Sources and dependencies

8 September 2026. Novelty and priority are unassessed.

Primary inputs

Tudor Dimofte and Stavros Garoufalidis, The quantum content of the gluing equations, arXiv:1202.6268v2 (25 October 2012), Section 2.6 and Section 4.6, provide the fixed actual incidence, meridian, and canonical longitude rows. These were checked in the preceding actual-data packet on 8 September 2026. Fresh retrieval was unavailable during this continuation. The present note displays the full input rows and checks its algebra directly; it does not claim a new primary-source or priority search.

The classical Euler quantum-dilogarithm expansion, recorded in NIST DLMF 17.5.1, supplies the one-sided seed coefficients and the meridian summation. Neither is a newly invented special function identity.

The cyclic symmetry of tetrahedron shape labels is part of the existing quantum-gluing framework. The present note proves the exact map and its left-unit relation identity in the declared root convention rather than attributing a new symmetry principle to FPRD.

Retained proof dependencies

• One-sided seed theorem: the required single recurrence, all-input convergence criterion, and correct-sided isotropic orbit averaging.
• Actual Nahm datum: source-fixed rows, exact native scalars, and preceding sign theorem.
• Phase-free theorem: the original-polarization region p/q>-4 and the general complete-square degree identity.
• Native local collar comparison: the named six-root peripheral operators at generic parameter, based on Panitch–Park, Garoufalidis–Yu and Chen–Kricker. Its original hypotheses and source-reading caveats remain in force.
• Generic filling reconstruction: the filled vacuum maps to 1-l² in the native meridian cokernel; the skein-category sources are credited there.

These retained FPRD notes are AI-assisted dependencies, not external validation. The current explicit cyclic map is an isomorphism within the declared sector presentations. It is not a proof that projection from the full six-root quotient into that sector is injective.

Intellectual influences and external scope

The preceding degree-gap argument suggested transporting the full polynomial and integral-support calculation rather than changing coefficient signs. The moment normal form makes the meridian question an exact recurrence-module question; its eight-relation certificate is given explicitly and lifted back into the quantum algebra.

The earlier GHHR normal-surface/index strategy influenced the broader program. No GHHR boundary-inversion theorem, conjectural separate-cusp symmetry, or new angular certificate is a formal dependency of the present proof.

Panitch–Romo Conjecture 7.1 remains the retained unrestricted nonzero-target problem. This note treats one specified actual triangulation. Its zero meridian vacuum is a zero class in a nonzero central-sector quotient, not a zero full target, and is not a counterexample to that conjecture. The note does not qualify for new author outreach.