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Barred response spaces and the actual obstruction rank budget

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

proven

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    Natural Language Statement

    Lemma

    Quotient the nominal spaces by frozen pins, selected incident keys, and private channel directions. Individual primal projections have kernel budget K+14, giving the paper's component obstruction bound 2K+28.

    Concept map
    52 concepts
    100%
    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 spacesBoolean 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 spacesRank 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 annihilator_component_rank proven

    2 barred_projection proven

    3 individual_key_dimension proven

    Lean source view on GitHub

    1import Lax342547.PairedAnnihilators
    2import Lax342547.PairedKeys
    3
    4/-!
    5---
    6title: Barred response spaces and the actual obstruction rank budget
    7type: lemma
    8---
    9Quotient the nominal spaces by frozen pins, selected incident keys, and
    10private channel directions. Individual primal projections have kernel
    11budget K+14, giving the paper's component obstruction bound 2K+28.
    12-/
    13
    14namespace Lax342547.BarredSpaces
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.CutProfiles Lax342547.PairedWitnesses Lax342547.PairedKeys
    18open Lax342547.ExactPins Lax342547.ProjectedPins Lax342547.TableSpaces
    19open Lax342547.PairedAnnihilators
    20
    21variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    22
    23noncomputable def individualDirection (W : Lists k n b degree r hr) (i : Fin 2)
    24 (e : Component (Tag k)) (x : Σ z, Fin (W.length i z)) : Vector k n b degree := by
    25 classical
    26 exact if (W.left i x.1 x.2).tag ∈ e.val then (W.left i x.1 x.2).vector else 0
    27
    28def individualKeys (W : Lists k n b degree r hr) (i : Fin 2) (e : Component (Tag k)) :
    29 Submodule Binary (Vector k n b degree) :=
    30 Submodule.span Binary (Set.range (individualDirection W i e))
    31
    32def barred (W : Lists k n b degree r hr)
    33 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    34 (U : Fin 2 → Submodule Binary (H → Binary)) (a : Component (Tag k) × Bool) :
    35 Submodule Binary (Nominal (Coordinate k n b degree) H) :=
    36 P.space a ⊔ keys (H := H) W a.1 ⊔ ⨆ z, (U z).map (channelEmbedding z)
    37
    38axiom individual_key_dimension (W : Lists k n b degree r hr) (i : Fin 2)
    39 (e : Component (Tag k)) : Module.finrank Binary (individualKeys W i e) ≤ 14
    40
    41axiom barred_projection (W : Lists k n b degree r hr)
    42 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    43 (U : Fin 2 → Submodule Binary (H → Binary)) (a : Component (Tag k) × Bool) (i : Fin 2) :
    44 (barred W P U a).map (primalProjection i) ≤ projected P i a ⊔ individualKeys W i a.1
    45
    46axiom annihilator_component_rank [Fintype H] {K : ℕ}
    47 (W : Lists k n b degree r hr)
    48 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    49 (hK : P.rank ≤ K) (U V : Fin 2 → Submodule Binary (H → Binary))
    50 (e : Component (Tag k)) (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, (M i).rank ≤ 2 * K + 28
    53
    54end Lax342547.BarredSpaces
    55
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