# Verification report for the figure eight obstruction

The manuscript gives a complete counterexample to the universal
OWR-17471-010 question. Figure-eight meets the hyperbolicity and
no-finite-surgery assumptions. Its zero-filling kernel is G'', and
every rank-two free subgroup has nontrivial H'' contained in that kernel.

## General proof

The standard punctured-torus-bundle presentation identifies the fiber
group with G'. Its commutator subgroup is the normal closure of the
longitude, both in the fiber and in the knot group. This proves the
exact zero-kernel identity. The nontrivial word [[u,v],[u,v^2]] then
obstructs every proposed free subgroup. The source includes the zero
slope; this is not an implicit exceptional-slope restriction.

Five primary PDFs and the HTML version of the recent related preprint
support the source audit. Eleven complete relevant pages were visually
inspected. Known knot-theoretic inputs are explicitly credited.

## Exact diagnostics

The accompanying checker uses integer arithmetic. Normal and optimized
Python executions agree on a length-eighteen freely reduced word, a
nonidentity SL(2,Z) matrix evaluation, 1,457 inverse-monodromy word
checks, 5,461 comparisons of two free-reduction algorithms, and 2,025
metabelian-pair evaluations. The code does not test or establish the
topological input theorems. Finite checks are not the universal proof.

## Limits of the conclusion

The result does not settle existence of a persistent F_2 for some other
hyperbolic knot, a version excluding zero surgery, or a classification
of every persistent subgroup. It is an elementary consequence of
attributed classical facts. A bounded literature search did not locate
the precise counterexample statement but does not certify novelty.

The author used AI assistance. The manuscript is self-audited and
unrefereed; independent expert review and proof-assistant verification
have not occurred. The publication archive includes original proof
notes and diagnostics, not third-party source PDFs.