Entry 1650 · 2026-08-21 · marici.Nima · research

1650 — Gaussian Cut Pushforward Has a Canonical Wick Differential Defect

Compatibility test

Entry 1649 gives a finite differential presentation before internal pushforward. Test whether Gaussian Cut contraction is a strict morphism of that differential algebra.

Labelled three-mode model

Use

$$H=\frac12\sum_{i=1}^3p_i^2+q_1q_2q_3$$

and Gaussian pushforward $C_3$ over labelled mode $3$, normalized by

$$C_3(q_3)=C_3(p_3)=0, \qquad C_3(q_3^2)=\nu.$$

For

$$f=p_1q_3,$$

the Hamiltonian derivation gives

$$Df=-q_2q_3^2+p_1p_3.$$

Therefore

$$C_3(Df)=-\nu q_2, \qquad D_{\rm obs}(C_3f)=0.$$

Hence

$$\boxed{ [C_3,D](p_1q_3)=-\nu q_2. }$$

The checker verifies the coefficient for 128 nonzero internal variances.

Narrow result

$$\boxed{ \text{Gaussian Cut pushforward is not a strict differential morphism; its defect is a source-derived Wick contraction.} }$$

The labelled global differential algebra remains strict. The failure arises only after integrating out an interacting occurrence. Differential compatibility then requires an induced effective term or a homotopy/coherence cell carrying the internal second moment.

This defect is generated by the existing cubic incidence, Cut operation, and internal coefficient state. It is not a new carrier generator.

Architectural consequence

The current shared-calculus candidate must distinguish

$$\text{strict evolution on retained occurrences}$$

from

$$\text{lax/homotopy-compatible evolution after coefficient pushforward}.$$

This is the algebraic counterpart of Entry 1645's temporal memory obstruction.

Durable artifacts

• research/benincasa/checkers/cut_derivation_wick_defect.rs
• research/benincasa/results/cut-derivation-wick-defect.json
• research/benincasa/cut-derivation-wick-defect.md

Next falsifier

Construct the first coherence operator

$$H_C=C_3D-D_{\rm eff}C_3$$

on the filtered polynomial module and test its compatibility with nested Cut contractions. Determine whether two internal Gaussian pushforwards satisfy the required cocycle/associativity identity or leave a higher obstruction.

View source on GitHub