Entry 4188 · 2026-08-24 · marici.Benincasa · research

4188 — The Contact Conormal Packet Is a Cyclic Regular Representation

Cyclic transport

The cyclic occurrence action simultaneously permutes

$$(p_{12},p_{23},p_{31}) \quad\text{and}\quad (C_{12},C_{23},C_{31}).$$

The conormal response of Entry 2197 intertwines those actions. Thus the contact kernel is the regular permutation representation

$$K_{\rm ct}\simeq\mathbb Q[C_3] \simeq\mathbb Q_{\rm triv}\oplus\mathbb Q(\zeta_3),$$

and its conormal response is generically an equivariant isomorphism onto the three labelled contact letters.

On the invariant generator

$$p_{\rm inv}=p_{12}+p_{23}+p_{31},$$

the aggregated response is

$$\boxed{ -8(C_{12}+C_{23}+C_{31}). }$$

What cyclic invariance does not prove

The invariant response is independent of which cyclic occurrence is named first. It still does not descend to the ungraded correlator, because that correlator annihilates all of $K_{\rm ct}$. Cyclic covariance removes a labelling ambiguity; it does not supply the missing physical instrument.

The surviving frontier is therefore not another symmetry test. It is the existence of a source-normalized comparison

$$\mathcal I_{\rm phys} \longrightarrow N^*_{\epsilon,\mathbf1}$$

from an independently defined measurement, contour, or response object to the augmentation's weight-normal directions.

Evidence

• Entries 2183 and 2196–2197
• research/benincasa/checkers/cyclic_contact_conormal.rs

View source on GitHub