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Common star atoms eliminate the diagonal predictor coefficient

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

proven

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    Natural Language Statement

    Lemma

    Actual equal shared atoms sum to zero across the odd tag star; a linear cut functional constant on those atoms must have constant zero.

    Concept map
    10 concepts; 5 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 common_atom_constant_zero proven

    2 common_atom_sum_zero proven

    Lean source view on GitHub

    1import Lax342547.Atoms
    2
    3/-!
    4---
    5title: Common star atoms eliminate the diagonal predictor coefficient
    6type: lemma
    7---
    8Actual equal shared atoms sum to zero across the odd tag star; a linear cut functional constant on those atoms must have constant zero.
    9-/
    10
    11namespace Lax342547.CommonAtoms
    12
    13open Lax342547.MomentSpace Lax342547.Atoms Lax342547.TagGeometry Lax342547.ConcreteCut Lax342547.ConcreteGeometry
    14open Lax342547.CutProfiles
    15open scoped BigOperators
    16
    17axiom common_atom_sum_zero {k n b degree : ℕ} (s : Fin b → Binary) (z : Base k n → Binary)
    18 (hz : ∀ l i, i ∉ allowedBase k n l → z i = 0) :
    19 (∑ l : Tag k, atom (degree := degree) l s z (hz l)) = 0
    20
    21axiom common_atom_constant_zero {k n b degree : ℕ} (s : Fin b → Binary) (z : Base k n → Binary)
    22 (hz : ∀ l i, i ∉ allowedBase k n l → z i = 0)
    23 (φ : Module.Dual Binary (Profile k n b degree)) (γ : Binary)
    24 (heq : ∀ l, φ (atom (degree := degree) l s z (hz l)) = γ) : γ = 0
    25
    26end Lax342547.CommonAtoms
    27
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