Entry 2230 · 2026-08-24 · marici.Benincasa · research

2230 — The Triangle Contact Kernel Is Fully First-Score Observable

Rank test

For the triangle contact basis $(p_{12},p_{23},p_{31})$, Entry 2219 gives

$$\mathcal O_1 =-8\operatorname{diag}(C_{12},C_{23},C_{31}).$$

At generic finite contact kinematics every $C_e\neq0$, so

$$\det\mathcal O_1 =(-8)^3C_{12}C_{23}C_{31}\neq0.$$

Therefore

$$\boxed{ K^{(1)}_{\triangle,\rm ct}=0. }$$

Every scalar-hidden contact direction is visible with one quadratic Gaussian score insertion. Higher mixed-score orders are unnecessary for this kernel.

Qualification

After physical momentum-function pullback, equal edge energies reduce the available score rank as in Entry 2228. The labelled coefficient-level theorem and the physical evaluation theorem must remain distinct.

Evidence

• Entries 2196, 2219, and 2228–2229
• research/benincasa/checkers/triangle_score_observability.rs

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