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Sparse unselected primal vectors in barred spaces lie in the pins

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

proven

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    Natural Language Statement

    Theorem

    Fresh selected keys cannot appear in a pin decomposition of an unselected sparse vector. Once the keys vanish, protected/private channel complements force the private part to vanish as well. This establishes the pin membership needed before testing the derivative table maps.

    Concept map
    65 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 basecoefficientsRetractions 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 3 statements. Each proof establishes one of them relative to its assumptions.

    1 individual_keys_zero proven

    2 private_primal_pin proven

    3 sparse_barred proven

    Lean source view on GitHub

    1import Lax342547.ResidualLabels
    2import Lax342547.SparseKeyCoefficients
    3
    4/-!
    5---
    6title: Sparse unselected primal vectors in barred spaces lie in the pins
    7type: theorem
    8---
    9Fresh selected keys cannot appear in a pin decomposition of an unselected
    10sparse vector. Once the keys vanish, protected/private channel
    11complements force the private part to vanish as well. This establishes
    12the pin membership needed before testing the derivative table maps.
    13-/
    14
    15namespace Lax342547.BarredElimination
    16
    17open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    18open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    19open Lax342547.ExactPins Lax342547.ProjectedPins Lax342547.TableSpaces
    20open Lax342547.PinLabelExclusions Lax342547.BarredSpaces Lax342547.ResidualLabels
    21open Lax342547.NominalPrimal Lax342547.SparsePins
    22
    23variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    24
    25axiom individual_keys_zero {R : ℕ} (W : Lists k n b degree r hr)
    26 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    27 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R)
    28 (i : Fin 2) (a : Component (Tag k) × Bool) (L : Finset (Fin b → Binary))
    29 (hL : L.card ≤ R) (v : (Fin b → Binary) → Option (Base k n) → Binary)
    30 (hdisjoint : Disjoint L (chosenLabels W i))
    31 (hfresh : ∀ z t, (W.left i z t).label ∉ A i)
    32 (q : Vector k n b degree) (hq : q ∈ individualKeys W i a.1)
    33 (hpin : (∑ s ∈ L, labelTensor (selectorEval (degree := degree)) s (v s)) - q ∈ projected P i a) :
    34 q = 0
    35
    36axiom private_primal_pin {Comp B H N : Type}
    37 (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    38 (U : Fin 2 → Submodule Binary (H → Binary)) (a : Comp × Bool)
    39 (hU : ∀ i, IsCompl (protectedChannel P i a) (U i))
    40 (v : (Fin 2 × (B ⊕ H)) → Binary) (hv : v ∈ primal)
    41 (h : v ∈ P.space a ⊔ ⨆ i, (U i).map (channelEmbedding i)) :
    42 v ∈ P.space a
    43
    44axiom sparse_barred {R : ℕ} [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 : Fin 2 → Submodule Binary (H → Binary)) (a : Component (Tag k) × Bool)
    49 (hU : ∀ i, IsCompl (protectedChannel P i a) (U i))
    50 (L : Fin 2 → Finset (Fin b → Binary)) (hL : ∀ i, (L i).card ≤ R)
    51 (hdisjoint : ∀ i, Disjoint (L i) (chosenLabels W i))
    52 (c : Fin 2 → (Fin b → Binary) → Option (Base k n) → Binary)
    53 (v : (Fin 2 × (Coordinate k n b degree ⊕ H)) → Binary) (hv : v ∈ primal)
    54 (hvec : ∀ i, primalProjection i v = ∑ s ∈ L i, labelTensor selectorEval s (c i s))
    55 (h : v ∈ barred W P U a) : v ∈ P.space a
    56
    57end Lax342547.BarredElimination
    58
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