# Verification of the critical group kernel theorem

The written proof establishes equality of the natural critical-group and
real-homology action kernels for every finite connected simple graph.
It uses the standard equivariant cut--flow quotient and a direct short-vector
argument. Edge reversal is treated separately. Faithfulness for minimum
degree at least two outside cycles follows from a credited known homology
theorem, not from the finite experiment.

The originating researcher's self-audit checks equivariance, integral cut
saturation, the squared-norm-three threshold, the reversed-edge coordinate,
the converse implication, and the exact scope of each corollary.
Parallel edges and pendant trees provide negative controls for omitted
hypotheses. No mathematical gap is presently known in the stated theorem.

Exact diagnostics:

- Fraction inverse and VF2: 577 bridgeless simple graph types through seven
  vertices; 10,598 automorphisms; only cycle rotations lie in a nontrivial kernel.
- Independent degree-block enumeration, integer Bareiss/cofactor arithmetic
  and signed cycle bases: 995 connected graph types, 13,286 automorphisms,
  4,879 fundamental cycle-basis vectors; no kernel disagreement.
- Every one of the 577 exploratory graph results was independently reproduced.
- Normal and optimized interpreter runs agree; checks do not rely on assert.
- A three-parallel-edge banana transposition fixes the critical group while
  changing a norm-two flow, confirming that simplicity is necessary.

The attached scripts and reference outputs permit replay. These are finite
computational diagnostics and an originating-researcher mathematical audit,
not independent expert review or a machine-checked formal proof.

The source record remains an open-ended research direction. This manuscript
contains a complete scoped theorem, with no claim to settle every module or
harmonic multigraph-action question in AIM item 41.