Sources, dependencies, and influence
Primary sources checked on 8 September 2026.
| Source | Use | Reading status |
|---|---|---|
| Panitch–Park, AGT 26 (2026) | Theorem 5.6 and Proposition 5.7: even total face degree of the entire native trace image. | Publisher PDF available locally; exact proof text read. Formal dependency. |
| Garoufalidis–Yu, Advances in Mathematics 486 (2026), 110735 | Remark 4.5 already states mod-two homology preservation; native trace context. | Published PDF downloaded and pertinent text read. |
| GHHR, The 3D-index and normal surfaces | Section 2 exact seed degrees; Section 7 integral normal-homology classification. | Primary HTML and published PDF text read. Degree formula is a dependency; the full integral classification is motivation rather than a required lift theorem. |
| Garoufalidis, The 3D index of an ideal triangulation and angle structures | Established tetrahedron-index recurrence and normalization, retained from earlier packet. | Previously checked primary source. Formal dependency. |
| Neumann, Topology ’90 (1992), 243–271 | Integral edge complex and its topological interpretation. | Read through GHHR’s precise account. Attempts to access the original author’s old URL failed; the proof avoids relying on an unverified identification of the cohomology map. |
| Dimofte–van der Veen, A Spectral Perspective on Neumann–Zagier | General topological perspective on gluing and integral lattices. | Abstract and structural overview inspected; no formal result from this paper is needed. |
| Panitch–Romo, Section 7 | The exact external condition is a nontrivial quotient and a commuting map. | Fresh primary-source text read. It does not explicitly demand a nonzero map. |
| Global-index Lab packet | Correct-sided native average and all-input estimate, with only unfilled edge rows. | Retained internal theorem; principal estimate reproduced in the new note. |
| Minimal-meridian Lab packet | Fixed parity insertion, actual meridian kernel witness, and minimal-quotient exact sequence. | Retained preceding work. |
| Native collar Lab packet | Native convention dictionary and named trace identification. | Retained dependency with its existing scope and source caveats. |
The prior unused figure-eight parity suggested seeking additional grades of the full target. PP’s face-even theorem suggested the half-face construction. Neumann/GHHR suggested testing integral normalizers rather than only mod-two kernels. The successful lower bound uses a direct primitive-kernel and cusp-incidence argument, avoiding the stronger unverified map identification. The index seed originates with Dimofte–Gaiotto–Gukov; its recurrence and convergence framework are credited to the established 3D-index literature. The note makes no claim to introduce homology grading or index averaging.