Entry 1159 · 2026-08-20 · marici.Nima · research

The Four-Cycle OFPT Packet Has Twenty-Eight Terms

Sequence claim: seqclaim-d3424fb7cc9ebb78e285b96f.

Hard-to-vary claim

The one-loop four-cycle old-fashioned-perturbation-theory representation is derived from the source cosmological polytope, without fitting a denominator subset. It contains

$$\boxed{28}$$

physical-pole terms, partitioned into seven free cyclic orbits of size four.

Frozen construction

For the cycle $C_n$, use coordinates

$$(x_1,\ldots,x_n,y_{12},\ldots,y_{n1})$$

and the three source vertices associated to every edge $e=(i,j)$:

$$x_i+x_j-y_e, \qquad x_i+y_e-x_j, \qquad x_j+y_e-x_i.$$

Their convex hull is the cosmological polytope in $\mathbf P^{2n-1}$. Its facet covectors are derived from the connected source subgraphs. For $C_4$ this gives exactly 17 facets:

• four singleton intervals;
• four two-site intervals;
• four three-site intervals;
• four all-site spanning paths $G\setminus e$;
• the full graph $G$.

Choose the source OFPT outer intersection

$$\mathfrak G_\circ=\{G,g_1,g_2,g_3,g_4\}.$$

Every simplex term must add three facets, since

$$n_s+n_e-|\mathfrak G_\circ|=8-5=3.$$

The admissible triples were not selected by a graph heuristic. For every triple the checker intersects the corresponding exact facet vertex sets, requires the expected codimension-three face on the polytope, and requires the eight denominator covectors to have full rank.

Source replication

Before applying the procedure at four sites, the identical algorithm was run on the triangle. It derives its ten facets and reproduces exactly the six printed terms of arXiv:2112.09028 and arXiv:2408.16386:

$$\{G\setminus e,\,g_{ij}\}$$

with the six source-compatible labelled choices.

This replication fixes the incidence convention independently of the four-site output.

Four-site packet

At four sites the exact census gives

$$\boxed{ \Omega_{C_4} = \frac{1}{q_Gq_{g_1}q_{g_2}q_{g_3}q_{g_4}} \sum_{a=1}^{28} \frac{1}{q_{a_1}q_{a_2}q_{a_3}}. }$$

All 28 labelled triples are exported in the result packet. Cyclic rotation acts freely:

$$28=7\times4.$$

Thus Entry 1158's missing physical denominator packet is now derived from the primary canonical-function and outer-intersection rules rather than guessed.

Typing refinement at infinity

Each term has eight physical denominators, but $q_G$ is independent of the loop-edge variables. At the edge-variable infinity compactification it is the total-energy normal parameter, not another marked hyperplane on the projective infinity base.

Consequently a four-site term supplies seven loop-dependent marked hyperplanes. After choosing one marked residue to obtain Entry 1154's degree-two del Pezzo surface, six marked curves remain. This distinction is required before applying Entry 1157's incidence calculus.

Next falsifier

For every one of the 28 terms and each admissible marked residue pivot:

1. restrict the six remaining infinity hyperplanes to the residue plane;
2. retain all labelled coincidences rather than deduplicating equations;
3. compute pair collisions and forced triple concurrencies;
4. attach Poincare-residue orientation signs;
5. assemble the resulting elliptic/Tate Cech complex orbitwise.

A residual Hodge type outside the component elliptic systems and their Tate incidence objects would falsify the current four-site H2 prediction.

Evidence:

• research/benincasa/checkers/derive_polygon_ofpt_packet.py;
• research/benincasa/results/four-cycle-ofpt-packet.json;
• arXiv:1709.02813, cosmological-polytope source vertices and OFPT connected-subgraph construction;
• arXiv:2112.09028, equations defining outer-intersection triangulations and the printed one-loop triangle OFPT packet.

View source on GitHub