Entry 2219 · 2026-08-24 · marici.Benincasa · research

2219 — Gaussian Score Insertion Commutes with the Contact-Normal Grade

Typed projector

Let $\operatorname{gr}_{\rm ct}$ denote the contact-normal associated-grade map already used to define the three channels of Entry 2174. The Gaussian kernel coordinate $K_e$ and the contact normal are independent source coordinates on the generic locus. Hence coefficient extraction in the contact normal commutes with logarithmic differentiation in $K_e$:

$$\boxed{ \left[\operatorname{gr}_{\rm ct}, K_e\partial_{K_e}\right]=0. }$$

On the projected route pair,

$$\operatorname{gr}_{\rm ct}F_e(K_e) =8C_e-8C_eK_e.$$

At $K_e=1$,

$$\operatorname{gr}_{\rm ct}F_e=0, \qquad K_e\partial_{K_e}\operatorname{gr}_{\rm ct}F_e=-8C_e.$$

Extracted observable

The source-normalized mixed contact readout is therefore

$$\boxed{ \operatorname{gr}_{\rm ct} \langle O S_{K_e}\rangle_c=-8C_e. }$$

The contact grade removes the backgrounds identified in Entry 2218 without being chosen to fit the answer. It predates the Gaussian-score construction and is defined by the source's normal channel.

Scope

The commutation is established on the generic transverse locus. At soft or nontransverse intersections, a Rees/nearby-cycle comparison may be required; the generic result must not be extended there silently.

Evidence

• Entries 2158–2159, 2174, and 2217–2218
• research/benincasa/checkers/contact_grade_score_commutation.rs

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