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Exact pure-flavor tester room and a law-uniform Gram mass floor

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

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    Natural Language Statement

    Lemma

    The paper's pure flavor retains exactly k*(2k+1-|T0|)*r tester pairs. Prescribing their affine coefficients gives a positive Gram mass floor uniform over tags and excluded sets and independent of the free dimension.

    Concept map
    48 concepts; 1 descendant hidden
    100%
    Exact-image bounds for independent affinecolumnsIndependent nominal affine slices from thefree blockConditioned affine-slice variance under theactual raw lawPoint atoms and finite flavor distributionsWalsh operator bounds with explicit bilinearrankConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formActual conditioned affine coefficientdependence boundsExact finite conditioning of separated FouriertestsExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsQuantitative comparison after cross-batchGram conditioningIndependent characters of actual mutualGram entriesCut profiles and the constant kernelFinite empirical variance from atwo-coefficient comparisonEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionFinite linear images and their uniform-lawdensity boundsFinite independent sampling and vertexexception tailsUniform positive Gram mass from boundedtester prescriptionsAffine-slice comparison for every actual atomflavorFinite scalar agreement from binarycharacter boundsAmbient symmetries and frame marginalsUniform tuple images on prescribed GramorbitsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesActual reference-batch laws for mutual GramtestsTwo-batch comparison across independentcomponent groupsThe binary hole relationExact joining of two prescribed Gram orbitsJoint-injectivity loss for two actual referencebatchesBoolean point moments with restricted basecoordinatesSquared restriction cost for independent uniteventsExact pure-flavor tester room and alaw-uniform Gram mass floorWalsh bounds for independent image lawsand separated phasesBounded tests under the actual two-batchGram lawActual raw frame observations with uniformtwo-batch decayRaw matrix frames and their tensorrealizationActual grouped raw observations witharbitrary common testsThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsImage caps inside original retained cellsBit rank and Walsh bounds for globalvector-slot pairingsMajority intersections in the cyclic taggeometryAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 7 statements. Each proof establishes one of them relative to its assumptions.

    1 allowed_drivers_card proven

    2 pure_gram_cell_mass proven

    3 pure_pairs_card proven

    4 slots_card proven

    5 uniform_floor_bound proven

    6 uniform_pure_gram_mass proven

    7 uniformFloor_pos proven

    Lean source view on GitHub

    1import Lax342547.FixedSelectorGram
    2
    3/-!
    4---
    5title: Exact pure-flavor tester room and a law-uniform Gram mass floor
    6type: lemma
    7---
    8The paper's pure flavor retains exactly k*(2k+1-|T0|)*r tester pairs.
    9Prescribing their affine coefficients gives a positive Gram mass floor
    10uniform over tags and excluded sets and independent of the free dimension.
    11-/
    12
    13namespace Lax342547.PureTesterRoom
    14noncomputable section
    15open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.Atoms
    16open Lax342547.FixedSelectorGram Lax342547.FlavorAffineSlices Lax342547.AffineSliceIndependence
    17open Lax342547.ConcreteGeometry Lax342547.RetainedImages Lax342547.RealCellLaws
    18open scoped BigOperators ENNReal
    19set_option backward.isDefEq.respectTransparency false
    20
    21
    22def pureOutside {k : ℕ} (T₀ : Finset (Tag k)) : Flavor k := by
    23 classical
    24 exact .pure (Finset.univ.filter (fun p : Tag k × Tag k => p.2 ∉ T₀))
    25
    26def retainedBlocksEquiv {k : ℕ} (l : Tag k) (T₀ : Finset (Tag k)) :
    27 {q : TesterBlock k // blockRetained l (pureOutside T₀) q} ≃
    28 ({d : Tag k // l ∈ interval k d} × {t : Tag k // t ∉ T₀}) where
    29 toFun q := by
    30 rcases q with ⟨q,hq⟩
    31 cases q with
    32 | none => simp [blockRetained,pureOutside] at hq
    33 | some q =>
    34 have hh : l ∈ interval k q.1 ∧ q.2 ∉ T₀ := by simpa [blockRetained,pureOutside] using hq
    35 exact (⟨q.1,hh.1⟩,⟨q.2,hh.2⟩)
    36 invFun p := ⟨some (p.1.val,p.2.val),by simpa [blockRetained,pureOutside] using And.intro p.1.property p.2.property⟩
    37 left_inv q := by
    38 rcases q with ⟨q,hq⟩
    39 cases q with
    40 | none => simp [blockRetained,pureOutside] at hq
    41 | some q => rfl
    42 right_inv p := by rfl
    43
    44def uniformFloor (k r : ℕ) (D : Type) [Fintype D] : ℝ :=
    45 1/(2 : ℝ)^(2*k*(2*k+1)*r*Fintype.card (Option D))
    46
    47axiom allowed_drivers_card (k : ℕ) (l : Tag k) :
    48 by classical exact Fintype.card {d : Tag k // l ∈ interval k d} = k
    49
    50axiom pure_pairs_card {k r : ℕ} (l : Tag k) (T₀ : Finset (Tag k)) :
    51 Fintype.card (Pairs l (pureOutside T₀) r) = k*(2*k+1-T₀.card)*r
    52
    53axiom pure_gram_cell_mass {k n b degree r : ℕ} {D : Type} [Fintype D]
    54 (hr : 2*r ≤ n) (hd : 1 ≤ degree)
    55 (M : Fin b → Matrix (Fin n) (Fin n) Binary) (s : Fin b → Binary)
    56 (l : Tag k) (T₀ : Finset (Tag k))
    57 (hroom : Fintype.card (Option D) ≤ k*(2*k+1-T₀.card)*r)
    58 (G : Matrix (Option D) (Option D) Binary) :
    59 massFloor l (pureOutside T₀) r D ≤ cellMass
    60 (weights (PMF.uniformOfFintype (Coefficients n l (pureOutside T₀) D)))
    61 (fun V => (pointColumns s (value V)).transpose*selfGram hr hd M*pointColumns s (value V) = G)
    62
    63axiom slots_card {k r : ℕ} (l : Tag k) (f : Flavor k) :
    64 Fintype.card (Slots l f r) = 2*Fintype.card (Pairs l f r)
    65
    66axiom uniformFloor_pos (k r : ℕ) (D : Type) [Fintype D] :
    67 0 < uniformFloor k r D
    68
    69axiom uniform_floor_bound {k r : ℕ} (l : Tag k) (T₀ : Finset (Tag k))
    70 (D : Type) [Fintype D] : uniformFloor k r D ≤ massFloor l (pureOutside T₀) r D
    71
    72axiom uniform_pure_gram_mass {k n b degree r : ℕ} {D : Type} [Fintype D]
    73 (hr : 2*r ≤ n) (hd : 1 ≤ degree)
    74 (M : Fin b → Matrix (Fin n) (Fin n) Binary) (s : Fin b → Binary)
    75 (l : Tag k) (T₀ : Finset (Tag k))
    76 (hroom : Fintype.card (Option D) ≤ k*(2*k+1-T₀.card)*r)
    77 (G : Matrix (Option D) (Option D) Binary) :
    78 uniformFloor k r D ≤ cellMass
    79 (weights (PMF.uniformOfFintype (Coefficients n l (pureOutside T₀) D)))
    80 (fun V => (pointColumns s (value V)).transpose*selfGram hr hd M*pointColumns s (value V) = G)
    81
    82end
    83end Lax342547.PureTesterRoom
    84
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