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Theorem · finite sewing

QA-SEW-T1

Finite abelian sewing by cycle circulation

Exact statement

An edge receipt on a finite connected graph is killed by vertex corrections exactly when its circulation vanishes on a cycle-space basis.

StatusProved and verifier-backed
External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

The general calculus treats a failed lift as a cocycle, its cohomology class as the obstruction, and a primitive—when one exists—as the receipt that repairs the presentation.

Proof or evidence

A spanning-tree construction gives the potential; independent cycle checks include a nonzero triangle obstruction.

Self-contained cohomological sewing proof and evidence →

Verification notes

The exact archived proofs were reconstructed, the displayed hypotheses and degree conventions were checked, the original finite controls were replayed, and an independent implementation tested the connecting, graph-transport, staged, and missing-subcomplex boundaries.

Limitations

  • The coefficient system is abelian and transport must be included when it is nontrivial.

Open work

Use this as the degree-one control case for every new sewing doctrine.