Datasets:

roseking
/

openAI-math-dataset

import Mathlib.Algebra.BigOperators.Field
import OAI.Computability.UniqueGames.Decoding.AdviceFibersLemmas
import OAI.Computability.UniqueGames.Decoding.DimensionSelection
import OAI.Computability.UniqueGames.Decoding.MatrixCoordinates
import OAI.Computability.UniqueGames.Decoding.PrivateStrategyLemmas
import OAI.Computability.UniqueGames.Decoding.SparseLawLemmas
import OAI.Computability.UniqueGames.Decoding.TableKeysGame
import OAI.Computability.UniqueGames.Foundations.SamplingLemmas
import OAI.Computability.UniqueGames.Inverse.KMSAnalyticHybridEnergyLemmas
import OAI.Computability.UniqueGames.Inverse.RowErasureMatrixLemmas
import OAI.Computability.UniqueGames.Reduction.ActualSourceLemmas

namespace OAI

section

noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Integration.BinaryLinear
open UniqueGamesTheorem.Reduction UniqueGamesTheorem.Reduction.ActualSource
open UniqueGamesTheorem.Fourier.MatrixNoise

namespace UniqueGamesTheorem.Decoder.ActualSpectral

open TableKeysGame PositiveMultiplier NoiseSampler SurrogateFibers

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

theorem expect_prod {A B : Type*} [Fintype A] [Fintype B] (f : A × B → ℝ) :
    (𝔼 p, f p) = 𝔼 a, 𝔼 b, f (a, b) := by
  simpa only [Finset.univ_product_univ] using
    (Finset.expect_product Finset.univ Finset.univ f)

theorem expect_eq_uniform_sum {I : Type*} [Fintype I] (f : I → ℝ) :
    (𝔼 i, f i) = ∑ i, (Fintype.card I : ℝ)⁻¹ * f i := by
  rw [Fintype.expect_eq_sum_div_card, ← Finset.mul_sum]
  ring

/-- The finite noise law's multiplier is exactly the rational kernel event
of the same gadget, cast to the real numbers. -/
theorem eigenvalue_eq_kernelProbability {s d : Nat} (g : SplitGadget s d)
    {P : Type} [AddCommGroup P] [Module F2 P] (S : Ambient s d →ₗ[F2] P) :
    linearNoiseEigenvalue (samplerLaw g.noise) S =
      (Integration.SplitGadget.kernelProbability g S : ℝ) := by
  rw [← sample_kernel_eq_eigenvalue]
  unfold Integration.SplitGadget.kernelProbability
  rw [rat_expect_cast]
  simp only [apply_ite, Rat.cast_one, Rat.cast_zero]

/-- The four independent samples of the actual single-orbit test. -/
theorem acceptance_nested (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) :
    (acceptanceProbability S k g label : ℝ) =
      𝔼 U : Question S k, 𝔼 X : Map k s d, 𝔼 i : g.NoiseIndex, 𝔼 l : Dual k,
        if unfolded S k s d label (U, X) =
          unfolded S k s d label (U, X + l.smulRight (g.noise i))
        then (1 : ℝ) else 0 := by
  rw [acceptanceProbability_eq_test, rat_expect_cast]
  simp only [apply_ite, Rat.cast_one, Rat.cast_zero]
  simp only [expect_prod, leftQuery, rightQuery, ActualGame.leftQuery, ActualGame.rightQuery]
  apply Finset.expect_congr rfl
  intro U _
  apply Finset.expect_congr rfl
  intro X _
  apply Finset.expect_congr rfl
  intro i _
  apply Finset.expect_congr rfl
  intro l _
  by_cases h : unfolded S k s d label (U, X) =
      unfolded S k s d label (U, X + l.smulRight (g.noise i)) <;> simp [h]

/-- No auxiliary side labeling is introduced: the same restored table occurs
on both sides of the actual equality test. -/
theorem acceptance_eq_equality_mean (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) :
    (acceptanceProbability S k g label : ℝ) =
      𝔼 U : Question S k,
        equalityAcceptance (samplerLaw g.noise) (fun X : Map k s d =>
          unfolded S k s d label (U, X)) := by
  rw [acceptance_nested]
  apply Finset.expect_congr rfl
  intro U _
  unfold equalityAcceptance
  apply Finset.expect_congr rfl
  intro X _
  rw [← sample_expect_eq]
  apply Finset.expect_congr rfl
  intro i _
  apply Finset.expect_congr rfl
  intro l _
  split_ifs <;> rfl

def fiberAcceptance (S : Source) (k s d : Nat)
    (label : Fin (vertexCount S k s d) → Fin (2 ^ s))
    (U : Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) : ℝ :=
  shortcodeAcceptance (fun M : ActualHomogeneous.E k →ₗ[F2] Alphabet s =>
    unfolded S k s d label (U, M.prod T))

theorem fiberAcceptance_le_one (S : Source) (k s d : Nat)
    (label : Fin (vertexCount S k s d) → Fin (2 ^ s))
    (U : Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) :
    fiberAcceptance S k s d label U T ≤ 1 := by
  unfold fiberAcceptance shortcodeAcceptance
  apply Finset.expect_le Finset.univ_nonempty
  intro M _
  apply Finset.expect_le Finset.univ_nonempty
  intro a _
  apply Finset.expect_le Finset.univ_nonempty
  intro l _
  split_ifs <;> norm_num

theorem weighted_fiber_mean_lower (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (label : Fin (vertexCount S k s d) → Fin (2 ^ s))
    (μ : Question S k → ℝ) (r : Nat)
    (hμ : ∀ U, 0 ≤ μ U) (hμsum : ∑ U, μ U = 1)
    (hkernel : ∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k,
      r ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range →
      Integration.SplitGadget.kernelProbability g L ≤ 7 / 8)
    (haccept : (99 : ℝ) / 100 ≤ ∑ U, μ U *
      equalityAcceptance (samplerLaw g.noise)
        (fun X : Map k s d => unfolded S k s d label (U, X))) :
    (9 : ℝ) / 10 * ((2 : ℝ) ^ r)⁻¹ ≤
      ∑ U, μ U * (𝔼 T, fiberAcceptance S k s d label U T) := by
  have h := matrix_spectral_comparison μ (samplerLaw g.noise) (alphabetEmbedding s d)
    (fun U (X : Map k s d) => unfolded S k s d label (U, X)) r
    hμ hμsum (samplerLaw_nonneg g.noise) (samplerLaw_normalized g.noise)
    (fun L hL => by
      rw [eigenvalue_eq_kernelProbability]
      have hc := (Rat.cast_le (K := ℝ)).mpr (hkernel L hL)
      simpa only [Rat.cast_div, Rat.cast_ofNat] using hc) haccept
  simpa only [alphabetEmbedding, surrogate_acceptance_eq_fiber_average,
    fiberAcceptance] using h

theorem actual_fiber_mean_lower (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) (r : Nat)
    (hkernel : ∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k,
      r ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range →
      Integration.SplitGadget.kernelProbability g L ≤ 7 / 8)
    (haccept : (99 : ℚ) / 100 ≤ acceptanceProbability S k g label) :
    (9 : ℝ) / 10 * ((2 : ℝ) ^ r)⁻¹ ≤
      𝔼 U : Question S k, 𝔼 T, fiberAcceptance S k s d label U T := by
  rw [expect_eq_uniform_sum]
  apply weighted_fiber_mean_lower S k g label
    (fun _ => (Fintype.card (Question S k) : ℝ)⁻¹) r
  · intro U
    positivity
  · rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
    apply mul_inv_cancel₀
    exact_mod_cast Fintype.card_ne_zero (α := Question S k)
  · exact hkernel
  · have hc := (Rat.cast_le (K := ℝ)).mpr haccept
    rw [acceptance_eq_equality_mean, expect_eq_uniform_sum] at hc
    simpa only [Rat.cast_div, Rat.cast_ofNat] using hc

def eta (r : Nat) : ℝ := ((2 : ℝ) ^ r)⁻¹ / 16

theorem eta_pos (r : Nat) : 0 < eta r := by unfold eta; positivity

/-- At least `10 η` of the actual `(U,T)` pairs have shortcode acceptance
at least `4 η`, exactly the input needed by row erasure. -/
theorem actual_good_fiber_mass (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) (r : Nat)
    (hkernel : ∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k,
      r ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range →
      Integration.SplitGadget.kernelProbability g L ≤ 7 / 8)
    (haccept : (99 : ℚ) / 100 ≤ acceptanceProbability S k g label) :
    10 * eta r ≤ 𝔼 U : Question S k,
      𝔼 T : ActualHomogeneous.E k →ₗ[F2] Vector d,
        if 4 * eta r ≤ fiberAcceptance S k s d label U T then (1 : ℝ) else 0 := by
  let I := Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d)
  let μ : I → ℝ := fun _ => (Fintype.card I : ℝ)⁻¹
  let p : I → ℝ := fun a => fiberAcceptance S k s d label a.1 a.2
  have hmean := actual_fiber_mean_lower S k g label r hkernel haccept
  have hmean' : (72 : ℝ) / 5 * eta r ≤ ∑ a, μ a * p a := by
    rw [← expect_eq_uniform_sum, expect_prod]
    unfold eta
    nlinarith
  have h := good_fiber_mass μ p (eta r) (fun _ => by positivity)
    (by simp [μ, Fintype.card_ne_zero]) (eta_pos r).le
    (fun a => fiberAcceptance_le_one S k s d label a.1 a.2) hmean'
  change 10 * eta r ≤
    ∑ a : Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d),
      (Fintype.card (Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d)) : ℝ)⁻¹ *
        (if 4 * eta r ≤ fiberAcceptance S k s d label a.1 a.2 then 1 else 0) at h
  rw [← expect_eq_uniform_sum, expect_prod] at h
  exact h

end UniqueGamesTheorem.Decoder.ActualSpectral

end

end

section

namespace UniqueGamesTheorem.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 UniqueGamesTheorem.Decoder.ActualGoodRows

end

section

/-! The two full-table events used in the projected-witness transfer. Their
sample space contains actual occurrence tuples and complete affine tables,
but no projection choice. The selected witness depends on row advice only.
-/

namespace UniqueGamesTheorem.Decoder.ActualEvents

open Integration.BinaryLinear Reduction ActualSource Foundations.Games
open Inverse Inverse.RowErasure
open Inverse.Shortcode (Mat)
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

abbrev Context (S : Source) (k d : ℕ) :=
  ActualGame.Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d)

abbrev Sample (S : Source) (k s d r : ℕ) :=
  Context S k d × (RowErasureDescriptions.RowMap s r × Mat s (1 + 2 * k))

abbrev FullTableSample (S : Source) (k s d r : ℕ) :=
  (ActualGame.Question S k × (Alphabet s →ₗ[F2] Vector r)) × ActualGame.Map k s d

/-- A bijection merely rearranges the full sampled table and changes its
matrix coordinates. It never appends the hidden projection to this law. -/
def fullTableEquiv (S : Source) (k s d r : ℕ) :
    FullTableSample S k s d r ≃ Sample S k s d r where
  toFun p := ((p.1.1, (LinearMap.snd F2 (Alphabet s) (Vector d)).comp p.2),
    (LinearMap.toMatrix' p.1.2,
      MatrixCoordinates.mapEquiv k s ((LinearMap.fst F2 (Alphabet s) (Vector d)).comp p.2)))
  invFun q := ((q.1.1, Matrix.toLin' q.2.1),
    ((MatrixCoordinates.mapEquiv k s).symm q.2.2).prod q.1.2)
  left_inv p := by
    rcases p with ⟨⟨U, A⟩, P⟩
    apply Prod.ext
    · simp
    · apply LinearMap.ext
      intro x
      simp only [LinearEquiv.symm_apply_apply, LinearMap.prod_apply]
      rfl
  right_inv q := by
    rcases q with ⟨⟨U, T⟩, A, M⟩
    apply Prod.ext
    · apply Prod.ext rfl
      apply LinearMap.ext
      intro x
      rfl
    · apply Prod.ext
      · exact LinearMap.toMatrix'_toLin' A
      · change MatrixCoordinates.mapEquiv k s
          ((LinearMap.fst F2 (Alphabet s) (Vector d)).comp
            (((MatrixCoordinates.mapEquiv k s).symm M).prod T)) = M
        have hf : (LinearMap.fst F2 (Alphabet s) (Vector d)).comp
            (((MatrixCoordinates.mapEquiv k s).symm M).prod T) =
              (MatrixCoordinates.mapEquiv k s).symm M := by
          apply LinearMap.ext
          intro x
          rfl
        rw [hf, LinearEquiv.apply_symm_apply]

def unrestricted (S : Source) (k s d r : ℕ) : FiniteDistribution (Sample S k s d r) :=
  FiniteDistribution.uniform _

def sliceEvent (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
    (α : ℝ) (p : Sample S k s d r) : Bool :=
  decide ((RowErasureMatrix.family s (1 + 2 * k) r).selectedEvent
    (ActualGoodRows.table S k s d labeling p.1.1 p.1.2) α p.2.1 p.2.2)

def targetHit (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
    (α : ℝ) (p : Sample S k s d r) : Bool :=
  decide ((RowErasureMatrix.family s (1 + 2 * k) r).selectedMatch
    (ActualGoodRows.table S k s d labeling p.1.1 p.1.2) α p.2.1 p.2.2)

theorem targetHit_implies_slice (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s))
    (α : ℝ) (p : Sample S k s d r)
    (h : targetHit S k s d r labeling α p = true) :
    sliceEvent S k s d r labeling α p = true := by
  exact of_decide_eq_true h |> fun hm => decide_eq_true
    ((RowErasureMatrix.family s (1 + 2 * k) r).selectedMatch_event _ α _ _ hm)

theorem uniform_probability_expect {Ω : Type*} [Fintype Ω] [Nonempty Ω]
    (p : Ω → Bool) :
    (FiniteDistribution.uniform Ω).probability p =
      𝔼 x : Ω, if p x then (1 : ℝ) else 0 := by
  simp only [FiniteDistribution.probability, FiniteDistribution.uniform,
    Fintype.expect_eq_sum_div_card, Finset.sum_div, ite_div, one_div, zero_div]

theorem unrestricted_slice_mass (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α : ℝ) :
    (unrestricted S k s d r).probability (sliceEvent S k s d r labeling α) =
      𝔼 q : Context S k d,
        RowErasureMatrix.selectedEventMass r
          (ActualGoodRows.table S k s d labeling q.1 q.2) α := by
  unfold unrestricted
  rw [uniform_probability_expect, ActualSpectral.expect_prod]
  apply Finset.expect_congr rfl
  intro q _
  rw [ActualSpectral.expect_prod]
  simp only [sliceEvent, RowErasureMatrix.selectedEventMass, uniformMass, indicator,
    decide_eq_true_eq]

theorem unrestricted_match_mass (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α : ℝ) :
    (unrestricted S k s d r).probability
      (fun p => sliceEvent S k s d r labeling α p && targetHit S k s d r labeling α p) =
      𝔼 q : Context S k d,
        RowErasureMatrix.selectedMatchMass r
          (ActualGoodRows.table S k s d labeling q.1 q.2) α := by
  have he : (fun p => sliceEvent S k s d r labeling α p && targetHit S k s d r labeling α p) =
      targetHit S k s d r labeling α := by
    funext p
    by_cases h : targetHit S k s d r labeling α p = true
    · simp [h, targetHit_implies_slice S k s d r labeling α p h]
    · simp [Bool.eq_false_iff.mpr h]
  rw [he]
  unfold unrestricted
  rw [uniform_probability_expect, ActualSpectral.expect_prod]
  apply Finset.expect_congr rfl
  intro q _
  rw [ActualSpectral.expect_prod]
  simp only [targetHit, RowErasureMatrix.selectedMatchMass, uniformMass, indicator,
    decide_eq_true_eq]

/-- The full-table slice event pays exactly one bounded column-fiber factor. -/
theorem slice_probability_lower (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α g₀ : ℝ)
    (hgood : g₀ ≤ ActualGoodRows.adviceMass S k s d r labeling α) :
    g₀ * (1 / (2 : ℝ) ^ (s * r)) ≤
      (unrestricted S k s d r).probability (sliceEvent S k s d r labeling α) := by
  rw [unrestricted_slice_mass]
  have h := RowErasureMatrix.selectedEventMass_contextual_ge (r := r)
    (fun q : Context S k d => ActualGoodRows.table S k s d labeling q.1 q.2) α g₀
    (by simpa only [ActualGoodRows.adviceMass, ActualSpectral.expect_prod] using hgood)
  simpa only [mul_comm] using h

theorem target_probability_lower (S : Source) (k s d r : ℕ)
    (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α : ℝ) :
    (α / 2) * (unrestricted S k s d r).probability (sliceEvent S k s d r labeling α) ≤
      (unrestricted S k s d r).probability
        (fun p => sliceEvent S k s d r labeling α p && targetHit S k s d r labeling α p) := by
  rw [unrestricted_slice_mass, unrestricted_match_mass]
  exact RowErasureMatrix.selectedMatchMass_contextual_ge (r := r)
    (fun q : Context S k d => ActualGoodRows.table S k s d labeling q.1 q.2) α

end
end UniqueGamesTheorem.Decoder.ActualEvents

end

section

namespace UniqueGamesTheorem.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 UniqueGamesTheorem.Decoder.MatrixGapChoice

end

end OAI