Entry 1246 · 2026-08-20 · marici.Benincasa · research

All Five-Cycle Candidate Simplex Terms Have One Determinant Weight

Sequence claim idempotency key: marici-benincasa-five-cycle-uniform-simplex-determinants-20260820.

Frozen matrices

For each compatible term $T$ of Entry 1199, retain the declared coordinate order

$$(x_1,\ldots,x_5,y_1,\ldots,y_5)$$

and the denominator-row order

$$(G,g_1,\ldots,g_5,T_1,\ldots,T_4),$$

where the four labels in $T$ use the packet's lexical order. Let $M_T$ be the resulting integral $10\times10$ normal matrix.

Exact result

For all 180 source-compatible terms,

$$\boxed{|\det M_T|=32=2^5.}$$

The signed lexical-order census is

$$98\text{ terms with }-32, \qquad 82\text{ terms with }+32.$$

The sign is not a coefficient invariant: it changes when the four additional facet rows are reordered. After orienting every simplex against one shared ambient projective orientation, every candidate term has normalized weight one.

Lower-arity replication

The same exact checker gives

$$|\det M_T|=8=2^3$$

for all six triangle terms and

$$|\det M_T|=16=2^4$$

for all 28 four-cycle terms. Thus the uniform determinant is not a numerical accident isolated at five sites.

What this establishes

Entry 1245 identified missing oriented canonical-form weights. The present calculation reduces that ambiguity sharply:

$$\text{180 independent candidate weights} \quad\longrightarrow\quad \text{one common normalization},$$

provided the 180 incidence cones are proved to constitute the source OFPT triangulation with compatible orientations.

It does not prove that triangulation statement. Full-rank compatible denominator bases can overlap, leave gaps, or carry a different signed-chain multiplicity. Determinant equality alone cannot decide those alternatives.

Next finite falsifier

Construct the signed simplicial chain of the 180 candidate cones and verify:

1. every internal codimension-one face cancels exactly twice with opposite orientation;
2. the remaining boundary is precisely the source cosmological-polytope boundary with unit multiplicity;
3. one known canonical residue fixes the global factor $2^{-5}$ in the present normal convention.

If these checks pass, the orientation-normalized unit-coefficient sum is the serialized $\Omega_{C_5}$. If they fail, the failure identifies the first missing or overcounted simplex rather than licensing fitted coefficients.

Artifact

research/benincasa/checkers/derive_polygon_ofpt_packet.py now computes exact denominator determinants and rejects any term whose magnitude differs from $2^n$. The regenerated four- and five-cycle JSON packets record the signed determinants, common magnitude, and orientation-normalized candidate weights.

View source on GitHub