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Explicit low-rank matrices for the whole-space gradient target

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

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    Natural Language Statement

    Lemma

    The adjoints of the concrete mixer sandwiches have ranks bounded by the witness component ranks. Combining both orientations with the routed tester matrices constructs the actual target matrices, proves the rank 15r+14J|E|, and represents the whole profile-pair target exactly.

    Concept map
    86 concepts
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    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 channelsA 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 boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsBaseline contractions on the selected atomsThe 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 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 framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesRank-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 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 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 spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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

    1 mixer_matrix_rank proven

    2 mixer_matrix_representation proven

    3 sandwich_adjoint proven

    4 target_family_representation proven

    5 target_matrix_rank proven

    6 target_matrix_representation proven

    7 witness_component_rank proven

    Lean source view on GitHub

    1import Lax342547.TargetTesters
    2
    3/-!
    4---
    5title: Explicit low-rank matrices for the whole-space gradient target
    6type: lemma
    7---
    8The adjoints of the concrete mixer sandwiches have ranks bounded by the witness component ranks. Combining both orientations with the routed tester matrices constructs the actual target matrices, proves the rank 15r+14J|E|, and represents the whole profile-pair target exactly.
    9-/
    10
    11namespace Lax342547.TargetMatrices
    12
    13open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    14open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.Atoms
    15open Lax342547.PairedWitnesses Lax342547.WitnessAtoms Lax342547.TargetTesters
    16
    17axiom sandwich_adjoint {B : Type} [Fintype B] (M W L R : Matrix B B Binary) :
    18 matrixPair M (L * W * R.transpose) = matrixPair (L.transpose * M * R) W
    19
    20noncomputable def mixerMatrix {k n b degree J : ℕ}
    21 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    22 (v : Profile k n b degree) (e : Component (Tag k)) : Moment k n b degree :=
    23 (∑ j, ∑ f, (L j f e).transpose * v.val f * R j f e) +
    24 ∑ j, ∑ f, L j e f * v.val f * (R j e f).transpose
    25
    26axiom mixer_matrix_representation {k n b degree J : ℕ}
    27 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    28 (v x : Profile k n b degree) :
    29 (∑ e, matrixPair (mixerMatrix L R v e) (x.val e)) = mixingForm L R v x + mixingForm L R x v
    30
    31axiom witness_component_rank {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    32 (W : Lists k n b degree r hr) (i z : Fin 2) (e : Component (Tag k)) :
    33 ((leftWitness W i z).val e).rank ≤ 7
    34
    35axiom mixer_matrix_rank {k n b degree J : ℕ}
    36 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    37 (v : Profile k n b degree) (S : ℕ) (hS : ∀ f, (v.val f).rank ≤ S)
    38 (e : Component (Tag k)) : (mixerMatrix L R v e).rank ≤
    39 2 * J * Fintype.card (Component (Tag k)) * S
    40
    41noncomputable def targetMatrix {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    42 (W : Lists k n b degree r hr)
    43 (D : Testers (k := k) (b := b) (degree := degree) hr)
    44 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    45 (i z : Fin 2) (e : Component (Tag k)) : Moment k n b degree :=
    46 witnessTesterMatrix W D i z e + mixerMatrix L R (leftWitness W i z) e
    47
    48axiom target_matrix_rank {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    49 (W : Lists k n b degree r hr)
    50 (D : Testers (k := k) (b := b) (degree := degree) hr)
    51 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    52 (i z : Fin 2) (e : Component (Tag k)) :
    53 (targetMatrix W D L R i z e).rank ≤ 15 * r + 14 * J * Fintype.card (Component (Tag k))
    54
    55axiom target_matrix_representation {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    56 (W : Lists k n b degree r hr)
    57 (D : Testers (k := k) (b := b) (degree := degree) hr)
    58 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    59 (i z : Fin 2) (hodd : (∑ j, diagonalBit hr (W.left i z j)) = 1)
    60 (x : Profile k n b degree) :
    61 (∑ e, matrixPair (targetMatrix W D L R i z e) (x.val e)) =
    62 (D.role + gradient D L R (leftWitness W i z)) x
    63
    64axiom target_family_representation {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    65 (W : Lists k n b degree r hr)
    66 (D : Testers (k := k) (b := b) (degree := degree) hr)
    67 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    68 (z : Fin 2) (hodd : ∀ i, (∑ j, diagonalBit hr (W.left i z j)) = 1) :
    69 Lax342547.ResidualRealization.matrixResponse (Profile k n b degree) (fun i => targetMatrix W D L R i z) =
    70 ∑ i, (D.role + gradient D L R (leftWitness W i z)).comp (LinearMap.proj i)
    71
    72end Lax342547.TargetMatrices
    73
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