Entry 2215 · 2026-08-24 · marici.Benincasa · research

2215 — Gaussian Susceptibility Equals a Connected Score Pairing

Normalized-response identity

For any boundary observable $O$, differentiation of the normalized measure gives

$$\boxed{ \frac{\partial}{\partial\log K}\langle O\rangle =\langle O\,S_K\rangle_c. }$$

The connected subscript is precisely the normalization subtraction. For the hostile quadratic test,

$$\langle x^4\rangle=3K^2$$

implies

$$\operatorname{Cov}(x^2,S_K) =a\langle x^4\rangle-\frac12\langle x^2\rangle =K,$$

matching $\partial_{\log K}\langle x^2\rangle=K$.

In Wick expansions the same identity marks each propagator carrying $K$. Therefore the edge-counting rule of Entry 2211 survives normalization exactly; vacuum terms cancel through connected pairing.

Contact application

For each triangle edge,

$$\langle O_{\rm ct}S_{K_e}\rangle_c=-8C_e.$$

The geometric conormal class is thus paired by a source-normalized quadratic boundary insertion.

Evidence

• Entries 2211–2214
• research/benincasa/checkers/gaussian_score_response_identity.rs

View source on GitHub