# Verification report

Accepted complete counterexample to the universal multiplicative core-volume bound.
Written proof: explicit classical Schottky family, positive volumes tending to zero via Bonahon's theorem, constant systole arcosh(31), distinct proper parity kernels, exact ratio 1/V(G). The paper also proves a finite-cover identity and a thickened-core bound.
Scope limitation: arbitrary C(G) bounds in one fixed ambient group are not settled. Whole-source closure increment: zero.
Known mathematical gaps in stated theorems: none. Originating-researcher self-audit complete. Independent review and formal verification: not performed. Absolute priority: not certified.

Exact supporting checker: symbolic polynomial matrix calculation, six face separations, connected parity covering and ranks 3,3,5, and all 36 subgroup pairs of S_3 for the double-coset identity. These checks are not a computation of hyperbolic volumes and do not replace the general written proofs.
Checker SHA-256: ece5f9f670fc246f5335f629a167e86b2949642945af1268ed966275e9f977d1.

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