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Joint-injectivity loss for two actual reference batches

Lax342547.JointGramInjection · concepts/Lax342547/JointGramInjection.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Concatenating both column lists turns the ambient reference experiment into a uniform matrix pair. The separate Gram-orbit caps transfer column failure to the actual product law; Fourier normalization also bounds that loss after conditioning on reciprocal Gram entries.

    Concept map
    19 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 ambient_failure proven

    2 conditioned_reference_failure proven

    3 reference_failure proven

    Lean source view on GitHub

    1import Lax342547.GramOrbitDensity
    2
    3/-!
    4---
    5title: Joint-injectivity loss for two actual reference batches
    6type: lemma
    7---
    8Concatenating both column lists turns the ambient reference experiment into
    9a uniform matrix pair. The separate Gram-orbit caps transfer column failure
    10to the actual product law; Fourier normalization also bounds that loss after
    11conditioning on reciprocal Gram entries.
    12-/
    13
    14namespace Lax342547.JointGramInjection
    15noncomputable section
    16open Lax342547.MomentSpace Lax342547.FrameTuples Lax342547.RealCellLaws
    17open Lax342547.GramOrbitDensity Lax342547.RetainedImages
    18open scoped BigOperators ENNReal
    19
    20variable {I J K L N : Type} [Fintype I] [Fintype J] [Fintype K] [Fintype L] [Fintype N]
    21 [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [DecidableEq N]
    22
    23def ambientEquiv :
    24 ((N × (I ⊕ J) → Binary) × (N × (K ⊕ L) → Binary)) ≃
    25 (Matrix N (I ⊕ K) Binary × Matrix N (J ⊕ L) Binary) where
    26 toFun z := (Matrix.fromCols (batchEquiv.symm z.1).1 (batchEquiv.symm z.2).1,
    27 Matrix.fromCols (batchEquiv.symm z.1).2 (batchEquiv.symm z.2).2)
    28 invFun z := (batchEquiv (z.1.submatrix id Sum.inl,z.2.submatrix id Sum.inl),
    29 batchEquiv (z.1.submatrix id Sum.inr,z.2.submatrix id Sum.inr))
    30 left_inv z := by apply Prod.ext <;> funext (n,s) <;> cases s <;> rfl
    31 right_inv z := by apply Prod.ext <;> ext n (i | k) <;> rfl
    32
    33def Good (z : (N × (I ⊕ J) → Binary) × (N × (K ⊕ L) → Binary)) :=
    34 Function.Injective (ambientEquiv z).1.mulVec ∧ Function.Injective (ambientEquiv z).2.mulVec
    35
    36axiom ambient_failure [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [DecidableEq N] :
    37 (PMF.uniformOfFintype ((N × (I ⊕ J) → Binary) × (N × (K ⊕ L) → Binary))).toOuterMeasure
    38 {z | ¬ Good z} ≤
    39 ((2 : ℝ≥0∞)^(Fintype.card I+Fintype.card K)+(2 : ℝ≥0∞)^(Fintype.card J+Fintype.card L))/
    40 (2 : ℝ≥0∞)^Fintype.card N
    41
    42axiom reference_failure [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L] [DecidableEq N] (G : Matrix I J Binary) (H : Matrix K L Binary)
    43 [Nonempty (Orbit N G)] [Nonempty (Orbit N H)]
    44 (hN₁ : Fintype.card I+Fintype.card J+1 ≤ Fintype.card N)
    45 (hN₂ : Fintype.card K+Fintype.card L+1 ≤ Fintype.card N) :
    46 cellMass (fun z : (N × (I ⊕ J) → Binary) × (N × (K ⊕ L) → Binary) =>
    47 batchLaw (N := N) G z.1*batchLaw (N := N) H z.2) (fun z => ¬ Good z) ≤
    48 ((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    49 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))*
    50 (((2 : ℝ)^(Fintype.card I+Fintype.card K)+(2 : ℝ)^(Fintype.card J+Fintype.card L))/
    51 (2 : ℝ)^Fintype.card N)
    52
    53axiom conditioned_reference_failure {I J K L : Type} [Fintype I] [Fintype J]
    54 [Fintype K] [Fintype L] [DecidableEq I] [DecidableEq J] [DecidableEq K] [DecidableEq L]
    55 (n : ℕ) (G : Matrix I J Binary) (H : Matrix K L Binary)
    56 [Nonempty (Orbit (Fin n) G)] [Nonempty (Orbit (Fin n) H)]
    57 (target : Lax342547.CrossGramBasis.Slots I J K L → Binary)
    58 (hN₁ : Fintype.card I+Fintype.card J+1 ≤ n)
    59 (hN₂ : Fintype.card K+Fintype.card L+1 ≤ n)
    60 (hsmall : ((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    61 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))/Real.sqrt ((2 : ℝ)^n) ≤
    62 1/(2 : ℝ)^(Fintype.card (Lax342547.CrossGramBasis.Slots I J K L)+1)) :
    63 cellMass (fun z : (Fin n × (I ⊕ J) → Binary) × (Fin n × (K ⊕ L) → Binary) =>
    64 batchLaw G z.1*batchLaw H z.2)
    65 (fun z => Lax342547.CrossBatchMixing.crossBits Lax342547.CrossGramBasis.tests n z = target ∧ ¬ Good z)/
    66 Lax342547.FourierTests.patternMass (fun z => batchLaw G z.1*batchLaw H z.2)
    67 (Lax342547.CrossBatchMixing.crossBits Lax342547.CrossGramBasis.tests n) target ≤
    68 (2 : ℝ)^(Fintype.card (Lax342547.CrossGramBasis.Slots I J K L)+1)*
    69 (((2 : ℝ)^(Fintype.card I*Fintype.card J+2))*
    70 ((2 : ℝ)^(Fintype.card K*Fintype.card L+2))*
    71 (((2 : ℝ)^(Fintype.card I+Fintype.card K)+(2 : ℝ)^(Fintype.card J+Fintype.card L))/(2 : ℝ)^n))
    72
    73end
    74end Lax342547.JointGramInjection
    75
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