Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Rank-one derivative tests are scalar pairings of residual contractions

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    A rank-one change in either channel map contracts to the baseline evaluation of the corresponding matrix contraction. Transposition exchanges the two orientations without a symmetry assumption.

    Concept map
    67 concepts
    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 budgetThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsConcrete 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 plusframeCut profiles and the constant kernelRank-one derivative tests are scalar pairingsof residual contractionsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesThe 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 basecoordinatesPrimal 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 framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawCoupled 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 labelsRank loss under restriction of a bilinear formSelected 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 pinvectorsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 left_rank_one proven

    2 paired_transpose proven

    3 right_rank_one proven

    4 transpose_contraction proven

    Lean source view on GitHub

    1import Lax342547.TensorContractions
    2import Lax342547.PrimalContractions
    3
    4/-!
    5---
    6title: Rank-one derivative tests are scalar pairings of residual contractions
    7type: theorem
    8---
    9A rank-one change in either channel map contracts to the baseline
    10evaluation of the corresponding matrix contraction. Transposition
    11exchanges the two orientations without a symmetry assumption.
    12-/
    13
    14namespace Lax342547.DerivativeContractions
    15
    16open Lax342547.MomentSpace Lax342547.PrimalContractions Lax342547.ConcreteGeometry
    17open Lax342547.TensorContractions
    18
    19axiom right_rank_one {B H : Type} [Fintype B] [Fintype H]
    20 (M : Matrix B B Binary) (f : (B → Binary) →ₗ[Binary] (H → Binary))
    21 (ψ : Module.Dual Binary (B → Binary)) (c : H → Binary) :
    22 ambient f (ψ.smulRight c) M = dotProduct (f (M.mulVecLin (coordinates ψ))) c
    23
    24axiom transpose_contraction {B H : Type} [Fintype B] [Fintype H]
    25 (M : Matrix B B Binary) (f g : (B → Binary) →ₗ[Binary] (H → Binary)) :
    26 ambient f g M = ambient g f M.transpose
    27
    28axiom left_rank_one {B H : Type} [Fintype B] [Fintype H]
    29 (M : Matrix B B Binary) (g : (B → Binary) →ₗ[Binary] (H → Binary))
    30 (φ : Module.Dual Binary (B → Binary)) (c : H → Binary) :
    31 ambient (φ.smulRight c) g M = dotProduct c (g (M.transpose.mulVecLin (coordinates φ)))
    32
    33open Lax342547.PairedAnnihilators
    34
    35axiom paired_transpose {B H : Type} [Fintype B]
    36 (M : Fin 2 → Matrix B B Binary) (D E : Submodule Binary (Nominal B H))
    37 (h : PairedAnnihilates M D E) : PairedAnnihilates (fun i => (M i).transpose) E D
    38
    39end Lax342547.DerivativeContractions
    40
    Show ProofShow ProofShow ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…