# Verification report for Minimal Geodesics under Finite Covers

The accepted scope is a complete counterexample family to the prescribed
simple-projection clause of Theorem 2.5 in Contreras and Mazzucchelli,
DOI 10.1112/plms.70085, and a sufficient descent proposition. The
primitive projected curves have exactly m double points for every m>=1.
The examples work with both manifolds orientable for every n>=3.
The repair proves equality of averaged-action infima under a finite
cover when the minimizing class is pulled back from the base. Its
dynamical corollary uses the credited non-cover Theorem A.

The source's exact minimality definition, theorem and simplicity convention
were checked in the PDF and the current publisher text. Six archived
source PDF hashes match their acquisition manifest. The proof audit
checks freeness, orientation, global action minimality, primitive period,
collision multiplicities, cohomology transport and first variation.

The standard-library checker check_cover.py passed 30,744 exact rational
assertions for 125 family members in each of two Python runtimes, 3.13
and 3.12, with identical output hashes and no failures. It supports the
explicit formulas but does not replace the written general proof.
A runtime replay is not independent mathematical review.

The native desktop LaTeX compiler accepted the manuscript. The public PDF
was exported using already-installed cached Tectonic. All four rendered
pages were inspected; no clipped text, overlaps, broken mathematics or
missing references were observed. The final log contains no overfull
or underfull boxes, undefined references or compilation warnings.

The original AIM entropy-density question remains unresolved here and
the frozen record closure increment is zero. The cited Theorem A and
Corollary B are not refuted; its general Corollary 2.6 is not decided.
The added descent condition is sufficient, not claimed necessary.
The elementary ingredients are standard, and a bounded literature
search establishes no absolute priority guarantee.

The originating researcher accepted the exact stated scope after detailed
self-audit. This AI-assisted preprint is unrefereed. No independent
human review or proof-assistant verification has occurred. DOI assignment
or website publication does not alter those qualifications.