Source ledger — positive geometric filling and PP branch conjugacy

Checked 8 September 2026. This packet contains the Lab’s proof and code, not copies of the primary papers. Source locations below identify exactly which inputs were used; their authors retain credit for those results.

Formal inputs

• Garoufalidis–Yu, arXiv:2406.04918v1, Section 3.3, equations (29)–(31), including exact independent-edge ideal generation; Section 5.2, symplectic exponent vectors; Section 5.5, (89)–(92) and Theorem 5.7. This is the identified square-root target and its actual boundary-linear trace, not the extended six-root target. Primary text. The PDF text retained by the preceding attack was inspected at these sections. Its previously recorded SHA-256 is 634e4d8ad74b391c4d4a8a4558c015d887db1ee3b69c43d22dc7a69c4e480e8a.
• Walter D. Neumann, Combinatorics of Triangulations and the Chern–Simons Invariant for Hyperbolic 3-Manifolds, Topology ’90 (1992), Sections 2–3 and Theorem 4.1. Formal input: the symplectic edge/peripheral complex. Geometric conventions: shape logarithms, the genuine surgery equation, and the complementary-slope core-length formula preceding (3.9). The original author-hosted PDF was opened and the relevant passages inspected on this date. Primary text.
• Stavros Garoufalidis and Christian K. Zickert, The symplectic properties of the PGL(n,C)-gluing equations, arXiv:1310.2497; Corollaries 2.6–2.7 and Remark 2.8 in the author-hosted text. Formal input/explicit reformulation for n=2: edge-row isotropy, orthogonality to peripheral rows, peripheral intersection pairing, and edge-row rank. The author-hosted PDF was opened and these statements were inspected. Primary text.
• Panitch–Park, arXiv:2509.09100v1, Theorem 2.20, Remark 2.21 and footnote 10, Definition 2.22 and footnote 11, Theorem 2.23. Formal input: the local face trace on triangle and bigon generators, the local i/−i branch switch, and the identified target relations. Primary text. The complete HTML/text retained by the preceding attack was inspected at these locations; fresh web opening was too large to render. The previously recorded HTML SHA-256 is 39777b6fb307cba58a0b254e6f56bb40630716cdff1247ef003c9591a49b6529.

Geometric existence and scope

• Carlo Petronio and Joan Porti, Negatively Oriented Ideal Triangulations and a Proof of Thurston’s Hyperbolic Dehn Filling Theorem, arXiv:math/9901045v1, Theorem 0.1 and Sections 2–3. Primary proof of the classical hyperbolic surgery input. Its introduction explicitly separates genuine positive triangulations from partially flat/mixed-orientation ones. We use strict positivity only when supplied; no universal existence of a positive triangulation is inferred. Primary text.
• Panitch–Romo, arXiv:2609.02706v1 (2 September 2026), Section 7, Conjecture 7.1, checked live. The objective is stated for arbitrary ideally triangulated oriented manifolds and fillings. It is not restricted to the positive chamber covered here. The source records ongoing work by Samuel Panitch; no priority over that work is claimed. Primary text.

Earlier proved dependencies

The current geometric corollary uses Theorems 8.1–8.2 of the preceding Lab note. The exact cokernel maps and the commuting-operator syzygy proof are there. Its generic filling architecture in turn uses Gunningham–Jordan–Vazirani’s torus-category generation, Gunningham–Jordan–Safronov excision, and Frohman–Gelca’s solid-torus relations. The earlier ledger contains the version/section audit for those inputs. These are not re-proved or newly claimed by this run.

Approach influence versus contribution

• The definite-Lagrangian viewpoint is directly motivated by the Neumann–Zagier symplectic formulation of shape equations. The current matrix lemma and Hermitian formula are proved explicitly, not attributed as an unlocated theorem from those sources.
• Geometric logarithmic surgery conventions motivate the half-log root lift. Principal-root, edge-sign, and filling-sign computations are given directly; no unproved representation-lifting theorem is invoked.
• PP explicitly supplied the local i/−i scaling observation. The current addition is the Eulerian dual-graph choice and its proof of compatibility with opposite-root identification and global gluing. Existence of an Euler orientation for an even multigraph is elementary classical graph theory, not a Lab innovation.
• The mixed-orientation folded-shape example is proved algebraically and is not claimed to be a manifold or a counterexample to the conjecture.

No external novelty/priority assessment is claimed. The paper-level quantum-trace comparison, global six-root peripheral normalization, arbitrary-filling nonvanishing, and triangulation naturality of a chosen branch conjugacy remain outside the proved scope.