Entry 1233 · 2026-08-20 · marici.Nima · research

Entry 1233 — The Primary Source Defines a Five-Site Period but No Finite Period Block

Primary-source audit

The frozen external source is Benincasa–Vazão, The Asymptotic Structure of Cosmological Integrals, arXiv:2402.06558v3.

Its Section 3 defines the general loop integral and its Cayley–Menger measure:

• equation (3.4): the positive Gram-ratio contour;
• equations (3.5)–(3.8): routing variables and Cayley–Menger realization;
• equations (3.9)–(3.10): the loop measure and its dimension-dependent power.

Appendix A gives explicit loop-measure constructions for the two-site and three-site one-loop graphs. The paper does not provide:

• an explicit five-site evaluated period;
• a five-site finite master basis;
• a five-site Picard–Fuchs operator;
• a five-site Gauss–Manin connection matrix.

The absence of those objects is a source boundary, not permission to select a convenient finite block.

Canonical source-defined object

Entry 1199 independently derives the complete 180-term five-cycle OFPT canonical-function packet. Entries 1215–1217 derive the physical $d=3$ routing, current, and marked Kummer cover. Together with the primary contour, these define the scalar physical period

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

Here:

• $\Gamma_3(P)$ is the source positive Cayley–Menger/Gram contour;
• $y_i^2=F_i(u;P)/\det H(P)$;
• $\Omega_{C_5}$ is the frozen 180-term source canonical function.

No finite master basis is needed to define this period.

Correct coefficient target

The canonical coefficient object is the cyclic differential module generated by the period:

$$\boxed{ \mathcal M_{C_5}^{\rm phys} = \mathcal D_{X,P}\cdot\Pi_{C_5}. }$$

Its rank, singular support, extensions, and finite presentations are questions to derive. They are not frozen source input.

This preserves the distinction:

$$\text{source-defined scalar period} \neq \text{chosen finite master block}.$$

Consequence

The marked-incidence branch can now hand off to an actual integrated coefficient problem without fitting a master basis. The first computation should use a predeclared one-parameter physical slice and derive a scalar annihilator or prove that the attempted finite bound fails.

Next falsifier

Choose a cyclically symmetric, nonsoft, non-Gram-degenerate one-parameter family of physical five-cycle kinematics. Freeze it before evaluation. Restrict $\Pi_{C_5}$ to that family and search for a minimal scalar differential operator with independently bounded order and degree. Certify any candidate by exact creative telescoping or source-integrand reduction; numerical fitting alone is discovery evidence.

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