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Pure obstructions on the actual pair of cut profiles

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

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    Natural Language Statement

    Lemma

    The independent component responses on barred nominal quotients define functionals on the pair of actual cut profiles. Annihilation gives the 2K+28 component-rank bound and hence a bounded label decomposition. Derivative responses and the final obstruction-space reduction remain separate obligations.

    Concept map
    54 concepts
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    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsBarred 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 supportBoolean 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 recipeequationsRetractions 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 cutprofilesRank 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 eventsRank loss under restriction of a bilinear formA selected affine ray determines its momentblockNumerical 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 taggeometryPure bilinear responses detect quotienttensorsWell-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 component_annihilation proven

    2 component_labels proven

    3 component_rank proven

    Lean source view on GitHub

    1import Lax342547.BarredSpaces
    2import Lax342547.LabelDecomposition
    3
    4/-!
    5---
    6title: Pure obstructions on the actual pair of cut profiles
    7type: lemma
    8---
    9The independent component responses on barred nominal quotients define
    10functionals on the pair of actual cut profiles. Annihilation gives the
    112K+28 component-rank bound and hence a bounded label decomposition.
    12Derivative responses and the final obstruction-space reduction remain
    13separate obligations.
    14-/
    15
    16namespace Lax342547.PureObstructions
    17
    18open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    19open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    20open Lax342547.ExactPins Lax342547.PairedAnnihilators Lax342547.BarredSpaces
    21open Lax342547.BaseMoments
    22
    23variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    24
    25abbrev Parameters (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 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary)) :=
    28 ∀ e, (Nominal (Coordinate k n b degree) H ⧸ barred W P (U e) (e, true)) →ₗ[Binary]
    29 (Nominal (Coordinate k n b degree) H ⧸ barred W P (V e) (e, false)) →ₗ[Binary] Binary
    30
    31def endpointComponent (e : Component (Tag k)) :
    32 (Fin 2 → Profile k n b degree) →ₗ[Binary] (Fin 2 → Moment k n b degree) where
    33 toFun x i := (x i).val e
    34 map_add' _ _ := rfl
    35 map_smul' _ _ := rfl
    36
    37noncomputable def response (W : Lists k n b degree r hr)
    38 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    39 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    40 (β : Parameters W P U V) : (Fin 2 → Profile k n b degree) →ₗ[Binary] Binary :=
    41 ∑ e, (pureResponse (barred W P (U e) (e, true))
    42 (barred W P (V e) (e, false)) (β e)).comp (endpointComponent e)
    43
    44def Annihilates (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 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    47 (x : Fin 2 → Profile k n b degree) : Prop := ∀ β, response W P U V β x = 0
    48
    49axiom component_annihilation (W : Lists k n b degree r hr)
    50 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    51 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    52 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P U V x)
    53 (e : Component (Tag k)) :
    54 PairedAnnihilates (endpointComponent e x)
    55 (barred W P (U e) (e, true)) (barred W P (V e) (e, false))
    56
    57axiom component_rank [Fintype H] {K : ℕ} (W : Lists k n b degree r hr)
    58 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    59 (hK : P.rank ≤ K) (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    60 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P U V x) :
    61 ∀ i e, ((x i).val e).rank ≤ 2 * K + 28
    62
    63axiom component_labels [Fintype H] {K : ℕ} (W : Lists k n b degree r hr)
    64 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    65 (hK : P.rank ≤ K) (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    66 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P U V x)
    67 (hdegree : 6 * (2 * K + 28) + 4 ≤ degree) :
    68 ∀ i e, ∃ L : Finset (Fin b → Binary),
    69 ∃ Z : (Fin b → Binary) → Matrix (Option (Base k n)) (Option (Base k n)) Binary,
    70 L.card ≤ 2 * K + 28 ∧ (∀ s ∈ L, IsBaseMoment (Z s)) ∧
    71 (∑ s ∈ L, (Z s).rank) ≤ 2 * K + 28 ∧
    72 ∀ a c, (x i).val e a c = ∑ s ∈ L, selectorEval s a.1 * selectorEval s c.1 * Z s a.2 c.2
    73
    74end Lax342547.PureObstructions
    75
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