# Verification report

Date: 2026-08-30

## Result

`PASS` - the source compiles, the exact checker passes, and all five PDF
pages pass visual review.

## Mathematical audit

- The neutral affine group has translation lattice Lambda direct-sum Lambda,
  is discrete, cocompact, and free.
- The split form has signature (n,n) and is invariant under diagonal
  orthogonal holonomy.
- N(u,v)=(0,u) is nonzero, square-zero, parallel, and self-adjoint.
- Parallel vectors on a flat quotient are precisely holonomy-fixed constant
  vectors; the doubled fixed space is E^H direct-sum E^H=0.
- For an index-two subgroup K, H/ell(K) is an image of Gamma-hat/K. Its index
  divides both 2 and odd |H|, so every connected double cover retains H.
- The explicit four characters are nontrivial and have trivial common
  kernel; the affine freeness table covers all eight nonidentity elements.
- In the rank-one boundary, q(Bx,By)=g(x,By) is well-defined, symmetric,
  nondegenerate, and parallel; orienting the line kills its sign holonomy.

## Reproducibility

Run `python3 reproducibility/check_counterexample.py`. It verifies the
character kernel, all freeness certificates, holonomy parity, split
signature, square-zero identity, and self-adjoint matrix identity.

Finite computation verifies the printed explicit data but does not replace
the symbolic doubling and double-cover proofs.

## Build and PDF QA

- Engine: Tectonic 0.17.0 with BibTeX
- Pages: 5, US Letter (612 x 792 pt)
- Undefined references/citations: 0
- Overfull/underfull boxes: 0
- LaTeX errors: 0
- Visual inspection: all 5 pages at 144 dpi; affine-action table, formulas,
  bibliography, margins, and page numbers are legible and unclipped

## 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/