Actual-incidence two-angle vacuum theorem

Read FPRD-actual-incidence-two-angle-vacuum-2026-09-08.md for the theorem and proof, then hostile-audit.md for the hypothesis boundary. FPRD-research-loop-state.md contains the continuing mathematical state.

The result detects the actual named generic filled vacuum under a complete strict angular certificate and a strict filling angular certificate. It includes an explicit all-sufficiently-large-slopes consequence. The unrestricted Panitch–Romo conjecture remains open.

Reproduction

Python 3 and its standard library suffice:

python3 verify-actual-incidence.py --output exact-results.json
python3 build-release.py

The verifier checks exact universal polynomial identities and a local fractional-lift calibration, not a finite truncation of a claimed infinite identity. The calibration is not an asserted triangulation. The proof, rather than a search or computational count, establishes the uniform theorem.

Optional supplied-certificate checker

python3 verify-actual-incidence.py --certificate supplied-native-data.json

The JSON fields are:

Field	Meaning
`tetrahedra`	Number $N$ of tetrahedra in the supplied actual triangulation.
`edges_six_root`	Full list of $N$ rows of length $6N$ , with entries 0 or 1; each local bare edge occurs in exactly one row.
`basis_a_six_root`, `basis_d_six_root`	Actual native additive peripheral rows of length $6N$ , with pairing $+2$ after collapse.
`other_cusp_basis_six_root`	Optional flat list of similarly paired basis rows for every other cusp.
`theta_complete`	$3N$ positive rationals, encoded as integers or fraction strings, satisfying exact complete angular equations.
`slope`	One supplied primitive integer pair $[p,q]$ . No slope list is generated.
`theta_filling`	Optional supplied filling-angle certificate. If absent, the theorem’s explicit deformation is used.

The six-root order per block is $(z,z',z'',y,y',y'')$ . The three-coordinate collapse sums opposite entries, and angle coordinates use that three-coordinate order. The named complementary longitude is not changed by the checker; its required native action and source dictionary are hypotheses of the proof.

The checker verifies the indicated affine, positivity, incidence-partition, and skew constraints. It is not a topological triangulation recognizer: source provenance and actual face-pairing/peripheral realization must be established independently. A negative certificate result is not a vanishing result for the target or its vacuum.

Integrity

build-release.py regenerates the exact output, an artifact checksum manifest, and a deterministic archive. It rereads every archived member and verifies its bytes and checksums. The archive excludes itself and its external checksum/validation records. Source snapshots contain bibliographic and integrity metadata, not redistributed papers.