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Endpoint projections of paired pure-response annihilators

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

proven

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    Natural Language Statement

    Lemma

    A shared bilinear response on the two nominal quotient spaces still tests each endpoint separately: project onto its primal block before choosing the bilinear form. This is the first step of the §6 obstruction argument, with channel and mixed pin coordinates retained.

    Concept map
    15 concepts
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    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 endpoint_projection proven

    2 endpoint_rank proven

    Lean source view on GitHub

    1import Lax342547.TensorAnnihilators
    2import Lax342547.TableSpaces
    3
    4/-!
    5---
    6title: Endpoint projections of paired pure-response annihilators
    7type: lemma
    8---
    9A shared bilinear response on the two nominal quotient spaces still
    10tests each endpoint separately: project onto its primal block before
    11choosing the bilinear form. This is the first step of the §6 obstruction
    12argument, with channel and mixed pin coordinates retained.
    13-/
    14
    15namespace Lax342547.PairedAnnihilators
    16
    17open Lax342547.MomentSpace Lax342547.ConcreteGeometry
    18open Lax342547.ProjectedPins Lax342547.TableSpaces Lax342547.TensorAnnihilators
    19
    20abbrev Nominal (B H : Type) := (Fin 2 × (B ⊕ H)) → Binary
    21
    22noncomputable def pureResponse {B H : Type} [Fintype B]
    23 (D E : Submodule Binary (Nominal B H))
    24 (β : (Nominal B H ⧸ D) →ₗ[Binary] (Nominal B H ⧸ E) →ₗ[Binary] Binary) :
    25 (Fin 2 → Matrix B B Binary) →ₗ[Binary] Binary := by
    26 classical
    27 exact ∑ i, (matrixPair (LinearMap.BilinForm.toMatrix'
    28 (β.compl₁₂ (D.mkQ.comp (primalEmbedding i))
    29 (E.mkQ.comp (primalEmbedding i))))).comp (LinearMap.proj i)
    30
    31def PairedAnnihilates {B H : Type} [Fintype B]
    32 (M : Fin 2 → Matrix B B Binary) (D E : Submodule Binary (Nominal B H)) : Prop :=
    33 ∀ β, pureResponse D E β M = 0
    34
    35axiom endpoint_projection {B H : Type} [Fintype B]
    36 (M : Fin 2 → Matrix B B Binary) (D E : Submodule Binary (Nominal B H))
    37 (h : PairedAnnihilates M D E) (i : Fin 2)
    38 (S T : Submodule Binary (B → Binary))
    39 (hD : D.map (primalProjection i) ≤ S) (hE : E.map (primalProjection i) ≤ T) :
    40 PureAnnihilates (M i) S T
    41
    42axiom endpoint_rank {B H : Type} [Fintype B] [DecidableEq B]
    43 (M : Fin 2 → Matrix B B Binary) (D E : Submodule Binary (Nominal B H))
    44 (h : PairedAnnihilates M D E) (i : Fin 2)
    45 (S T : Submodule Binary (B → Binary))
    46 (hD : D.map (primalProjection i) ≤ S) (hE : E.map (primalProjection i) ≤ T) :
    47 (M i).rank ≤ Module.finrank Binary S + Module.finrank Binary T
    48
    49end Lax342547.PairedAnnihilators
    50
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