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Rank-controlled factorization through allowed orthogonal channels

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

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    Natural Language Statement

    Lemma

    A pure quotient form factors through the actual dot pairing whenever its rank fits. Orthogonality to protected channels, baseline values and bounded derivative values costs at most K+2R on each side. The resulting allowed maps realize the pure form exactly and have all required vanishing cross pairings; simultaneous gradient assembly and target routing remain separate.

    Concept map
    83 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 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 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 9 statements. Each proof establishes one of them relative to its assumptions.

    1 allowed_pure_factor proven

    2 channel_pairing_rank proven

    3 factor_in_channel_spaces proven

    4 factor_through_pairing proven

    5 perpendicular_dimension proven

    6 remaining_values_allowed proven

    7 remaining_values_dimension proven

    8 remaining_values_factor proven

    9 remaining_values_orthogonal proven

    Lean source view on GitHub

    1import Lax342547.CompressedResidual
    2import Mathlib.LinearAlgebra.Dual.Lemmas
    3import Mathlib.LinearAlgebra.Dimension.Free
    4import Mathlib.Algebra.Module.Projective
    5
    6/-!
    7---
    8title: Rank-controlled factorization through allowed orthogonal channels
    9type: lemma
    10---
    11A pure quotient form factors through the actual dot pairing whenever its rank fits. Orthogonality to protected channels, baseline values and bounded derivative values costs at most K+2R on each side. The resulting allowed maps realize the pure form exactly and have all required vanishing cross pairings; simultaneous gradient assembly and target routing remain separate.
    12-/
    13
    14namespace Lax342547.ChannelFactorization
    15
    16open Lax342547.MomentSpace Lax342547.ChannelChanges
    17open Lax342547.PairedAnnihilators Lax342547.ResponseMatrices
    18
    19axiom factor_through_pairing {X Y U V : Type}
    20 [AddCommGroup X] [Module Binary X] [AddCommGroup Y] [Module Binary Y]
    21 [AddCommGroup U] [Module Binary U] [AddCommGroup V] [Module Binary V]
    22 [FiniteDimensional Binary Y] [FiniteDimensional Binary V]
    23 (β : X →ₗ[Binary] Y →ₗ[Binary] Binary) (π : U →ₗ[Binary] V →ₗ[Binary] Binary)
    24 (hr : Module.finrank Binary (LinearMap.range β) ≤ Module.finrank Binary (LinearMap.range π)) :
    25 ∃ f : X →ₗ[Binary] U, ∃ g : Y →ₗ[Binary] V, π.compl₁₂ f g = β
    26
    27axiom factor_in_channel_spaces {X Y H : Type} [Fintype H]
    28 [AddCommGroup X] [Module Binary X] [AddCommGroup Y] [Module Binary Y]
    29 [FiniteDimensional Binary Y]
    30 (S T : Submodule Binary (H → Binary)) (β : X →ₗ[Binary] Y →ₗ[Binary] Binary)
    31 (hr : Module.finrank Binary (LinearMap.range β) ≤ Module.finrank Binary
    32 (LinearMap.range ((dotProductBilin Binary Binary).compl₁₂ S.subtype T.subtype))) :
    33 ∃ f : X →ₗ[Binary] S, ∃ g : Y →ₗ[Binary] T,
    34 (dotProductBilin Binary Binary).compl₁₂ (S.subtype.comp f) (T.subtype.comp g) = β
    35
    36axiom perpendicular_dimension {H : Type} [Fintype H] (S : Submodule Binary (H → Binary)) :
    37 Module.finrank Binary S + Module.finrank Binary (perpendicular S) = Fintype.card H
    38
    39axiom channel_pairing_rank {H : Type} [Fintype H]
    40 (S T : Submodule Binary (H → Binary)) {cS cT : ℕ}
    41 (hS : Fintype.card H ≤ Module.finrank Binary S + cS)
    42 (hT : Fintype.card H ≤ Module.finrank Binary T + cT) :
    43 Fintype.card H ≤ Module.finrank Binary
    44 (LinearMap.range ((dotProductBilin Binary Binary).compl₁₂ S.subtype T.subtype)) + cS + cT
    45
    46noncomputable def remainingValues {A B H : Type} [Fintype H]
    47 [AddCommGroup A] [Module Binary A] [AddCommGroup B] [Module Binary B]
    48 (S : Submodule Binary (H → Binary)) (G : A →ₗ[Binary] (H → Binary))
    49 (D : B →ₗ[Binary] (H → Binary)) : Submodule Binary (H → Binary) :=
    50 perpendicular (S ⊔ LinearMap.range G ⊔ LinearMap.range D)
    51
    52axiom remaining_values_dimension {A B H : Type} [Fintype H]
    53 [AddCommGroup A] [Module Binary A] [AddCommGroup B] [Module Binary B]
    54 (S : Submodule Binary (H → Binary)) (G : A →ₗ[Binary] (H → Binary))
    55 (D : B →ₗ[Binary] (H → Binary)) {K R : ℕ}
    56 (hS : Module.finrank Binary S ≤ K)
    57 (hG : Module.finrank Binary (LinearMap.range G) ≤ R)
    58 (hD : Module.finrank Binary (LinearMap.range D) ≤ R) :
    59 Fintype.card H ≤ Module.finrank Binary (remainingValues S G D) + (K + 2 * R)
    60
    61axiom remaining_values_allowed {A B H : Type} [Fintype H]
    62 [AddCommGroup A] [Module Binary A] [AddCommGroup B] [Module Binary B]
    63 (S : Submodule Binary (H → Binary)) (G : A →ₗ[Binary] (H → Binary))
    64 (D : B →ₗ[Binary] (H → Binary)) : remainingValues S G D ≤ perpendicular S
    65
    66axiom remaining_values_orthogonal {A B H : Type} [Fintype H]
    67 [AddCommGroup A] [Module Binary A] [AddCommGroup B] [Module Binary B]
    68 (S : Submodule Binary (H → Binary)) (G : A →ₗ[Binary] (H → Binary))
    69 (D : B →ₗ[Binary] (H → Binary)) (v : remainingValues S G D) :
    70 (∀ a, dotProduct v.val (G a) = 0) ∧ (∀ b, dotProduct v.val (D b) = 0)
    71
    72axiom remaining_values_factor {X Y A B C D H : Type} [Fintype H]
    73 [AddCommGroup X] [Module Binary X] [AddCommGroup Y] [Module Binary Y]
    74 [AddCommGroup A] [Module Binary A] [AddCommGroup B] [Module Binary B]
    75 [AddCommGroup C] [Module Binary C] [AddCommGroup D] [Module Binary D]
    76 [FiniteDimensional Binary Y]
    77 (S T : Submodule Binary (H → Binary))
    78 (G : A →ₗ[Binary] (H → Binary)) (dG : B →ₗ[Binary] (H → Binary))
    79 (F : C →ₗ[Binary] (H → Binary)) (dF : D →ₗ[Binary] (H → Binary))
    80 (β : X →ₗ[Binary] Y →ₗ[Binary] Binary) {K R : ℕ}
    81 (hS : Module.finrank Binary S ≤ K) (hT : Module.finrank Binary T ≤ K)
    82 (hG : Module.finrank Binary (LinearMap.range G) ≤ R)
    83 (hdG : Module.finrank Binary (LinearMap.range dG) ≤ R)
    84 (hF : Module.finrank Binary (LinearMap.range F) ≤ R)
    85 (hdF : Module.finrank Binary (LinearMap.range dF) ≤ R)
    86 (hcapacity : Module.finrank Binary (LinearMap.range β) + 2 * K + 4 * R ≤ Fintype.card H) :
    87 ∃ f : X →ₗ[Binary] remainingValues S G dG,
    88 ∃ g : Y →ₗ[Binary] remainingValues T F dF,
    89 (dotProductBilin Binary Binary).compl₁₂
    90 ((remainingValues S G dG).subtype.comp f) ((remainingValues T F dF).subtype.comp g) = β
    91
    92axiom allowed_pure_factor {B H A C E F : Type} [Fintype B] [Fintype H]
    93 [AddCommGroup A] [Module Binary A] [AddCommGroup C] [Module Binary C]
    94 [AddCommGroup E] [Module Binary E] [AddCommGroup F] [Module Binary F]
    95 (D Q : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    96 (G : A →ₗ[Binary] (H → Binary)) (dG : C →ₗ[Binary] (H → Binary))
    97 (G' : E →ₗ[Binary] (H → Binary)) (dG' : F →ₗ[Binary] (H → Binary))
    98 (β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ Q) →ₗ[Binary] Binary) {K R : ℕ}
    99 (hS : Module.finrank Binary S ≤ K) (hT : Module.finrank Binary T ≤ K)
    100 (hG : Module.finrank Binary (LinearMap.range G) ≤ R)
    101 (hdG : Module.finrank Binary (LinearMap.range dG) ≤ R)
    102 (hG' : Module.finrank Binary (LinearMap.range G') ≤ R)
    103 (hdG' : Module.finrank Binary (LinearMap.range dG') ≤ R)
    104 (hcapacity : Module.finrank Binary (LinearMap.range β) + 2 * K + 4 * R ≤ Fintype.card H) :
    105 ∃ δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S,
    106 ∃ ε : (Nominal B H ⧸ Q) →ₗ[Binary] perpendicular T,
    107 extraProduct D Q S T δ ε = β ∧
    108 (∀ x a, dotProduct (δ x).val (G a) = 0) ∧
    109 (∀ x c, dotProduct (δ x).val (dG c) = 0) ∧
    110 (∀ y e, dotProduct (ε y).val (G' e) = 0) ∧
    111 (∀ y f, dotProduct (ε y).val (dG' f) = 0)
    112
    113end Lax342547.ChannelFactorization
    114
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