# Verification and scope report

Manuscript: A Sharp Three-Cycle Bound for Two-Zone Affine Systems with a
Virtual Focus. Author: Alper Ferudun, Mercury Software GmbH.
Version 1.0, 10 October 2026. Language: English.

## Mathematical scope

The canonical systems have d,e>0, t²<4d, z²<4e, actz nonzero, and a>0 or
c<0. There are at most three isolated crossing cycles, including
nonhyperbolic cycles, for every real offset b. Known Huan-Yang/Llibre-Ponce
examples establish sharpness. The theorem does not include sliding cycles,
non-focus configurations, or all entries of Gasull Problem 17. It does not
increment closure of the combined frozen record AMR-046-0017.

## What was checked

The written proof was re-derived and subjected to an originating-researcher
self-audit covering: positive quadratic normalization; common quadrant;
the no-lost-fiber Cramer argument; unique contact-curve interval; the
Jacobian sign certificate; mirrored contact fields; conic component counts
before and after clipping; separating half-map endpoints; energy sign;
generic-parameter regularity; conic degeneracy; multiple-zero unfolding;
zero/negative offsets; and exchange of the two subsystems.

The algebra checkers validate 27 geometry identities/controls and 20
monotonicity/control conditions, supplemented by 522 applicable saved exact
rational regressions and the inherited 27 Fraction-Sturm/identity checks.
They use exceptions rather than removable assertions. Normal and optimized
outputs are compared byte-for-byte, including from the portable source
directory. These checks are supporting calculations, not formal validation
of the analytic and topological reasoning.

The manuscript was compiled successfully with the desktop LaTeX compiler
and exported with Tectonic. The eight rendered pages were visually reviewed
for equations, references, typography and pagination. The author-only
heading, institutional footnote and prose AI disclosure follow the author's
instructions. Source hashes and ZIP membership are recorded in the manifests.

## Review and attribution limits

The proof is accepted internally for exactly its stated scope. It is not
independently peer reviewed or proof-assistant verified. No absolute
priority or exhaustive novelty certification is claimed. The half-map
framework and extended Khovanskii theorem are credited to the cited prior
work; so are the endpoint-sum sign and the sharpness example. The detailed
source comparison is included in source_and_novelty_review.md.

This report establishes local package checks only. A local file, reserved
identifier or drafted notice does not prove a public deposit, active DOI
resolver, site deployment, Hugging Face posting or search-engine indexing.