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Joint retained-cell characters across all components

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

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    Natural Language Statement

    Lemma

    Componentwise quotient projections and rank factors combine into one joint Walsh image. No independence between components or nominal orientations is imposed; rank sums control the complete character.

    Concept map
    99 concepts; 8 descendants hidden
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsSparse unselected primal vectors in barredspaces lie in the pinsBarred response spaces and the actualobstruction rank budgetCompression preserving allowed pointmoments and the actual cut domainThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsBounded allowed derivatives preserving theactual linearized responseAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsCoefficient phases as dot products ofrank-factor imagesJoint retained-cell characters across allcomponentsA bounded-rank pure correction for theactual scalar recipeConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formNumerical recipes prescribe the concretegradients on witness atomsThe affine minus-column law at a fixed plusframeCompression that fixes pin/key vectors andpreserves target matricesCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesFull response obstructions are effectiveprofiles plus selected atomsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsFinite scalar agreement from binarycharacter boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsExact agreement of zero quotient charactersUnary records determine every frozen crossentryBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesScalar agreement after quotient characterestimatesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partEndpoint projections of paired pure-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsSparse pin exclusions with arbitrary basecoefficientsSparse residual contractions belong to theactual primal pinsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesSplit quotient projections preserve coefficientcharactersPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesWalsh bounds for independent image lawsand separated phasesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank factors independent modulo the actualfrozen spacesRank of a tensor killed in two quotientspacesFull rank coefficient factorizationsRank-controlled pure forms on the actualbarred quotientsBounded baselines for both actual crossorientationsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawGradient residuals vanish on effective profilesand selected atomsCoupled scalar recipes give consistent atomgradientsJoint reference image caps across both signsand all drawsThe full reference cap for exact pin eventsRemoving selected atoms leaves onlyunselected component labelsThe bounded pure remainder of an actualscalar recipeMatrix representations and the boundedresidual rank ingredientsUnrestricted linearized solutions for actualscalar recipesRank loss under restriction of a bilinear formActual cross Gram characters on frozenunary cellsImage caps inside original retained cellsSelected tensor blocks of actual pureobstructionsA selected affine ray determines its momentblockSubtracting selected atoms preserves pureannihilationSelected label coefficients agree across thecut profileInterpolation of finitely many binary selectorlabelsNumerical cross tables, injection flags, andunary admissibilityUnselected sparse vectors cannot concealfresh key coefficientsA uniform label budget for all sparse pinvectorsLow-rank tester routing along tag starsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryExplicit low-rank matrices for thewhole-space gradient targetThe fifteen-rank witness tester targetPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesOrthogonality and finite Walsh correlationboundsWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 block_factor_pair proven

    2 block_factor_phase proven

    3 block_image_dimension proven

    4 component_character_bound proven

    5 retained_component_character_dichotomy proven

    Lean source view on GitHub

    1import Lax342547.RetainedCharacters
    2
    3/-!
    4---
    5title: Joint retained-cell characters across all components
    6type: lemma
    7---
    8Componentwise quotient projections and rank factors combine into one joint Walsh image. No independence between components or nominal orientations is imposed; rank sums control the complete character.
    9-/
    10
    11namespace Lax342547.ComponentCharacters
    12
    13open Lax342547.ConcreteGeometry
    14open Lax342547.MomentSpace Lax342547.Walsh Lax342547.CoefficientPhase
    15open Lax342547.PushforwardWalsh Lax342547.RetainedCharacters
    16
    17noncomputable def blockImage {Axis N : Type} {I R : Axis → Type}
    18 [Fintype N] [∀ a, Fintype (I a)]
    19 (A : ∀ a, Matrix (I a) (R a) Binary) (X : ∀ a, Matrix N (I a) Binary) :
    20 ((Σ a, R a) × N) → Binary := fun v => factorImage (A v.1.1) (X v.1.1) (v.1.2,v.2)
    21
    22axiom block_factor_pair {Axis N : Type} {I R : Axis → Type}
    23 [Fintype Axis] [Fintype N] [∀ a, Fintype (I a)] [∀ a, Fintype (R a)]
    24 (A B : ∀ a, Matrix (I a) (R a) Binary) (X Y : ∀ a, Matrix N (I a) Binary) :
    25 (∑ a, coefficientPair (A a * (B a).transpose) (X a) (Y a)) =
    26 dotProduct (blockImage A X) (blockImage B Y)
    27
    28axiom block_factor_phase {Axis N : Type} {I R : Axis → Type}
    29 [Fintype Axis] [Fintype N] [∀ a, Fintype (I a)] [∀ a, Fintype (R a)]
    30 (A B : ∀ a, Matrix (I a) (R a) Binary) (X Y : ∀ a, Matrix N (I a) Binary) :
    31 sign (∑ a, coefficientPair (A a * (B a).transpose) (X a) (Y a)) =
    32 phase (blockImage A X) (blockImage B Y)
    33
    34axiom block_image_dimension {Axis N : Type} {R : Axis → Type}
    35 [Fintype Axis] [Fintype N] [∀ a, Fintype (R a)] :
    36 Fintype.card ((Σ a, R a) × N) = (∑ a, Fintype.card (R a)) * Fintype.card N
    37
    38axiom component_character_bound {ΩA ΩB Axis N : Type} {I R : Axis → Type}
    39 [Fintype ΩA] [Fintype ΩB] [Fintype Axis] [Fintype N]
    40 [∀ a, Fintype (I a)] [∀ a, Fintype (R a)]
    41 (α : ΩA → ℝ) (β : ΩB → ℝ)
    42 (X : ΩA → ∀ a, Matrix N (I a) Binary) (Y : ΩB → ∀ a, Matrix N (I a) Binary)
    43 (A B : ∀ a, Matrix (I a) (R a) Binary) (c : Binary)
    44 (hα : ∀ x, 0 ≤ α x) (hβ : ∀ y, 0 ≤ β y)
    45 (hαsum : ∑ x, α x ≤ 1) (hβsum : ∑ y, β y ≤ 1)
    46 (hcapA : ∀ x, push α (fun ω => blockImage A (X ω)) x ≤
    47 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, Fintype.card (R a))*Fintype.card N)))
    48 (hcapB : ∀ y, push β (fun ω => blockImage B (Y ω)) y ≤
    49 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, Fintype.card (R a))*Fintype.card N))) :
    50 |∑ x, ∑ y, α x * β y * sign ((∑ a,
    51 coefficientPair (A a * (B a).transpose) (X x a) (Y y a)) + c)| ≤
    52 (2 : ℝ)^(-(45/100 : ℝ)*((∑ a, Fintype.card (R a))*Fintype.card N))
    53
    54axiom retained_component_character_dichotomy {ΩA ΩB Axis N : Type} {I : Axis → Type}
    55 [Fintype ΩA] [Fintype ΩB] [Fintype Axis] [Fintype N] [∀ a, Fintype (I a)]
    56 (α : ΩA → ℝ) (β : ΩB → ℝ)
    57 (X : ΩA → ∀ a, Matrix N (I a) Binary) (Y : ΩB → ∀ a, Matrix N (I a) Binary)
    58 (D E : ∀ a, Submodule Binary (I a → Binary))
    59 (T : ∀ a, LinearMap.BilinForm Binary (I a → Binary))
    60 (hα : ∀ x, 0 ≤ α x) (hβ : ∀ y, 0 ≤ β y)
    61 (hαsum : ∑ x, α x = 1) (hβsum : ∑ y, β y = 1)
    62 (hD : ∀ x y a v, v ∈ D a → ∀ w, actualForm (X x a) (Y y a) v w = T a v w)
    63 (hE : ∀ x y a v w, w ∈ E a → actualForm (X x a) (Y y a) v w = T a v w)
    64 (hcapA : ∀ (r : Axis → ℕ) (A : ∀ a, Matrix (I a) (Fin (r a)) Binary),
    65 (∀ a, Function.Injective ((D a).mkQ.comp (A a).mulVecLin)) →
    66 ∀ x, push α (fun ω => blockImage A (X ω)) x ≤
    67 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    68 (hcapB : ∀ (r : Axis → ℕ) (B : ∀ a, Matrix (I a) (Fin (r a)) Binary),
    69 (∀ a, Function.Injective ((E a).mkQ.comp (B a).mulVecLin)) →
    70 ∀ y, push β (fun ω => blockImage B (Y ω)) y ≤
    71 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    72 (C : ∀ a, Matrix (I a) (I a) Binary) : by
    73 classical
    74 let mean := ∑ x, ∑ y, α x * β y * sign (∑ a,
    75 (coefficientPair (C a) (X x a) (Y y a) +
    76 matrixPair (LinearMap.BilinForm.toMatrix' (T a)) (C a)))
    77 exact mean = 1 ∨ |mean| ≤ (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card N)
    78
    79end Lax342547.ComponentCharacters
    80
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