le_one (by positivity)).mpr hp unfold ActualSpectral.eta rw [← one_div] linarith theorem exists_inverse_parameters (hinverse : InversePrinciple) (rstar : ℕ) : Nonempty (InverseParameters rstar) := by obtain ⟨α, hα, hαone, r, _hr, s₀, hinv⟩ := hinverse (ActualSpectral.eta rstar) (ActualSpectral.eta_pos rstar) (eta_lt_one rstar) obtain ⟨s, hs, hrs, hsmall, _htiny⟩ := DimensionSelection.exists_logical_dimension hα (ActualSpectral.eta_pos rstar) r (max rstar s₀) have hsstar : rstar ≤ s := (le_max_left _ _).trans hs have hs₀ : s₀ ≤ s := (le_max_right _ _).trans hs obtain ⟨m₀, hm₀⟩ := hinv s hs₀ have halphabet : 1 / (2 : ℝ) ^ s ≤ α / 8 := by have hp : (2 : ℝ) ^ (s - r) ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) (Nat.sub_le s r) exact (one_div_le_one_div_of_le (by positivity) hp).trans hsmall.le obtain ⟨k₀, hk₀⟩ := ActualGoodRows.exists_actual_advice_threshold rstar α hα hαone s r hrs halphabet let threshold := max m₀ k₀ have hi (k : ℕ) (hk : threshold ≤ k) : ∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ equalityAcceptance f → HasAffineSlice f α r := by apply hm₀ (1 + 2 * k) have hm : m₀ ≤ k := (le_max_left _ _).trans hk omega refine ⟨{ α := α positive := hα bounded := hαone r := r s := s s_large := hsstar r_small := hrs trivial_small := hsmall tupleThreshold := threshold inverse := hi advice := ?_ }⟩ intro k hk S d g labeling hkernel haccept exact hk₀ k ((le_max_right _ _).trans hk) (hi k hk) S d g labeling hkernel haccept /-- After all fixed dimensions are known, the cube simultaneously exceeds the inverse threshold, reduces full-table variation, and beats the decoder mass. -/ theorem exists_cube {rstar : ℕ} (P : InverseParameters rstar) (d : ℕ) : ∃ m : ℕ, 0 < m ∧ P.tupleThreshold ≤ m ^ 3 ∧ 0 ≤ 1 / (m : ℝ) ^ 2 ∧ 1 / (m : ℝ) ^ 2 ≤ 1 ∧ Foundations.Information.totalVariation (SparseLaw.independentWeights (SparseLaw.mixture (1 / (m : ℝ) ^ 2) (SparseLaw.singletonPairWeights (Ambient P.s d))) (m ^ 3)) (SparseLaw.uniformWeights (Fin (m ^ 3) → Ambient P.s d × Ambient P.s d)) ≤ ConstantSelection.variationBudget P.α (ActualGoodRows.goodRowMass rstar P.s P.r) P.s P.r ∧ (1 - ((1 / (m : ℝ) ^ 2) / (2 : ℝ) ^ (d + P.r)) / 3600) ^ (m ^ 3) < ConstantSelection.decodingMass P.α (ActualGoodRows.goodRowMass rstar P.s P.r) P.s P.r := by obtain ⟨m, hlarge, hm, htv, hrate⟩ := ConstantSelection.exists_matrix_cube (Ambient P.s d) P.positive (ActualGoodRows.goodRowMass_pos rstar P.s P.r) P.s P.r d P.tupleThreshold have hmreal : (1 : ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_iff.mpr hm) have hmsq : (1 : ℝ) ≤ (m : ℝ) ^ 2 := one_le_pow₀ hmreal refine ⟨m, hm, hlarge.trans (Nat.le_self_pow (by decide : (3 : ℕ) ≠ 0) m), by positivity, ?_, htv, hrate⟩ exact (div_le_one (by positivity)).mpr hmsq end end MaxCutGames.Decoder.MatrixGapChoice end OAI