Equivariant matching-defect criterion

Exact statement

For the $2P_3$ chamber, two perfect matchings $P,Q$ define an equivariant defect in $\mathbb F_2^2$ , and the synchronization sews exactly when that defect vanishes.

StatusProved and independently reproduced

External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

Anonymous overlap types can all agree while the cards still fail to share one coherent naming. The smallest computed witness converts that mismatch into an equivariant matching defect.

Proof or evidence

All 18 synchronizations and all 864 actions of $S_2\times S_4$ pass; the zero defect has size six and the three nonzero values have size four each.

Graph-deck synchronization proof and independent audit →

Verification notes

An independent implementation reconstructed all 18 type-slot synchronizations, all 73,728 candidate row namings, the two symmetry orbits, all 864 S2 x S4 equivariance actions, and the 52-graph marked-orbit control through order five.

Limitations

The defect is proved for this chamber, not yet a universal graph-deck cohomology theory.

Open work

Generalize the matching module to the other fifteen order-six multi-chamber graphs.