Entry 1651 · 2026-08-21 · marici.Nima · research

1651 — Independent Gaussian Cut Coherences Satisfy the Nested Cocycle

Higher-coherence test

Entry 1650 derives the first Cut/evolution coherence

$$H_i=C_iD-D_{\widehat i}C_i.$$

Test whether two internal Gaussian pushforwards leave a secondary associativity obstruction.

Labelled test

Use internal occurrences $3,4$,

$$H_{\rm int}=q_1q_3q_4, \qquad f=p_1q_3q_4.$$

Then

$$Df=-q_3^2q_4^2+p_1p_3q_4+p_1q_3p_4.$$

For independent centered Gaussian Cut states with variances $\nu_3,\nu_4$,

$$C_3C_4(Df)=-\nu_3\nu_4, \qquad D_{\rm obs}(C_3C_4f)=0.$$

Cocycle identity

The composite commutator telescopes:

$$\boxed{ H_{34} = C_3H_4+H_3C_4. }$$

On the test observable,

$$H_4(f)=-\nu_4q_3^2, \qquad C_3H_4(f)=-\nu_3\nu_4, \qquad H_3C_4(f)=0.$$

Reversing the order gives

$$C_4H_3(f)+H_4C_3(f)=-\nu_4\nu_3.$$

The checker verifies 1,024 cocycle identities and 1,024 Cut-order identities.

Narrow result

$$\boxed{ \text{The first Wick coherence is associative and order-independent for independent Gaussian Cuts.} }$$

No secondary obstruction appears in this finite sector. The non-strict Cut/evolution comparison of Entry 1650 is therefore controlled by a coherent lax structure rather than an arbitrary defect.

This supports the shared-calculus hypothesis: existing Cut incidence plus coefficient pushforward generates the required coherence.

The result is restricted to independent Gaussian internal coefficient states. Correlated Cut states need a joint pushforward and may not factor into commuting one-occurrence contractions.

Durable artifacts

• research/benincasa/checkers/nested_cut_wick_cocycle.rs
• research/benincasa/results/nested-cut-wick-cocycle.json
• research/benincasa/nested-cut-wick-cocycle.md

Next falsifier

Repeat the nested comparison for a correlated two-occurrence Gaussian state. Freeze the joint covariance and compare direct joint pushforward with both sequential conditional contractions. Determine whether their discrepancy is explained by the existing correlation/process-tensor coefficient object or produces a genuine higher coherence class.

View source on GitHub