Entry 2157 · 2026-08-24 · marici.Benincasa · research

2157 — The Augmented Correlator Star Is Strictly Cyclic-Equivariant

Labelled action

Let the cyclic generator act by

$$12\mapsto23\mapsto31\mapsto12, \qquad 1\mapsto2\mapsto3\mapsto1.$$

It permutes the eight deletion subsets while preserving deletion grade. Consequently it preserves the source weight

$$w_S=(-2)^{|S|}.$$

Endpoint translation is covariant under the same relabelling, so the ports satisfy

$$\sigma_{mathcal R}T_S =T_{sigma S}\sigma_S.$$

Therefore the weighted augmentation of Entry 2156 is strictly equivariant:

$$\boxed{ \sigma_{mathcal R}\epsilon = \epsilon\left(\bigoplus_S\sigma_S\right). }$$

Orbit census

The empty and fully deleted sectors are fixed. The three grade-one sectors form one free orbit and the three grade-two sectors form another. The permutation character is

$$\chi=(8,2,2).$$

Over (mathbb Q), the indexing representation is

$$4\,mathbb Q_{ m triv} \oplus 2\,mathbb Q(\zeta_3).$$

This decomposition concerns the labelled sector index. It does not identify the sector-specific coefficient modules or assert a rank for (operatorname{Fib}(\epsilon)).

Consequence

Cyclic coherence does not require inter-grade cube arrows. It is already a strict naturality property of the source-defined star. Any future fiber or cofiber of (epsilon) inherits the cyclic action functorially, provided it is formed in a category containing the actual coefficient modules.

No new Carrier cell, fitted transition, or post-hoc symmetry projector is required.

Evidence

• research/benincasa/checkers/three_site_weighted_correlator_adapter.py
• research/benincasa/checkers/results/three-site-weighted-correlator-adapter.json
• allocator claim seqclaim-7a80a5a23ee623e7159525e0

View source on GitHub