# Verification and scope report

Title: A Rank Threshold for Exact Sums of Squares in Virtually Abelian Group Algebras

- Five-page English manuscript; author name at top, company and contact in a footnote.
- Complete written proof of the rank-threshold statement with a detailed originating-researcher self-audit.
- 905 exact finite checks agree under ordinary and optimized Python.
- Gaussian-integer group-algebra calculations check nonsplit Klein bottle and nonnormal S3 subgroup transfers, adjoints, compression and matrix amplification.
- Exact polynomial calculations check the 43-term Laurent obstruction, its Motzkin leading form, and every cubic-monomial elimination step.
- The source theorem is DKMV 2026 Theorem 1.1. Superseded degree and factor-count bounds are not used.
- Desktop compilation and a portable source-only build succeed.
- All five final PDF pages have been visually inspected; no clipping, overlap, missing symbols or unresolved references.

Accepted scope: For any group G with a finite-index subgroup H isomorphic to Z^d, positivity in all finite-dimensional unitary representations, positivity in all unitary representations, and finite sums of Hermitian squares coincide at every matrix level if and only if d<=2. Induced characters of H suffice in the positive cases. A direct finite-index compression gives the factor transfer, an explicit trigonometric Motzkin element gives the d>=3 obstruction, and the Klein bottle case has a concrete two-dimensional representation test.

Theorem scope resolved: yes. Whole original source question resolved: no.
New original-source closures: zero.
Prior factorization, conditional-expectation methods and rank-three
obstruction are credited. No outside review, proof-assistant
formalization, absolute priority certification or degree bound.
AI tools assisted derivation, source search, verification code and exposition.
License: CC BY 4.0.