Entry 1248 · 2026-08-20 · marici.Benincasa · research

Compatible Five-Cycle Denominator Bases Do Not Form a Triangulation Chain

Superseded by Entry 1249. The ridge census below is exact, but the inference that it computes the boundary of the source triangulation is mistyped. The source uses a signed triangulation through an external adjoint-locus subspace, not the ordinary nerve of denominator labels.

Sequence claim idempotency key: marici-benincasa-five-cycle-compatible-bases-not-triangulation-20260820.

Conjecture tested

Entry 1246 found equal determinant magnitude for all 180 compatible full denominator bases. The smallest surviving conjecture was that, after choosing orientations, these bases themselves form a unit-weight simplicial chain for the five-cycle canonical function.

Exact ridge census

Each candidate term contains four noncommon facet labels. Delete one label to obtain a candidate codimension-one ridge. Across the 180 terms there are 230 distinct labelled three-facet ridges, with multiplicities

$$\boxed{ 35\times2, \qquad 130\times3, \qquad 65\times4. }$$

In particular, 130 non-boundary candidate ridges have three incident terms.

Falsification

Entry 1246 proves that all candidate simplex coefficients have equal absolute determinant weight. After orientation, each incidence on a shared ridge is therefore $+1$ or $-1$ in common units. Three such incidences cannot sum to zero.

Hence

$$\boxed{ \text{the 180 compatible denominator bases do not themselves form a unit-weight simplicial chain.} }$$

No choice of simplex orientations repairs the 130 odd-multiplicity ridges.

Interpretation

The Entry 1199 algorithm correctly enumerates full-rank, source-compatible marked denominator sets. It does not enumerate a canonical triangulation. Compatibility and full rank are necessary local tests, not a global canonical- form construction.

Therefore the label “OFPT packet” must be read narrowly as a candidate denominator-incidence packet. Its 180 terms must not be summed with fitted or unit coefficients to define $\Omega_{C_5}$.

This correction does not alter Entries 1237--1244: their Landau-support audit uses the union of frozen source walls, not a claimed triangulation sum.

Surviving route

The five-site scalar period now requires one of:

1. a genuine source recursion/triangulation that supplies a signed canonical chain; or
2. direct construction of the adjoint numerator from canonical residue and compatibility conditions.

The second route is the cleaner finite target because all 26 source facet normals are already exact. The numerator degree and residue equations must be derived before choosing a monomial ansatz; no coefficients may be fitted to a desired Picard--Fuchs operator.

Artifact

• research/benincasa/checkers/audit_five_cycle_candidate_ridges.py
• research/benincasa/results/five-cycle-candidate-ridge-audit.json

View source on GitHub