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Matrix representations and the bounded residual rank ingredients

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

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    Natural Language Statement

    Lemma

    Actual cut-profile functionals have component matrix representations. Projected forms descend through the actual barred nominal quotients, with a per-individual removal cost at most 2K+28. Baseline and derivative contractions have bounded rank, and the derivative product is a permitted pure quotient response. These ingredients give the paper residual matrix rank estimate before the remaining compression and channel realization.

    Concept map
    76 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 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 quotientspacesBounded 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 19 statements. Each proof establishes one of them relative to its assumptions.

    1 actual_pure_projection proven

    2 ambient_matrix_representation proven

    3 baseline_coordinate_rank proven

    4 bounded_residual_rank proven

    5 coefficients_product proven

    6 coefficients_rank proven

    7 comp_left_rank proven

    8 comp_right_rank proven

    9 extra_product_response proven

    10 matrix_representation proven

    11 projected_endpoint_response proven

    12 projected_paired_response proven

    13 projected_pin_key_dimension proven

    14 projection_difference proven

    15 projection_difference_rank proven

    16 quotient_projection proven

    17 rank_sub_le proven

    18 restriction_rank proven

    19 toMatrix_rank proven

    Lean source view on GitHub

    1import Lax342547.BoundedDerivatives
    2import Mathlib.LinearAlgebra.Matrix.Rank
    3
    4/-!
    5---
    6title: Matrix representations and the bounded residual rank ingredients
    7type: lemma
    8---
    9Actual cut-profile functionals have component matrix representations.
    10Projected forms descend through the actual barred nominal quotients, with
    11a per-individual removal cost at most 2K+28. Baseline and derivative
    12contractions have bounded rank, and the derivative product is a permitted
    13pure quotient response. These ingredients give the paper residual matrix
    14rank estimate before the remaining compression and channel realization.
    15-/
    16
    17namespace Lax342547.ResponseMatrices
    18
    19open Lax342547.MomentSpace Lax342547.ConcreteGeometry
    20open Lax342547.TensorContractions
    21open Lax342547.PairedAnnihilators Lax342547.TableSpaces
    22open Lax342547.ProjectedPins
    23open Lax342547.TagGeometry Lax342547.ConcreteCut Lax342547.PairedWitnesses Lax342547.CutProfiles
    24open Lax342547.ExactPins Lax342547.BarredSpaces
    25open Lax342547.DerivativeResponses Lax342547.ChannelChanges Lax342547.TableContractions
    26open Lax342547.NominalPrimal Lax342547.TensorAnnihilators
    27
    28noncomputable def coefficients {B H : Type} [Fintype B] [Fintype H]
    29 (f g : (B → Binary) →ₗ[Binary] (H → Binary)) : Matrix B B Binary := by
    30 classical
    31 exact LinearMap.BilinForm.toMatrix' ((dotProductBilin Binary Binary).compl₁₂ f g)
    32
    33noncomputable def extraProduct {B H : Type} [Fintype H]
    34 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    35 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    36 (ε : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T) :
    37 (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary :=
    38 (dotProductBilin Binary Binary).compl₁₂ ((perpendicular S).subtype.comp δ)
    39 ((perpendicular T).subtype.comp ε)
    40
    41noncomputable def residualCoefficients {B H : Type} [Fintype B] [Fintype H]
    42 (M Q : Matrix B B Binary) (f g df dg : (B → Binary) →ₗ[Binary] (H → Binary)) :
    43 Matrix B B Binary :=
    44 M - Q - coefficients f g - coefficients df g - coefficients f dg - coefficients df dg
    45
    46axiom coefficients_product {B H : Type} [Fintype B] [Fintype H]
    47 (f g : (B → Binary) →ₗ[Binary] (H → Binary)) : by
    48 classical
    49 exact coefficients f g = (LinearMap.toMatrix' f).transpose * LinearMap.toMatrix' g
    50
    51axiom toMatrix_rank {B H : Type} [Fintype B] [Fintype H]
    52 (f : (B → Binary) →ₗ[Binary] (H → Binary)) : by
    53 classical
    54 exact (LinearMap.toMatrix' f).rank = Module.finrank Binary (LinearMap.range f)
    55
    56axiom coefficients_rank {B H : Type} [Fintype B] [Fintype H]
    57 (f g : (B → Binary) →ₗ[Binary] (H → Binary)) :
    58 (coefficients f g).rank ≤ min (Module.finrank Binary (LinearMap.range f))
    59 (Module.finrank Binary (LinearMap.range g))
    60
    61axiom projection_difference {B : Type} [Fintype B] [DecidableEq B]
    62 (A M C : Matrix B B Binary) :
    63 M - A.transpose * M * C = (1 - A).transpose * M + A.transpose * M * (1 - C)
    64
    65axiom projection_difference_rank {B : Type} [Fintype B] [DecidableEq B]
    66 (A M C : Matrix B B Binary) :
    67 (M - A.transpose * M * C).rank ≤ (1 - A).rank + (1 - C).rank
    68
    69axiom quotient_projection {B : Type} [Fintype B]
    70 (S : Submodule Binary (B → Binary)) : by
    71 classical
    72 exact ∃ A : Matrix B B Binary, S ≤ LinearMap.ker A.mulVecLin ∧
    73 (1 - A).rank ≤ Module.finrank Binary S
    74
    75axiom projected_endpoint_response {B H : Type} [Fintype B]
    76 (D E : Submodule Binary (Nominal B H)) (i : Fin 2) (M A C : Matrix B B Binary)
    77 (hD : D.map (primalProjection i) ≤ LinearMap.ker A.mulVecLin)
    78 (hE : E.map (primalProjection i) ≤ LinearMap.ker C.mulVecLin) :
    79 ∃ β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary,
    80 pureResponse D E β = (matrixPair (A.transpose * M * C)).comp (LinearMap.proj i)
    81
    82axiom projected_paired_response {B H : Type} [Fintype B]
    83 (D E : Submodule Binary (Nominal B H)) (M A C : Fin 2 → Matrix B B Binary)
    84 (hD : ∀ i, D.map (primalProjection i) ≤ LinearMap.ker (A i).mulVecLin)
    85 (hE : ∀ i, E.map (primalProjection i) ≤ LinearMap.ker (C i).mulVecLin) :
    86 ∃ β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary,
    87 pureResponse D E β = ∑ i, (matrixPair ((A i).transpose * M i * C i)).comp (LinearMap.proj i)
    88
    89axiom projected_pin_key_dimension {k n b degree r K : ℕ} {hr : 2 * r ≤ n}
    90 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    91 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    92 (hK : P.rank ≤ K) (i : Fin 2) (a : Component (Tag k) × Bool) :
    93 Module.finrank Binary ↥(projected P i a ⊔ individualKeys W i a.1) ≤ K + 14
    94
    95axiom actual_pure_projection {k n b degree r K : ℕ} {hr : 2 * r ≤ n}
    96 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    97 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    98 (hK : P.rank ≤ K) (U V : Fin 2 → Submodule Binary (H → Binary))
    99 (e : Component (Tag k)) (M : Fin 2 → Moment k n b degree) :
    100 ∃ (A C : Fin 2 → Moment k n b degree)
    101 (β : (Nominal (Coordinate k n b degree) H ⧸ barred W P U (e, true)) →ₗ[Binary]
    102 (Nominal (Coordinate k n b degree) H ⧸ barred W P V (e, false)) →ₗ[Binary] Binary),
    103 pureResponse (barred W P U (e, true)) (barred W P V (e, false)) β =
    104 ∑ i, (matrixPair ((A i).transpose * M i * C i)).comp (LinearMap.proj i) ∧
    105 ∀ i, (M i - (A i).transpose * M i * C i).rank ≤ 2 * K + 28 ∧
    106 ((A i).transpose * M i * C i).rank ≤ (M i).rank
    107
    108axiom ambient_matrix_representation {Comp B : Type} [Fintype Comp] [Fintype B]
    109 (t : (Comp → Matrix B B Binary) →ₗ[Binary] Binary) :
    110 ∃ M : Comp → Matrix B B Binary, ∀ x, t x = ∑ e, matrixPair (M e) (x e)
    111
    112axiom matrix_representation {Comp B : Type} [Fintype Comp] [Fintype B]
    113 (X : Submodule Binary (Comp → Matrix B B Binary)) (t : X →ₗ[Binary] Binary) :
    114 ∃ M : Comp → Matrix B B Binary, ∀ x : X, t x = ∑ e, matrixPair (M e) (x.val e)
    115
    116axiom comp_left_rank {A V X : Type} [AddCommGroup A] [Module Binary A]
    117 [AddCommGroup V] [Module Binary V] [AddCommGroup X] [Module Binary X]
    118 [FiniteDimensional Binary X] (f : A →ₗ[Binary] X) (g : V →ₗ[Binary] A) :
    119 Module.finrank Binary (LinearMap.range (f.comp g)) ≤ Module.finrank Binary (LinearMap.range f)
    120
    121axiom comp_right_rank {A V X : Type} [AddCommGroup A] [Module Binary A]
    122 [AddCommGroup V] [Module Binary V] [AddCommGroup X] [Module Binary X]
    123 [FiniteDimensional Binary A] (f : A →ₗ[Binary] X) (g : V →ₗ[Binary] A) :
    124 Module.finrank Binary (LinearMap.range (f.comp g)) ≤ Module.finrank Binary (LinearMap.range g)
    125
    126axiom restriction_rank {B H : Type} [Fintype B] [Fintype H]
    127 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    128 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (i : Fin 2) :
    129 Module.finrank Binary (LinearMap.range (restriction D S δ i)) ≤
    130 Module.finrank Binary (LinearMap.range δ)
    131
    132axiom baseline_coordinate_rank {Comp B H : Type} [Fintype B] [Fintype H]
    133 (F : CrossForms Comp B H) (e : Comp) (z : Fin 2) (R : ℕ)
    134 (hplus : Module.finrank Binary (LinearMap.range ((F.forward e).compl₁₂
    135 (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ R)
    136 (hminus : Module.finrank Binary (LinearMap.range ((F.reverse e).flip.compl₁₂
    137 (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ R) :
    138 ∀ i, Module.finrank Binary (LinearMap.range (fullPlus F e i z)) ≤ R ∧
    139 Module.finrank Binary (LinearMap.range (fullMinus F e i z)) ≤ R
    140
    141axiom extra_product_response {B H : Type} [Fintype B] [Fintype H]
    142 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    143 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    144 (ε : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T) :
    145 pureResponse D E (extraProduct D E S T δ ε) =
    146 ∑ i, (matrixPair (coefficients (restriction D S δ i) (restriction E T ε i))).comp (LinearMap.proj i)
    147
    148axiom rank_sub_le {B : Type} [Fintype B] (M C : Matrix B B Binary) :
    149 (M - C).rank ≤ M.rank + C.rank
    150
    151axiom bounded_residual_rank {B H : Type} [Fintype B] [Fintype H] (K : ℕ)
    152 (M Q : Matrix B B Binary) (hM : (M - Q).rank ≤ 2 * K + 28)
    153 (f g df dg : (B → Binary) →ₗ[Binary] (H → Binary))
    154 (hf : Module.finrank Binary (LinearMap.range f) ≤ 3 * K + 28)
    155 (hdf : Module.finrank Binary (LinearMap.range df) ≤ 3 * K + 28) :
    156 (residualCoefficients M Q f g df dg).rank ≤ 14 * K + 140
    157
    158end Lax342547.ResponseMatrices
    159
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