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Compression on the actual barred nominal quotients

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

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    Natural Language Statement

    Lemma

    Base compression fixes frozen pins, selected keys, and private channels. It descends through the actual barred spaces with the paper quotient-rank bound, and precomposition bounds a pure bilinear form while retaining its response on compressed component matrices.

    Concept map
    80 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 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 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 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 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 8 statements. Each proof establishes one of them relative to its assumptions.

    1 barred_compression proven

    2 barred_fixed proven

    3 compressed_bilinear_rank proven

    4 nominalLift_fix proven

    5 nominalLift_primal proven

    6 nominalLift_sum proven

    7 pure_compressed proven

    8 quotient_lift_rank proven

    Lean source view on GitHub

    1import Lax342547.BaseCompression
    2import Lax342547.ResidualRealization
    3
    4/-!
    5---
    6title: Compression on the actual barred nominal quotients
    7type: lemma
    8---
    9Base compression fixes frozen pins, selected keys, and private channels. It descends through the actual barred spaces with the paper quotient-rank bound, and precomposition bounds a pure bilinear form while retaining its response on compressed component matrices.
    10-/
    11
    12namespace Lax342547.QuotientCompression
    13
    14open Lax342547.MomentSpace Lax342547.PairedAnnihilators
    15open Lax342547.TableSpaces Lax342547.ProjectedPins Lax342547.BaseCompression
    16open Lax342547.TagGeometry Lax342547.ConcreteGeometry Lax342547.ConcreteCut
    17open Lax342547.PairedWitnesses Lax342547.ExactPins Lax342547.BarredSpaces
    18open Lax342547.CutProfiles
    19
    20def nominalLift {B H : Type} (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary)) :
    21 Nominal B H →ₗ[Binary] Nominal B H where
    22 toFun v q := match q.2 with
    23 | Sum.inl b => f q.1 (primalProjection q.1 v) b
    24 | Sum.inr h => v (q.1, Sum.inr h)
    25 map_add' v w := by
    26 ext ⟨i, b | h⟩
    27 · exact congrFun ((f i).map_add _ _) b
    28 · rfl
    29 map_smul' c v := by
    30 ext ⟨i, b | h⟩
    31 · exact congrFun ((f i).map_smul c _) b
    32 · rfl
    33
    34axiom nominalLift_primal {B H : Type}
    35 (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary)) (i : Fin 2) (v : B → Binary) :
    36 nominalLift (H := H) f (primalEmbedding i v) = primalEmbedding i (f i v)
    37
    38axiom nominalLift_fix {B H : Type}
    39 (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary)) (v : Nominal B H)
    40 (h : ∀ i, f i (primalProjection i v) = primalProjection i v) : nominalLift f v = v
    41
    42axiom nominalLift_sum {B H : Type}
    43 (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary)) (v : Nominal B H) :
    44 nominalLift f v = ∑ i, (primalEmbedding i (f i (primalProjection i v)) +
    45 channelEmbedding i (channelProjection i v))
    46
    47axiom barred_fixed {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    48 (W : Lists k n b degree r hr)
    49 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    50 (U : Fin 2 → Submodule Binary (H → Binary)) (a : Component (Tag k) × Bool)
    51 (f : Fin 2 → Vector k n b degree →ₗ[Binary] Vector k n b degree)
    52 (hfix : ∀ i v, v ∈ projected P i a ⊔ individualKeys W i a.1 → f i v = v) :
    53 ∀ v ∈ barred W P U a, nominalLift f v = v
    54
    55axiom quotient_lift_rank {B H : Type} [Fintype B] [Fintype H]
    56 (D : Submodule Binary (Nominal B H))
    57 (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary))
    58 (hD : D ≤ D.comap (nominalLift f))
    59 (S U : Fin 2 → Submodule Binary (H → Binary))
    60 (hcompl : ∀ i, IsCompl (S i) (U i))
    61 (hprivate : ∀ i, (U i).map (channelEmbedding (B := B) i) ≤ D) :
    62 Module.finrank Binary (LinearMap.range (D.mapQ D (nominalLift f) hD)) ≤
    63 ∑ i, (Module.finrank Binary (LinearMap.range (f i)) + Module.finrank Binary (S i))
    64
    65axiom barred_compression {k n b degree r K Dblk : ℕ} {hr : 2 * r ≤ n}
    66 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    67 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    68 (hK : P.rank ≤ K) (a : Component (Tag k) × Bool)
    69 (U : Fin 2 → Submodule Binary (H → Binary))
    70 (hcompl : ∀ i, IsCompl (protectedChannel P i a) (U i))
    71 (A : Fin 2 → Block k → (Fin n → Binary) →ₗ[Binary] (Fin n → Binary))
    72 (hA : ∀ i q, Module.finrank Binary (LinearMap.range (A i q)) ≤ Dblk)
    73 (hfix : ∀ i v, v ∈ projected P i a ⊔ individualKeys W i a.1 →
    74 liftBase (blockMap (A i)) v = v) :
    75 ∃ q : (Nominal (Coordinate k n b degree) H ⧸ barred W P U a) →ₗ[Binary]
    76 (Nominal (Coordinate k n b degree) H ⧸ barred W P U a),
    77 (∀ v, q ((barred W P U a).mkQ v) = (barred W P U a).mkQ
    78 (nominalLift (fun i => liftBase (blockMap (A i))) v)) ∧
    79 Module.finrank Binary (LinearMap.range q) ≤
    80 2 * Fintype.card (SelectorCoordinates b degree) *
    81 (1 + ((2 * k + 1)^2 + 3) * Dblk) + 2 * K
    82
    83axiom compressed_bilinear_rank {A B : Type} [AddCommGroup A] [Module Binary A]
    84 [AddCommGroup B] [Module Binary B] [FiniteDimensional Binary A]
    85 (β : A →ₗ[Binary] B →ₗ[Binary] Binary) (q : A →ₗ[Binary] A) (p : B →ₗ[Binary] B) :
    86 Module.finrank Binary (LinearMap.range (β.compl₁₂ q p)) ≤
    87 Module.finrank Binary (LinearMap.range q)
    88
    89axiom pure_compressed {B H : Type} [Fintype B]
    90 (D E : Submodule Binary (Nominal B H))
    91 (β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary)
    92 (f : Fin 2 → (B → Binary) →ₗ[Binary] (B → Binary))
    93 (qD : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ D))
    94 (qE : (Nominal B H ⧸ E) →ₗ[Binary] (Nominal B H ⧸ E))
    95 (hD : ∀ v, qD (D.mkQ v) = D.mkQ (nominalLift f v))
    96 (hE : ∀ v, qE (E.mkQ v) = E.mkQ (nominalLift f v))
    97 (x : Fin 2 → Matrix B B Binary) :
    98 pureResponse D E (β.compl₁₂ qD qE) x =
    99 pureResponse D E β (fun i => tensorMap (f i) (x i))
    100
    101end Lax342547.QuotientCompression
    102
    Show ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow Proof
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