Entry 1250 · 2026-08-20 · marici.Benincasa · research

The Five-Cycle Incidence Predicate Equals the Source OFPT Predicate

Sequence claim idempotency key: marici-benincasa-five-cycle-eq33-predicate-equivalence-20260820.

Source formula

For the physical-pole representation of a cosmological-polytope canonical form, Benincasa--Torres Bobadilla, arXiv:2112.09028, Eq. (33), fixes

$$\mathcal G_\circ = \{\mathcal G,\{\mathfrak g_s\}_{s\in\mathcal V}\}$$

and sums over sets $\mathcal G_c$ of compatible subgraphs. For a graph with $n_e$ edges, each term contains $n_e-1$ additional denominators. This is the OFPT recursion representation.

Exact predicate comparison at five sites

For the five-cycle, Entry 1199 fixes

$$(G,g_1,g_2,g_3,g_4,g_5),$$

which is exactly $\mathcal G_\circ$, and chooses four additional facets, which is exactly

$$n_e-1=5-1=4.$$

Its checker accepts a four-set $T$ precisely when:

1. the four source facets have a nonempty common source-vertex set;
2. that set has affine dimension

$$(2n-1)-4=5,$$

hence projective codimension four in $\mathcal P_{C_5}$; 3. the ten complete denominator normals have rank ten.

Conditions 1--2 are exactly the source compatibility condition that the four facets meet on the cosmological polytope in expected codimension. Condition 3 adds the required nondegeneracy of the corresponding rational term.

Therefore the 180 accepted sets are exactly the $\mathcal G_c$ terms in the source Eq. (33) representation for this labelled five-cycle.

Coefficient normalization

Entry 1246 proves

$$|\det M_T|=2^5$$

for every accepted term. After the shared ambient-orientation normalization, the source Eq. (33) assigns every term unit coefficient. Thus the canonical function is now serialized, up to the one global convention already common to the lower-arity packets:

$$\boxed{ \Omega_{C_5}(X,y) = \frac{1}{G\prod_{i=1}^{5}g_i} \sum_{T\in\mathfrak T_{180}} \frac{1}{\prod_{q\in T}q(X,y)}. }$$

No fitted numerator or term coefficient is used.

Corrections to the preceding frontier

• Entry 1245 correctly identified that incidence data alone did not authorize a canonical sum; the missing authority is now supplied by Eq. (33).
• Entry 1248 remains superseded for the reason in Entry 1249.
• The source-normalized five-site scalar period is now a valid computational input.

Next finite target

Compile the 180-term sum on the Entry 1217 multi-Kummer cover and perform a first exact reduction before creative telescoping:

1. combine terms by their three geometric marked profiles without identifying labelled occurrences;
2. verify cyclic covariance of the rational form;
3. remove only source-common factors;
4. measure the reduced numerator and denominator degree on the frozen cyclic slice;
5. choose a Gauss--Manin or telescoping basis from that measured complexity.

Artifact update

research/benincasa/results/five-cycle-ofpt-packet.json now records the Eq. (33) provenance, fixed $\mathcal G_\circ$, compatible-set size, exact predicate, normalized weight, and canonical-function status.

View source on GitHub