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The 3K+28 baseline bound in the actual nominal coordinates

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

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    Natural Language Statement

    Theorem

    For the two stored pin spaces and independent primal key spaces, complete one cross form while retaining all frozen entries and the whole table. Both reciprocal primal-to-channel maps have rank at most 3K+28. The same form satisfies both bounds, so no independent completions are silently used for its two endpoint roles.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.NominalPrimal
    2import Lax342547.SmallTables
    3
    4/-!
    5---
    6title: The 3K+28 baseline bound in the actual nominal coordinates
    7type: theorem
    8---
    9For the two stored pin spaces and independent primal key spaces, complete
    10one cross form while retaining all frozen entries and the whole table.
    11Both reciprocal primal-to-channel maps have rank at most 3K+28. The same
    12form satisfies both bounds, so no independent completions are silently
    13used for its two endpoint roles.
    14-/
    15
    16namespace Lax342547.NominalBaselines
    17
    18open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.TableSpaces
    19open Lax342547.SmallTables Lax342547.NominalPrimal Lax342547.FrozenBaselines
    20
    21axiom nominal_baseline {Comp B H N : Type} [Fintype Comp] [Fintype B] [Fintype H]
    22 (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N) (a b : Comp × Bool)
    23 (Keys Keys' : Submodule Binary ((Fin 2 × (B ⊕ H)) → Binary)) (K : ℕ)
    24 (hP : P.rank ≤ K) (hQ : Q.rank ≤ K)
    25 (hkey : Keys ≤ primal) (hkey' : Keys' ≤ primal)
    26 (hcard : Module.finrank Binary Keys ≤ 28) (hcard' : Module.finrank Binary Keys' ≤ 28)
    27 (hfresh : Disjoint (tableSpace P a) Keys) (hfresh' : Disjoint (tableSpace Q b) Keys')
    28 (A : ((Fin 2 × (B ⊕ H)) → Binary) →ₗ[Binary]
    29 ((Fin 2 × (B ⊕ H)) → Binary) →ₗ[Binary] Binary)
    30 (T : tableSpace P a →ₗ[Binary] tableSpace Q b →ₗ[Binary] Binary)
    31 (h : Compatible (tableSpace P a) (P.space a) (tableSpace Q b) (Q.space b) A T) :
    32 ∃ F : ((Fin 2 × (B ⊕ H)) → Binary) →ₗ[Binary]
    33 ((Fin 2 × (B ⊕ H)) → Binary) →ₗ[Binary] Binary,
    34 ExtendsFrozen (P.space a) Keys (Q.space b) Keys' A F ∧
    35 (∀ v : tableSpace P a, ∀ w : tableSpace Q b, F v.val w.val = T v w) ∧
    36 ∀ z : Fin 2,
    37 Module.finrank Binary (LinearMap.range
    38 (F.compl₁₂ (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ 3 * K + 28 ∧
    39 Module.finrank Binary (LinearMap.range
    40 (F.flip.compl₁₂ (primal (B := B) (H := H)).subtype (channelEmbedding z))) ≤ 3 * K + 28
    41
    42end Lax342547.NominalBaselines
    43
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