Source ledger
Checked 8 September 2026. This is a bounded dependency check, not a comprehensive priority determination.
| Source | Exact role | Status used here |
|---|---|---|
| GHHR, The 3D-index and normal surfaces, published author copy | Theorem 7.1 and Section 7.1: integral kernel and image, normal peripheral rows. Definition 8.1: total boundary fibers. Published Remark 8.3, p. 312: simultaneous inversion. | Formal dependencies. Published local PDF text and fresh arXiv text checked. No independent-cusp identity inferred. |
| Frohman–Gelca, Skein modules and the noncommutative torus, author paper | Sections 3–4: inversion-fixed quantum torus algebra and the torus-skein embedding. | Formal dependency and influence on the fixed-subalgebra argument. Fresh primary lookup. |
| Garoufalidis–Yu, The 3d-index of the 3d-skein module via the quantum trace map, 6 August 2025 revision | Section 5.5, equation (94), Theorem 5.7: curve sums and peripheral action. Section 5.6: scope of descendant symmetry. | Formal dependency through the retained native dictionary. Stronger symmetry in equation (99) is not used as a theorem; chirality is not same-chamber formal inversion. |
| Dimofte–Gaiotto–Gukov, 3-Manifolds and 3d Indices | Index origin and equation (4.15) Weyl-symmetry motivation. | Intellectual source only for this attack. Enhancement expectation is not an independent-cusp proof. |
| Refined 3D index, equation (111) | Uses separate Weyl symmetries as a compatibility requirement in refined filling. | Inspected, not a formal dependency or a theorem establishing the required symmetry for our total GHHR index. |
| The FKB invariant is the 3d index | Possible alternate geometric interpretation. | Bounded scope check only; no boundary-insertion comparison imported. |
Formal retained FPRD dependencies: the global-index all-input convergence and correct-sided translate construction; the balanced-pairing integral-fiber and phase formulas; the six-root native collar/action dictionary. The preceding two-angle result is retained as a nonvanishing theorem in its own chamber, not used to infer the value of this balanced detector.
The intellectual contribution claimed by this note is limited to the multiple-cusp integral classification and its combination with the inversion-odd vacuum functional and the known torus-skein fixed algebra. Its novelty has not been assessed. No new index, generic symmetry theorem, quantum trace construction, or conjecture resolution is claimed.