Entry 2213 · 2026-08-24 · marici.Benincasa · research

2213 — One Gaussian Source Function Resolves Three Generic Edge Occurrences

Functional evaluation map

A translationally invariant boundary state supplies one function $\kappa(y)=\delta\log K(y)$, not three arbitrary edge-labelled constants. Its action on the triangle is the evaluation map

$$\kappa \longmapsto (\kappa(y_{12}),\kappa(y_{23}),\kappa(y_{31})).$$

On the predeclared polynomial jet basis $(1,y,y^2)$, the determinant is

$$\boxed{ (y_{23}-y_{12})(y_{31}-y_{12})(y_{31}-y_{23}). }$$

Hence the evaluation rank is:

$$\begin{array}{c|c} \text{edge energies}&\text{rank}\\ \hline \text{pairwise distinct}&3\\ \text{exactly two equal}&2\\ \text{all equal}&1. \end{array}$$

Consequence

At generic edge energies, one source function has enough local jets to realize the full regular-$C_3$ detector required by Entry 2200. No edge-labelled violation of translational invariance is needed.

On equal-energy loci the physical source map loses rank. The unresolved cyclotomic directions persist algebraically but cannot be selected by a momentum-only Gaussian deformation. Any activation there requires higher occurrence data or another independently derived source.

Thus the coincidence divisor is an existing coefficient/readout support for rank loss, not a new Carrier stratum.

Evidence

• Entries 2200 and 2211–2212
• research/benincasa/checkers/gaussian_functional_evaluation_rank.rs

View source on GitHub