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Prescribed Gram products routed into distinct retained tester pairs

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

proven

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    Natural Language Statement

    Lemma

    Each prescribed Gram product is routed to a distinct retained tester pair. The exact uniform tester restriction law then supplies a positive real conditioning mass without restricting the growing free coefficients.

    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 uniteventsWalsh 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 taggeometryPrescribed Gram products routed intodistinct retained tester pairsAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 product_gram_cell_mass proven

    2 routed_product_gram proven

    3 sum_extended_product proven

    Lean source view on GitHub

    1import Lax342547.FixedSelectorGram
    2
    3/-!
    4---
    5title: Prescribed Gram products routed into distinct retained tester pairs
    6type: lemma
    7---
    8Each prescribed Gram product is routed to a distinct retained tester
    9pair. The exact uniform tester restriction law then supplies a positive
    10real conditioning mass without restricting the growing free coefficients.
    11-/
    12
    13namespace Lax342547.TesterProductRouting
    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 productRouting {k r : ℕ} {l : Tag k} {f : Flavor k} {D I : Type}
    23 (c : I → Pairs l f r) (a b : I → Option D → Binary) :
    24 Matrix (Slots l f r) (Option D) Binary := by
    25 classical
    26 exact fun u j => if u.2.2 then Function.extend c (fun i => b i j) 0 (u.1,u.2.1)
    27 else Function.extend c (fun i => a i j) 0 (u.1,u.2.1)
    28
    29axiom sum_extended_product {I J : Type} [Fintype I] [Fintype J]
    30 (c : I → J) (hc : Function.Injective c) (a b : I → Binary) :
    31 (∑ j,Function.extend c a 0 j*Function.extend c b 0 j) = ∑ i,a i*b i
    32
    33axiom routed_product_gram {k r : ℕ} {l : Tag k} {f : Flavor k} {D I : Type}
    34 [Fintype I] (c : I → Pairs l f r) (hc : Function.Injective c)
    35 (a b : I → Option D → Binary) :
    36 restrictedGram (productRouting c a b) = fun x y => ∑ i,a i x*b i y
    37
    38axiom product_gram_cell_mass {k n b degree r : ℕ} {D I : Type} [Fintype D] [Fintype I]
    39 (hr : 2*r ≤ n) (hd : 1 ≤ degree)
    40 (M : Fin b → Matrix (Fin n) (Fin n) Binary) (s : Fin b → Binary)
    41 (l : Tag k) (f : Flavor k) (c : I → Pairs l f r) (hc : Function.Injective c)
    42 (a b' : I → Option D → Binary) :
    43 massFloor l f r D ≤ cellMass
    44 (weights (PMF.uniformOfFintype (Coefficients n l f D)))
    45 (fun V => (pointColumns s (value V)).transpose*selfGram hr hd M*pointColumns s (value V) =
    46 fun x y => ∑ i,a i x*b' i y)
    47
    48end
    49end Lax342547.TesterProductRouting
    50
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