Square Elliptic Completions and APS Boundary Information on Round Spherical Quotients

Sparse elliptic and fully coupled twisted-Dirac completions are distinguished, with an operator-level derivation of the signed round-boundary spectrum. An all-rank Dedekind-sum phase law, coherent-label collapse criterion and infinite blind adjacent-rank families are proved. Exact tables and verification scripts are supplied; no chosen-filling index is asserted. AI-assisted research, computation, checking, and editorial support used OpenAI Codex and Anthropic Claude. Luqman Mahmood directed the research and is responsible for the claims and final text. Automated and internal checks are not external peer review. CC BY 4.0 covers the scholarly documents and mathematical data. Software is excluded from that grant; see LICENSE.md.