# Verification report for the signed cycle criterion

The manuscript proves a necessary and sufficient CAT(0) criterion for
a -> a^epsilon and b_i -> a^{p_i} b_{sigma(i)}^{delta_i} a^{q_i}.
It constructs a compact two-dimensional locally CAT(0) model in the
positive cases and proves a proper-semisimple-action obstruction otherwise.
The full AIM free-by-cyclic characterization remains open.

The proof checks the cycle Tietze elimination, the exact square-root
relation including epsilon^l, the Klein-bottle orthogonality condition,
equal translation lengths, local-isometry attachments including
degree-two circle maps, and convex gluing in the universal cover.
The construction is for the original group, not only a finite-index
subgroup. The positive-power comparison is a separate cross-check
and gives power-invariance only within this family.

The included standard-library Python program tests 10,423 automorphisms
using exact integer and rational arithmetic and makes 254,789 checks.
It enumerates all one- and two-generator auxiliary parameter choices in
[-2,2], all corresponding signs/permutations, 320 seeded cases in higher
rank and three examples. Literal freely reduced substitutions are
compared with the compressed recurrence, inverse maps and the known
unsigned criterion after taking a positive power. Normal and optimized
Python runs agree exactly. These finite checks are not formal proof,
and they do not computationally certify CAT(0) geometry.

The six-page English source compiled with the built-in LaTeX compiler
and Tectonic. All six final pages were visually inspected, including
the theorem kept on one page, formulas, author footnote and references.
The release procedure additionally checks a clean-directory source-ZIP
build and portable execution of the supplied checker before readiness.

The originating researcher accepts the theorem after written self-audit.
No external reviewer or proof assistant has certified it. Wu--Ye's
unsigned case and the classical geometric machinery remain attributed.
Bounded literature comparison did not identify the full signed-cycle
formula in the checked sources; this is not an exhaustive novelty or
absolute-priority certificate. AI assistance is disclosed in the paper.