# Verification report

## Result

For every integer `m >= 2`, the manuscript gives a mapping torus `G_m` of a
linearly growing UPG automorphism of `F_(2m+1)` such that:

- `b_1(G_m) = m+2`;
- `Sigma^1(G_m)` is the complement of `x+i y=0`, `0 <= i <= m`;
- there are exactly `2m+2` connected components; and
- there are at least `m+1` component orbits under the full `Out(G_m)` action.

## Proof gates checked

1. The graph has free rank `2m+1`, and the suffix-reversing graph map is an
   explicit homotopy inverse.
2. The upper-triangular representative is UPG and genuinely linearly growing.
3. All mapping-torus square signs and graph-of-groups relations were checked.
4. Abelianization leaves exactly the coordinates
   `(x,y,z_0,z_2,...,z_m)` and gives `b_1=m+2`.
5. The graph of groups is finite, reduced, and nonascending, with `Z^2`
   vertex groups; the hypotheses of Cashen--Levitt Corollary 2.10 match.
6. Every edge and multiplicity is present in both the BNS and fiber-rank
   calculations.
7. The complement of `m+1` distinct central lines has exactly `2m+2`
   sectors, and the unrestricted stable-letter coordinates cannot join them.
8. Bass--Serre double-coset counting independently gives
   `rank ker(chi)=1+m|x|+sum_(i=1)^m |x+i y|`.
9. Pullback by any automorphism preserves primitive characters, components,
   and kernel isomorphism types, so the minimum kernel rank is invariant under
   the full outer automorphism group.
10. Integer minimization gives `1+2m` on the outer antipodal pair and
    `1+m(m+1)+2j^2` on the `j`th interior antipodal pair.

No critical proof gap survived the adversarial audit.  The exact checker
row-reduces the abelian relation matrix and enumerates every chamber minimum
for `2 <= m <= 30`; the general theorem itself is symbolic.

## Scope and novelty

The proof resolves the AIM request under its strongest natural unbounded
family interpretation.  It does not compute the exact orbit count, decide
whether antipodal members are exchanged, or construct a hyperbolic or fully
irreducible family.

Cashen--Levitt contains the graph-of-groups machinery and the closest
cycle-of-tori family. A final search located an anonymous public candidate,
released on 2026-08-31 with DOI `10.5281/zenodo.22201487`, containing the same
weighted-chord construction and conclusions. It is now cited explicitly.
No novelty or public-priority claim is made for this construction.

## Artifact status

- Mathematical dossier: complete
- Exact checker: passing
- Literature and direct-prior audits: updated through 2026-09-02
- LaTeX/BibTeX build: PASS, six letter-size pages, zero warning classes
- Exact checker: PASS for `2 <= m <= 30`
- Direct 144-DPI visual inspection: PASS on all six pages
- Upload archive: PASS; English paper sources and reproducibility files only
- Public release: approved by the author on 2026-09-02 under CC BY 4.0; the manuscript remains unrefereed

## Public release and license

The author approved public release on 2026-09-02. This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/