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Allowed channel changes and the actual table injection tests

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

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    Natural Language Statement

    Lemma

    Channel changes factor through a frozen nominal quotient and take values orthogonal to the opposite protected channel. They preserve the frozen entries and vanish on opposite primal inputs. Orthogonal channel tests and the actual table injection flags force a pinned primal vector to zero. DerivativeResponsesDerivativeResponses derives these tests from the full derivative response.

    Concept map
    67 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 correctionsAllowed 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 kernelExact 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 columnsThe 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 basecoordinatesPrimal 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 quotientspacesActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawCoupled 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 labelsRank 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 channel_entry proven

    2 column_injection_zero proven

    3 frozen proven

    4 opposite_primal proven

    5 perpendicular_detects proven

    6 pin_frozen proven

    7 row_injection_zero proven

    Lean source view on GitHub

    1import Lax342547.PrimalContractions
    2import Lax342547.TableContractions
    3import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
    4
    5/-!
    6---
    7title: Allowed channel changes and the actual table injection tests
    8type: lemma
    9---
    10Channel changes factor through a frozen nominal quotient and take values
    11orthogonal to the opposite protected channel. They preserve the frozen
    12entries and vanish on opposite primal inputs. Orthogonal channel tests
    13and the actual table injection flags force a pinned primal vector to zero.
    14`DerivativeResponses` derives these tests from the full derivative response.
    15-/
    16
    17namespace Lax342547.ChannelChanges
    18
    19open Lax342547.MomentSpace Lax342547.TableSpaces Lax342547.PairedAnnihilators
    20open Lax342547.PrimalContractions Lax342547.ExactPins Lax342547.ProjectedPins
    21open Lax342547.SmallTables Lax342547.TableContractions
    22open Lax342547.TensorAnnihilators
    23
    24def perpendicular {H : Type} [Fintype H] (S : Submodule Binary (H → Binary)) :
    25 Submodule Binary (H → Binary) :=
    26 LinearMap.BilinForm.orthogonal (dotProductBilin Binary Binary) S
    27
    28def channelChange {B H : Type} [Fintype H]
    29 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    30 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (z : Fin 2) :
    31 LinearMap.BilinForm Binary (Nominal B H) :=
    32 (dotProductBilin Binary Binary).compl₁₂
    33 (((perpendicular S).subtype.comp δ).comp D.mkQ) (channelProjection z)
    34
    35axiom frozen {B H : Type} [Fintype H]
    36 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    37 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (z : Fin 2)
    38 (v w : Nominal B H) (h : v ∈ D ∨ channelProjection z w ∈ S) :
    39 channelChange D S δ z v w = 0
    40
    41axiom opposite_primal {B H : Type} [Fintype H]
    42 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    43 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (z j : Fin 2)
    44 (v : Nominal B H) (w : B → Binary) :
    45 channelChange D S δ z v (primalEmbedding j w) = 0
    46
    47axiom pin_frozen {Comp B H N : Type} [Fintype H]
    48 (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    49 (D : Submodule Binary (Nominal B H)) (a : Comp × Bool) (z : Fin 2)
    50 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular (protectedChannel P z a))
    51 (v w : Nominal B H) (hw : w ∈ P.space a) :
    52 channelChange D (protectedChannel P z a) δ z v w = 0
    53
    54axiom perpendicular_detects {H : Type} [Fintype H]
    55 (S : Submodule Binary (H → Binary)) (v : H → Binary)
    56 (h : ∀ c ∈ perpendicular S, dotProduct v c = 0) : v ∈ S
    57
    58axiom row_injection_zero {Comp B H N : Type} [Fintype H]
    59 {P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N}
    60 (T : Table P Q) (hT : Injecting T) (F : CrossForms Comp B H) (hF : Extends F T)
    61 (e : Comp) (i z : Fin 2) (d : pinnedPrimal P i (e, true))
    62 (h : ∀ c ∈ perpendicular (protectedChannel Q z (e, true)),
    63 dotProduct (fullPlus F e i z d.val) c = 0) : d.val = 0
    64
    65axiom column_injection_zero {Comp B H N : Type} [Fintype H]
    66 {P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N}
    67 (T : Table P Q) (hT : Injecting T) (F : CrossForms Comp B H) (hF : Extends F T)
    68 (e : Comp) (i z : Fin 2) (d : pinnedPrimal P i (e, false))
    69 (h : ∀ c ∈ perpendicular (protectedChannel Q z (e, false)),
    70 dotProduct (fullMinus F e i z d.val) c = 0) : d.val = 0
    71
    72axiom channel_entry {B H : Type} [Fintype H]
    73 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    74 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (z : Fin 2)
    75 (v : Nominal B H) (h : H) : by
    76 classical
    77 exact channelChange D S δ z v (channelEmbedding z (Pi.single h 1)) =
    78 (δ (D.mkQ v)).val h
    79
    80end Lax342547.ChannelChanges
    81
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