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Actual paired-witness key spaces satisfy the baseline hypotheses

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

proven

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    Natural Language Statement

    Lemma

    At a component keep precisely the nominal primal rays belonging to incident selected atoms. Their span has dimension at most twenty-eight, is contained in the combined primal space, and is disjoint from every table space when the labels avoid the sparse exclusions.

    Concept map
    45 concepts
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsRank 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 spacesBoolean point moments with restricted basecoordinatesPrimal blocks of the nominal spaces andtheir bounded table partActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesPrimal projections and preservation ofeffective spacesActual 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 eventsNumerical cross tables, injection flags, andunary admissibilityA 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 taggeometryWell-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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 fresh_spans proven

    2 key_bounds proven

    Lean source view on GitHub

    1import Lax342547.PairedRecipes
    2import Lax342547.KeySpans
    3
    4/-!
    5---
    6title: Actual paired-witness key spaces satisfy the baseline hypotheses
    7type: lemma
    8---
    9At a component keep precisely the nominal primal rays belonging to
    10incident selected atoms. Their span has dimension at most twenty-eight,
    11is contained in the combined primal space, and is disjoint from every
    12table space when the labels avoid the sparse exclusions.
    13-/
    14
    15namespace Lax342547.PairedKeys
    16
    17open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    18open Lax342547.CutProfiles Lax342547.PairedWitnesses Lax342547.PairedRecipes
    19open Lax342547.ExactPins Lax342547.TableSpaces Lax342547.NominalPrimal Lax342547.KeySpans
    20
    21variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    22
    23abbrev Positions (W : Lists k n b degree r hr) := Σ i, Σ z, Fin (W.length i z)
    24
    25def labels (W : Lists k n b degree r hr) (i : Fin 2) (x : Σ z, Fin (W.length i z)) :=
    26 (W.left i x.1 x.2).label
    27
    28def bases (W : Lists k n b degree r hr) (i : Fin 2) (x : Σ z, Fin (W.length i z)) :=
    29 (W.left i x.1 x.2).base
    30
    31noncomputable def direction (W : Lists k n b degree r hr) (e : Component (Tag k))
    32 (x : Positions W) : (Fin 2 × (Coordinate k n b degree ⊕ H)) → Binary := by
    33 classical
    34 exact if (W.left x.1 x.2.1 x.2.2).tag ∈ e.val then ray (labels W) (bases W) x else 0
    35
    36def keys (W : Lists k n b degree r hr) (e : Component (Tag k)) :
    37 Submodule Binary ((Fin 2 × (Coordinate k n b degree ⊕ H)) → Binary) :=
    38 Submodule.span Binary (Set.range (direction (H := H) W e))
    39
    40axiom key_bounds [Fintype H] (W : Lists k n b degree r hr) (e : Component (Tag k)) :
    41 keys (H := H) W e ≤ primal ∧ Module.finrank Binary (keys (H := H) W e) ≤ 28
    42
    43axiom fresh_spans [Fintype H]
    44 (W : Lists k n b degree r hr)
    45 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    46 (A B : Fin 2 → Finset (Fin b → Binary)) (hA : Excludes P A) (hB : Excludes Q B)
    47 (hfresh : FreshAgainst W A B) :
    48 (∀ a, Disjoint (tableSpace P a) (keys (H := H) W a.1)) ∧
    49 ∀ a, Disjoint (tableSpace Q a) (keys (H := H) (flip W) a.1)
    50
    51end Lax342547.PairedKeys
    52
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