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Rank-controlled pure forms on the actual barred quotients

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

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    Natural Language Statement

    Lemma

    Projection away from the actual pin/key spaces yields pure quotient forms whose bilinear rank is at most the sum of the two individual target-matrix ranks. The projection loss remains at most 2K+28 per individual. This supplies the initial pure-form rank needed when restoring the compressed residual correction.

    Concept map
    77 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 budgetThe 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 testsConcrete 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 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 cutprofilesExtracting 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 labelsMatrix 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 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 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 actual_pure_projection proven

    2 bilinear_comp_rank proven

    3 linear_add_rank proven

    4 linear_sum_rank proven

    5 matrix_bilinear_rank proven

    6 projected_endpoint_response proven

    7 projected_paired_response proven

    Lean source view on GitHub

    1import Lax342547.ResponseMatrices
    2
    3/-!
    4---
    5title: Rank-controlled pure forms on the actual barred quotients
    6type: lemma
    7---
    8Projection away from the actual pin/key spaces yields pure quotient forms whose bilinear rank is at most the sum of the two individual target-matrix ranks. The projection loss remains at most 2K+28 per individual. This supplies the initial pure-form rank needed when restoring the compressed residual correction.
    9-/
    10
    11namespace Lax342547.RankedProjection
    12
    13open Lax342547.MomentSpace Lax342547.PairedAnnihilators Lax342547.TableSpaces
    14open Lax342547.ProjectedPins Lax342547.ResponseMatrices
    15open Lax342547.ConcreteGeometry
    16open Lax342547.TagGeometry Lax342547.ConcreteCut Lax342547.PairedWitnesses Lax342547.CutProfiles
    17open Lax342547.ExactPins Lax342547.BarredSpaces
    18
    19axiom bilinear_comp_rank {A B V X : Type} [AddCommGroup A] [Module Binary A]
    20 [AddCommGroup B] [Module Binary B] [AddCommGroup V] [Module Binary V]
    21 [AddCommGroup X] [Module Binary X] [FiniteDimensional Binary B]
    22 (β : A →ₗ[Binary] B →ₗ[Binary] Binary)
    23 (f : V →ₗ[Binary] A) (g : X →ₗ[Binary] B) :
    24 Module.finrank Binary (LinearMap.range (β.compl₁₂ f g)) ≤
    25 Module.finrank Binary (LinearMap.range β)
    26
    27axiom matrix_bilinear_rank {B : Type} [Fintype B] (M : Matrix B B Binary) :
    28 by
    29 classical
    30 exact Module.finrank Binary (LinearMap.range (Matrix.toBilin' M)) ≤ M.rank
    31
    32axiom linear_add_rank {V X : Type} [AddCommGroup V] [Module Binary V]
    33 [AddCommGroup X] [Module Binary X] [FiniteDimensional Binary X]
    34 (f g : V →ₗ[Binary] X) :
    35 Module.finrank Binary (LinearMap.range (f + g)) ≤
    36 Module.finrank Binary (LinearMap.range f) + Module.finrank Binary (LinearMap.range g)
    37
    38axiom linear_sum_rank {I V X : Type} [Fintype I] [AddCommGroup V] [Module Binary V]
    39 [AddCommGroup X] [Module Binary X] [FiniteDimensional Binary X]
    40 (f : I → V →ₗ[Binary] X) :
    41 Module.finrank Binary (LinearMap.range (∑ i, f i)) ≤
    42 ∑ i, Module.finrank Binary (LinearMap.range (f i))
    43
    44axiom projected_endpoint_response {B H : Type} [Fintype B]
    45 (D E : Submodule Binary (Nominal B H)) (i : Fin 2) (M A C : Matrix B B Binary)
    46 (hD : D.map (primalProjection i) ≤ LinearMap.ker A.mulVecLin)
    47 (hE : E.map (primalProjection i) ≤ LinearMap.ker C.mulVecLin) :
    48 ∃ β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary,
    49 pureResponse D E β = (matrixPair (A.transpose * M * C)).comp (LinearMap.proj i) ∧
    50 Module.finrank Binary (LinearMap.range β) ≤ M.rank
    51
    52axiom projected_paired_response {B H : Type} [Fintype B] [Fintype H]
    53 (D E : Submodule Binary (Nominal B H)) (M A C : Fin 2 → Matrix B B Binary)
    54 (hD : ∀ i, D.map (primalProjection i) ≤ LinearMap.ker (A i).mulVecLin)
    55 (hE : ∀ i, E.map (primalProjection i) ≤ LinearMap.ker (C i).mulVecLin) :
    56 ∃ β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary,
    57 pureResponse D E β = ∑ i, (matrixPair ((A i).transpose * M i * C i)).comp (LinearMap.proj i) ∧
    58 Module.finrank Binary (LinearMap.range β) ≤ ∑ i, (M i).rank
    59
    60axiom actual_pure_projection {k n b degree r K : ℕ} {hr : 2 * r ≤ n}
    61 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    62 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    63 (hK : P.rank ≤ K) (U V : Fin 2 → Submodule Binary (H → Binary))
    64 (e : Component (Tag k)) (M : Fin 2 → Moment k n b degree) :
    65 ∃ (A C : Fin 2 → Moment k n b degree)
    66 (β : (Nominal (Coordinate k n b degree) H ⧸ barred W P U (e, true)) →ₗ[Binary]
    67 (Nominal (Coordinate k n b degree) H ⧸ barred W P V (e, false)) →ₗ[Binary] Binary),
    68 pureResponse (barred W P U (e, true)) (barred W P V (e, false)) β =
    69 ∑ i, (matrixPair ((A i).transpose * M i * C i)).comp (LinearMap.proj i) ∧
    70 Module.finrank Binary (LinearMap.range β) ≤ ∑ i, (M i).rank ∧
    71 ∀ i, (M i - (A i).transpose * M i * C i).rank ≤ 2 * K + 28 ∧
    72 ((A i).transpose * M i * C i).rank ≤ (M i).rank
    73
    74end Lax342547.RankedProjection
    75
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