Entry 2211 · 2026-08-24 · marici.Benincasa · research

2211 — The Gaussian Two-Point Kernel Induces the Deletion Tangent

Source-derived family

The primary probability-distribution formula contracts every erased internal edge with

$$K(y)=\frac{1}{2\operatorname{Re}\psi_2(y)}.$$

Thus the coefficient of an erased-edge subset $S$ contains

$$K_S=\prod_{e\in S}K(y_e).$$

For a logarithmic variation $\kappa(y)=\delta\log K(y)$,

$$\delta\log K_S =\sum_{e\in S}\kappa(y_e).$$

In particular, differentiation with respect to the value at a labelled edge gives

$$\boxed{ \frac{\partial}{\partial\log K_e}\log K_S=1_{e\in S}. }$$

This is exactly the edge-deletion counting operator of Entries 2195 and 2207.

Meaning

The source has no independent deletion fugacity, but it does have a genuine coefficient object whose variation induces the same tangent: the Gaussian boundary covariance. Varying $\operatorname{Re}\psi_2$ while retaining a positive real part defines a nearby normalizable boundary-state packet.

The geometric orientation tangent therefore has a source-derived pullback from boundary-state coefficient space. It is no longer merely a formal weighted-geometry susceptibility.

Qualification

This is a state/coefficient deformation, not necessarily a deformation generated by changing one fixed cosmological bulk Lagrangian. Its physical status is a boundary-state susceptibility. Dynamical realizability remains a separate question.

Evidence

• Benincasa–Dian, arXiv:2401.05207, probability-distribution formula and erased-edge rules surrounding equations (2.8)–(2.14)
• Entries 2205–2210
• research/benincasa/checkers/gaussian_kernel_deletion_tangent.rs

View source on GitHub