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Sparse residual contractions belong to the actual primal pins

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

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    Natural Language Statement

    Lemma

    Pure nominal quotient responses put a tested individual contraction in the barred space. Unselected sparse labels eliminate its key and private terms, so the contraction lies in the actual primal pin intersection. DerivativeResponsesDerivativeResponses pairs this membership with the actual derivative tests and table injection flags in both reciprocal modes.

    Concept map
    66 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 correctionsConcrete 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 4 statements. Each proof establishes one of them relative to its assumptions.

    1 endpoint_vector proven

    2 sparse_contraction proven

    3 unselected_pin_contraction proven

    4 unselected_pin_contraction_mode proven

    Lean source view on GitHub

    1import Lax342547.BarredElimination
    2import Lax342547.TensorBlocks
    3
    4/-!
    5---
    6title: Sparse residual contractions belong to the actual primal pins
    7type: lemma
    8---
    9Pure nominal quotient responses put a tested individual contraction in
    10the barred space. Unselected sparse labels eliminate its key and private
    11terms, so the contraction lies in the actual primal pin intersection.
    12`DerivativeResponses` pairs this membership with the actual derivative
    13tests and table injection flags in both reciprocal modes.
    14-/
    15
    16namespace Lax342547.PrimalContractions
    17
    18open Lax342547.MomentSpace Lax342547.TableSpaces Lax342547.PairedAnnihilators
    19open Lax342547.TensorAnnihilators Lax342547.ConcreteGeometry Lax342547.ProjectedPins
    20open Lax342547.TagGeometry Lax342547.ConcreteCut Lax342547.CutProfiles
    21open Lax342547.PairedWitnesses Lax342547.ExactPins Lax342547.SparsePins
    22open Lax342547.PinLabelExclusions Lax342547.BarredSpaces Lax342547.ResidualLabels
    23
    24noncomputable def coordinates {B : Type} (ψ : Module.Dual Binary (B → Binary)) : B → Binary := by
    25 classical
    26 exact dualCoordinates ψ
    27
    28axiom endpoint_vector {B H : Type} [Fintype B]
    29 (M : Fin 2 → Matrix B B Binary) (D E : Submodule Binary (Nominal B H))
    30 (h : PairedAnnihilates M D E) (i : Fin 2)
    31 (T : Submodule Binary (B → Binary)) (hE : E.map (primalProjection i) ≤ T)
    32 (ψ : Module.Dual Binary (B → Binary)) (hψ : T ≤ LinearMap.ker ψ) :
    33 primalEmbedding i ((M i).mulVecLin (coordinates ψ)) ∈ D
    34
    35axiom sparse_contraction {Label Coord Base : Type} [Fintype Coord] [Fintype Base]
    36 (p : Label → Coord → Binary) (L : Finset Label)
    37 (Z : Label → Matrix Base Base Binary)
    38 (v : Coord × Base → Binary) :
    39 (∑ s ∈ L, Lax342547.TensorBlocks.block p s (Z s)).mulVecLin v =
    40 ∑ s ∈ L, Lax342547.SparsePins.labelTensor p s
    41 (fun a => ∑ c, p s c.1 * Z s a c.2 * v c)
    42
    43axiom unselected_pin_contraction {k n b degree r R : ℕ} {hr : 2 * r ≤ n}
    44 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    45 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    46 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R)
    47 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    48 (U V : Fin 2 → Submodule Binary (H → Binary)) (e : Component (Tag k))
    49 (hU : ∀ j, IsCompl (protectedChannel P j (e, true)) (U j))
    50 (M : Fin 2 → Moment k n b degree)
    51 (h : PairedAnnihilates M (barred W P U (e, true)) (barred W P V (e, false)))
    52 (i : Fin 2) (L : Finset (Fin b → Binary)) (hL : L.card ≤ R)
    53 (hdisjoint : Disjoint L (chosenLabels W i))
    54 (Z : (Fin b → Binary) → Matrix (Option (Base k n)) (Option (Base k n)) Binary)
    55 (hM : M i = ∑ s ∈ L, Lax342547.TensorBlocks.block selectorEval s (Z s))
    56 (ψ : Module.Dual Binary (Vector k n b degree))
    57 (hψ : projected P i (e, false) ⊔ individualKeys W i e ≤ LinearMap.ker ψ) :
    58 (M i).mulVecLin (coordinates ψ) ∈ pinnedPrimal P i (e, true)
    59
    60axiom unselected_pin_contraction_mode {k n b degree r R : 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 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R)
    64 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    65 (p : Bool) (U V : Fin 2 → Submodule Binary (H → Binary)) (e : Component (Tag k))
    66 (hU : ∀ j, IsCompl (protectedChannel P j (e, p)) (U j))
    67 (M : Fin 2 → Moment k n b degree)
    68 (h : PairedAnnihilates M (barred W P U (e, p)) (barred W P V (e, !p)))
    69 (i : Fin 2) (L : Finset (Fin b → Binary)) (hL : L.card ≤ R)
    70 (hdisjoint : Disjoint L (chosenLabels W i))
    71 (Z : (Fin b → Binary) → Matrix (Option (Base k n)) (Option (Base k n)) Binary)
    72 (hM : M i = ∑ s ∈ L, Lax342547.TensorBlocks.block selectorEval s (Z s))
    73 (ψ : Module.Dual Binary (Vector k n b degree))
    74 (hψ : projected P i (e, !p) ⊔ individualKeys W i e ≤ LinearMap.ker ψ) :
    75 (M i).mulVecLin (coordinates ψ) ∈ pinnedPrimal P i (e, p)
    76
    77end Lax342547.PrimalContractions
    78
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