Source and influence ledger
Checked 8 September 2026. Primary mathematical sources only; no author confirmation or external review assumed.
Contents
Formal dependencies
| Source | Exact inspected location | Role |
|---|---|---|
| Garoufalidis–Hodgson–Hoffman–Rubinstein, The 3D-index and normal surfaces, Illinois J. Math. 60 (2016), 289–352; published PDF | Theorem 7.1; Section 7.1 before Remark 7.5; Definition 8.1 | The integral normal-homology kernel/image theorem, zero mod-2 class of peripheral raw rows, and the precise extended-index definition. These make the unsaturated-lattice bridge possible. |
| Same published source | Remark 8.3, printed p. 312, PDF page 24 | . The final balanced vanishing uses this prior result only after the full-fiber identification. No independent proof of this source result is claimed. |
| Same source; arXiv v1 | Section 2; Section 8; Corollary 8.12; Appendix A | Exact tetrahedron translation/degree, formal Euler weights, and relation of complete strict angles to normal convergence. |
| FPRD actual-incidence packet | Sections 1–5 and next question | Native raw-row conventions, all-plus character consistency, prior injective normal-class map, and the preserved two-angle theorem. |
| FPRD longitude packet | Theorem 3.1 | The correct-sided left-seed translates and their exact shifted convergence character. |
| FPRD global detector packet | Theorem 4.1 and Sections 5–7 | Unordered convergence on every algebraic input, moment descent, and normalized complete-character unit. |
The retained seed recurrence and quadratic/skew convergence ideas originate in the DGG, Garoufalidis, GHHR, and Garoufalidis–Yu index literature. They are not new constructions of this note. The preceding packets contain their detailed formal dependency chains.
Sources checked for scope or influence, not imported as proof
| Source | Location | Boundary |
|---|---|---|
| Duan–Garoufalidis–Gu, The descendants of the 3d-index, arXiv:2301.00098v1 | Section 3.2, Conjecture 3.1 and (12)–(13); Section 3.3 | Its general factorization route to rotated-index symmetry is conjectural. It is not substituted for GHHR’s published boundary-index statement. The discussion of descendants influences the next possible route. |
| Celoria–Hodgson–Rubinstein, The 3D index and Dehn filling, arXiv:2509.09886v2 | Sections 6 and 13 | Existing surgery-index constructions and their stated closed-manifold limitations. The present balanced pairing is not identified with that surgery formula. No examples from its tables are used in the proof. |
| Panitch–Romo, On the length conjecture for twist knots, arXiv:2609.02706v1, 2 September 2026 | Conjecture 1.2 and Section 7, Conjecture 7.1 | Exact scope of the external question. The source explicitly states that the intrinsic filled trace is conjectural; the current note does not close it. |
Contribution and priority boundary
The present contribution is the exact all-orders native bridge through the full integral normal-homology fiber, its application to the balanced pairing, and the same-complete-character translate obstruction. The inversion identity, the normal-homology theorem, the index construction, and the general averaging method belong to prior literature. First priority of the particular synthesis is unassessed. Its value is to close a specific failed route in an ongoing attack, not to establish a new publication-level result.
Version care
The GHHR arXiv page has only v1, submitted 10 April 2016. The published author-hosted PDF was also inspected; it records final form received 17 December 2016 and prints Remark 8.3 on p. 312. Thus the vanishing conclusion does not rely only on an early preprint sentence. The author publication page links the printed copy; the byte hash is recorded in source-snapshots.json. Primary full texts are not redistributed in this research archive.