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Pure corrections realized by actual nonlinear channel products

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

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    Natural Language Statement

    Lemma

    Low-rank baseline values use the combined primal space. Orthogonal private factors add the desired pure form to the channel product while retaining the entire derivative response. The paper margin provides sufficient channel capacity. These changes retain their barred domains and permitted perpendicular outputs.

    Concept map
    87 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 partPure corrections realized by actual nonlinearchannel productsEndpoint 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 12 statements. Each proof establishes one of them relative to its assumptions.

    1 ambient_add_expansion proven

    2 ambient_add_left proven

    3 ambient_add_right proven

    4 ambient_orthogonal_zero proven

    5 component_response_unchanged proven

    6 extra_product_add proven

    7 nonlinear_component_correction proven

    8 paper_channel_capacity proven

    9 primal_minus_values proven

    10 primal_plus_values proven

    11 primal_values_rank proven

    12 restriction_add proven

    Lean source view on GitHub

    1import Lax342547.TargetMatrices
    2
    3/-!
    4---
    5title: Pure corrections realized by actual nonlinear channel products
    6type: lemma
    7---
    8Low-rank baseline values use the combined primal space. Orthogonal private factors add the desired pure form to the channel product while retaining the entire derivative response. The paper margin provides sufficient channel capacity. These changes retain their barred domains and permitted perpendicular outputs.
    9-/
    10
    11namespace Lax342547.NonlinearChannels
    12
    13open Lax342547.ConcreteCut
    14open Lax342547.MomentSpace Lax342547.PairedAnnihilators Lax342547.ChannelChanges
    15open Lax342547.ResponseMatrices Lax342547.TableSpaces Lax342547.TableContractions
    16open Lax342547.DerivativeResponses Lax342547.NominalPrimal Lax342547.TensorAnnihilators
    17
    18noncomputable def primalValues {B H : Type} [Fintype H]
    19 (F : LinearMap.BilinForm Binary (Nominal B H)) (z : Fin 2) :
    20 primal (B := B) (H := H) →ₗ[Binary] (H → Binary) := by
    21 classical
    22 exact dualCoordinates.comp (F.compl₁₂ primal.subtype (channelEmbedding z))
    23
    24axiom primal_values_rank {B H : Type} [Fintype B] [Fintype H]
    25 (F : LinearMap.BilinForm Binary (Nominal B H)) (z : Fin 2) :
    26 Module.finrank Binary (LinearMap.range (primalValues F z)) ≤
    27 Module.finrank Binary (LinearMap.range (F.compl₁₂ primal.subtype (channelEmbedding z)))
    28
    29def liftPrimal {B H : Type} (i : Fin 2) : (B → Binary) →ₗ[Binary] primal (B := B) (H := H) :=
    30 (primalEmbedding i).codRestrict primal (by intro v a h; simp [primalEmbedding])
    31
    32axiom primal_plus_values {Comp B H : Type} [Fintype H]
    33 (F : CrossForms Comp B H) (e : Comp) (i z : Fin 2) (v : B → Binary) :
    34 primalValues (F.forward e) z (liftPrimal i v) = fullPlus F e i z v
    35
    36axiom primal_minus_values {Comp B H : Type} [Fintype H]
    37 (F : CrossForms Comp B H) (e : Comp) (i z : Fin 2) (v : B → Binary) :
    38 primalValues (F.reverse e).flip z (liftPrimal i v) = fullMinus F e i z v
    39
    40axiom extra_product_add {B H : Type} [Fintype H]
    41 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    42 (δ f : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    43 (ε g : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T)
    44 (hfg : ∀ x y, dotProduct (f x).val (ε y).val = 0)
    45 (hgf : ∀ x y, dotProduct (g y).val (δ x).val = 0) :
    46 extraProduct D E S T (δ + f) (ε + g) =
    47 extraProduct D E S T δ ε + extraProduct D E S T f g
    48
    49axiom ambient_add_left {B H : Type} [Fintype B] [Fintype H]
    50 (f df g : (B → Binary) →ₗ[Binary] (H → Binary)) :
    51 Lax342547.TensorContractions.ambient (f + df) g =
    52 Lax342547.TensorContractions.ambient f g + Lax342547.TensorContractions.ambient df g
    53
    54axiom ambient_add_right {B H : Type} [Fintype B] [Fintype H]
    55 (f g dg : (B → Binary) →ₗ[Binary] (H → Binary)) :
    56 Lax342547.TensorContractions.ambient f (g + dg) =
    57 Lax342547.TensorContractions.ambient f g + Lax342547.TensorContractions.ambient f dg
    58
    59axiom ambient_orthogonal_zero {B H : Type} [Fintype B] [Fintype H]
    60 (f g : (B → Binary) →ₗ[Binary] (H → Binary))
    61 (h : ∀ v w, dotProduct (f v) (g w) = 0) :
    62 Lax342547.TensorContractions.ambient f g = 0
    63
    64axiom restriction_add {B H : Type} [Fintype H]
    65 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    66 (δ f : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (i : Fin 2) :
    67 restriction D S (δ + f) i = restriction D S δ i + restriction D S f i
    68
    69axiom component_response_unchanged {Comp B H : Type} [Fintype B] [Fintype H]
    70 (F : CrossForms Comp B H) (e : Comp) (z : Fin 2)
    71 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    72 (δ f : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    73 (ε g : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T)
    74 (hf : ∀ x (v : primal (B := B) (H := H)),
    75 dotProduct (f x).val (primalValues (F.reverse e).flip z v) = 0)
    76 (hg : ∀ y (v : primal (B := B) (H := H)),
    77 dotProduct (g y).val (primalValues (F.forward e) z v) = 0) :
    78 componentResponse F e z D E S T (δ + f) (ε + g) =
    79 componentResponse F e z D E S T δ ε
    80
    81axiom nonlinear_component_correction {Comp B H : Type} [Fintype B] [Fintype H]
    82 (F : CrossForms Comp B H) (e : Comp) (z : Fin 2)
    83 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    84 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    85 (ε : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T)
    86 (β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary)
    87 {K R : ℕ} (hS : Module.finrank Binary S ≤ K) (hT : Module.finrank Binary T ≤ K)
    88 (hplus : Module.finrank Binary (LinearMap.range ((F.forward e).compl₁₂
    89 (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ R)
    90 (hminus : Module.finrank Binary (LinearMap.range ((F.reverse e).flip.compl₁₂
    91 (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ R)
    92 (hδ : Module.finrank Binary (LinearMap.range δ) ≤ R)
    93 (hε : Module.finrank Binary (LinearMap.range ε) ≤ R)
    94 (hcapacity : Module.finrank Binary (LinearMap.range β) + 2 * K + 4 * R ≤ Fintype.card H) :
    95 ∃ δ' ε', componentResponse F e z D E S T δ' ε' = componentResponse F e z D E S T δ ε ∧
    96 extraProduct D E S T δ' ε' = β + extraProduct D E S T δ ε
    97
    98axiom paper_channel_capacity {r J components Dquo K R h rank : ℕ}
    99 (hrank : rank ≤ 30 * r + 28 * J * components + Dquo)
    100 (hmargin : 28 * J * components + Dquo + 2 * K + 4 * R < 970 * r)
    101 (hchannels : 1000 * r ≤ h) : rank + 2 * K + 4 * R ≤ h
    102
    103axiom ambient_add_expansion {B H : Type} [Fintype B] [Fintype H]
    104 (f g df dg : (B → Binary) →ₗ[Binary] (H → Binary)) :
    105 Lax342547.TensorContractions.ambient (f + df) (g + dg) =
    106 Lax342547.TensorContractions.ambient f g + Lax342547.TensorContractions.ambient f dg +
    107 Lax342547.TensorContractions.ambient df g + Lax342547.TensorContractions.ambient df dg
    108
    109end Lax342547.NonlinearChannels
    110
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