For every finite connected simple graph, we prove that the natural actions of its automorphism group on the finite critical group and on real first homology have the same kernel. The proof uses the cut-flow presentation and the absence of integral flows of squared norm at most two, with a separate coordinate argument for reversed edges. Combining this equality with known homological faithfulness yields a faithful critical-group action for every connected simple graph of minimum degree at least two other than a cycle. In particular, semiregularity is unnecessary for simple three-connected graphs. Parallel-edge and pendant-tree examples explain the hypotheses. This is a complete scoped theorem within AIM-COMBINATORICS-0131, not closure of the open-ended AIM research suggestion. The preprint is AI-assisted, self-audited and unrefereed.
