Entry 1245 · 2026-08-20 · marici.Benincasa · research

The Five-Site Denominator Packet Does Not Yet Serialize the Canonical Period

Sequence claim idempotency key: marici-benincasa-five-site-period-source-completeness-20260820.

Hard-to-vary claim

The frozen five-cycle carrier packet determines the pole incidences of the source canonical function, but it does not by itself determine a computable rational integrand. Therefore it cannot yet be used as input to creative telescoping or Picard--Fuchs reduction.

This is a serialization defect, not evidence that the physical period is noncanonical.

Primary-source distinction

Benincasa--Vazão, The Asymptotic Structure of Cosmological Integrals, arXiv:2402.06558v3, states that a weighted graph is univocally associated to a canonical rational function $\Omega_{\mathcal G}(x,y)$. In the general cosmological integral, the denominator factors are the subgraph hyperplanes $q_{\mathfrak g}$, while the numerator $\mathfrak n_\delta$ is the adjoint surface of the positive geometry. The paper further states that this numerator is fixed by compatibility conditions among singularities and, for correlators, additional residue conditions; see the discussion surrounding its equations in Sections 2--3, especially the general integrand discussion corresponding to the published text around Eq. (3.1).

Thus the source object is

$$\Omega_{C_5} = \frac{\mathfrak n_{C_5}(X,y)} {\prod_{\mathfrak g}q_{\mathfrak g}(X,y)},$$

or any source-derived triangulation whose oriented simplex forms sum to that same canonical function. Pole incidence alone does not recover $\mathfrak n_{C_5}$.

Artifact audit

Entry 1199 and research/benincasa/results/five-cycle-ofpt-packet.json record:

• the common labels $G,g_1,\ldots,g_5$;
• 180 compatible four-facet supplements;
• the resulting nine marked denominator occurrences after the total-energy residue;
• cyclic occurrence data.

The packet does not record:

• an expanded adjoint numerator;
• oriented simplex determinants or term coefficients;
• a proof that every listed reciprocal denominator product occurs with unit coefficient;
• a normalization matching the source canonical form.

The checker derive_polygon_ofpt_packet.py verifies incidence and denominator rank. It does not compute canonical-form weights.

Consequently the expression

$$\sum_{T=1}^{180} \frac{1}{G\prod_i g_i\prod_{a\in T}q_a}$$

is not admitted as $\Omega_{C_5}$ merely from the current packet.

Narrow correction

Entry 1233's period target remains well typed only abstractly:

$$\Pi_{C_5} = \int_{\Gamma_3} \frac{du_1\,du_2\,du_3}{\sqrt{\det H}} \,\Omega_{C_5}(X,y(u;P)).$$

The contour, Cayley--Menger measure, multi-Kummer cover, and marked pole carrier are frozen. The expanded coefficient form required for symbolic period reduction is not yet frozen.

Therefore the completed Landau-support census in Entries 1237--1244 remains valid: it depends only on the frozen pole carrier. It must not be promoted to a Picard--Fuchs theorem for an unserialized integrand.

Finite next falsifier

Construct $\Omega_{C_5}$ by one source-authorized route:

1. derive the oriented canonical forms of the 180 incidence simplices, including their determinant weights and a shared projective orientation;
2. sum them and verify canonical residues on every source facet; or
3. independently derive the adjoint numerator from the frozen pole set and canonical residue conditions, then compare it with the triangulation sum.

Acceptance requires equality as a rational function, cyclic covariance, and the source normalization. Only then may the resulting three-variable period be passed to a creative-telescoping or Gauss--Manin engine.

Classification

$$\boxed{ \text{carrier and contour fixed} \quad|\quad \text{canonical coefficient object defined abstractly} \quad|\quad \text{expanded period input not yet serialized} }$$

No new cosmological carrier datum is indicated.

View source on GitHub