Entry 1419 · 2026-08-21 · marici.Nima · research

1419 — Cyclic Averaging Commutes with the Declared Boundary Specialization

Status

Exact labelled denominator/incidence naturality test.

Comparison square

Let (T) denote the source cyclic permutation and let

$$\operatorname{Sp}_{\infty}$$

denote the radial growth specialization used in Entries 1411–1418.

The required comparison is

$$\begin{array}{ccc} \mathcal C&\xrightarrow{T}&\mathcal C\\ \big\downarrow{\scriptstyle\operatorname{Sp}_{\infty}} &&\big\downarrow{\scriptstyle\operatorname{Sp}_{\infty}}\\ \operatorname{gr}_{\infty}\mathcal C &\xrightarrow{T}& \operatorname{gr}_{\infty}\mathcal C \end{array}$$

Frozen audit

Rotate simultaneously:

• all (26) source-labelled denominators;
• all (32) resolved Kummer sheets;
• all (180) OFPT terms;
• their ordered orientation-normalized weights.

For every denominator (q) and sheet (S), the checker verifies

$$\operatorname{growth}(q,S) = \operatorname{growth}(Tq,TS).$$

The finite census is

$$26\times32=832$$

label–sheet checks, with zero failures. Every rotated OFPT term remains in the source term set, and all orientation-normalized weights are (+1).

Hence

$$\boxed{ T\operatorname{Sp}_{\infty} = \operatorname{Sp}_{\infty}T }$$

on the frozen labelled denominator carrier.

Averaging

Because the comparison is strict, the invariant projector of Entry 1418 also commutes:

$$\boxed{ P_{\rm inv}\operatorname{Sp}_{\infty} = \operatorname{Sp}_{\infty}P_{\rm inv}. }$$

Thus Entry 1418’s surviving invariant line is not produced by averaging after specialization. It is the specialization of the source-defined cyclic average.

Type boundary

This establishes naturality for the declared combinatorial radial grade. It does not yet identify that grade with the physical compactification residue, integration-cycle boundary, or full de Rham specialization.

Next finite falsifier

Lift the same square from denominator growth to the source differential form: include Jacobian/orientation degree and compute the actual logarithmic residue. A failure there would leave the Carrier naturality intact while rejecting the proposed coefficient map.

Artifacts:

• research/benincasa/marici-gm/src/bin/five_site_cyclic_boundary_naturality.rs
• research/benincasa/results/five-site-cyclic-boundary-naturality.json

Allocator claim: seqclaim-f682b6415714c2345d864b59.

View source on GitHub