# Verification report

## Result

The source gives a finite necessary-and-sufficient angle-data closure test
and proves that the actual combined angle image of a genus-`g` triangulated
polyhedral surface has dimension `3V-7 = E-1-6g`. The AIM conjecture is true
for `g=0` and false for every positive genus.

## Gates checked

1. The official AIM problem and the workshop report's genus clarification
   were inspected.
2. Hempel's own Definition 3 was checked: simplexwise linear and simplexwise
   injective, without global embedding.
3. The free seven-dimensional similarity action and normalized root-face
   slice were audited.
4. The dual-tree reconstruction was checked separately for new and repeated
   third vertices.
5. Correctness on the actual image, rather than on arbitrary inconsistent
   nearby inputs, was identified as the exact left-inverse obligation.
6. Differentiating the analytic left inverse supplies injective differential;
   no inference from set-theoretic injectivity is used.
7. Euler and face-edge incidence counts give the genus correction exactly.
8. Sine-law face scales give necessary and sufficient intrinsic gluing.
9. Vertex-occurrence closure plus non-tree hinge closure is necessary and
   sufficient for a global Hempel realization.
10. Fogelsanger's scope for connected triangulated closed surfaces was
    checked in a modern primary account.
11. Csaszar's primary paper supplies an embedded `(7,21,14)` torus.
12. The exact checker verifies the cyclic torus, orientability, genus, and a
    rank-14 normalized squared-distance Jacobian over the rationals.

No critical proof gap survived the audit. The Hempel elimination-pattern
conjecture is not used.

## Novelty and artifact status

The direct-prior search through 2026-09-02 did not locate the formula
`E-1-6g` or this dual-tree closure theorem. Novelty confidence is moderate-low
because the argument may be folklore. No absolute priority claim is
supported.

- Mathematical dossier: complete
- Literature and direct-prior audits: complete through 2026-09-02, with author contact advised
- Exact checker: passing
- LaTeX/BibTeX build: PASS, five letter-size pages, zero warning classes
- Direct 144-DPI visual inspection: PASS on all five pages
- Upload archive: PASS; English paper sources and reproducibility files only
- Public Zenodo/EulerSolve release: approved on 2026-09-02 under CC BY 4.0; arXiv status is separate

## 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/