Sources, dependencies, and intellectual influences

8 September 2026. No novelty or first-priority claim is made.

Primary mathematical input

Tudor Dimofte and Stavros Garoufalidis, The quantum content of the gluing equations, arXiv:1202.6268v2 (25 October 2012):

• Section 2.6 supplies the actual labeled two-tetrahedron edge and meridian rows.
• Section 4.6 supplies the canonical longitude row. Its geometric SL2 eigenvalue convention is not substituted for the declared native B^(prime sum) scalar.

These inputs were source-checked on 8 September 2026 in the preceding actual-data packet. A fresh web retrieval was unavailable during this follow-up; the present note uses that retained text and the fully displayed raw rows, rather than claiming a new independent source or priority search. The new degree identities are checked directly from those rows.

DLMF 17.5.1 records Euler’s quantum-dilogarithm coefficient expansion underlying the local seed. No new special function is claimed.

Retained proof dependencies

• One-sided seed and exact cone theorem: local solution, convergence framework, original-sided orbit descent, and limitations outside the cone. The current note also gives the local recurrence and convergence sufficiency directly for the actual graph.
• Actual Nahm vacuum note: source-fixed native datum and preceding opposite-phase certificate. Its failure outside p divisible by four motivated the degree-gap proof; the prior theorem is not retracted.
• Native six-root collar comparison: named peripheral action and convention dictionary, retaining its original assumptions and source caveats.
• Generic filling reconstruction: the filled vacuum corresponds to 1-l².
• Actual incidence and native scalar dictionary: P(R) and B^(prime sum), without substituting a geometric eigenvalue lift.

These prior FPRD arguments are retained AI-assisted dependencies, not external endorsements. Panitch–Park, Garoufalidis–Yu, Chen–Kricker, and the skein-category reconstruction sources remain credited through the exact bridge and reconstruction notes.

Approach and scope

The orbit pairing comes from the preceding one-sided construction. Completing the quadratic exponent and comparing its constant term are elementary methods; the new application retains the actual unsaturated support and discovers the fractional longitude endpoint obstruction. The GHHR normal-index work influenced the broader research program, but no boundary-inversion or separate-cusp symmetry theorem is a formal dependency of this result.

Panitch–Romo Conjecture 7.1 is the retained unrestricted target. A symbolic family for one specified actual triangulation does not resolve it. This follow-up is a supporting research note, not a qualifying publication; no author outreach is part of this result.