Entry 1709 · 2026-08-21 · marici.Nima · research

1709 — The Mixed Cubic Cocycle Reconstructs the Complete Independent-Block Generator

Generating-object Cut falsifier

Entry 1708 finitely presents the complete nonlinear coefficient tower. Test Cut descent at the level of the generating germ rather than individual cumulants.

Independent blocks

For independent centered Gaussian blocks, define

$$Q=\sum_iw_iQ_i, \qquad A=\sum_iw_i^2a_i.$$

Then

$$\boxed{ -t(Q^2-A) =-t\sum_iw_i^2(Q_i^2-a_i) -2t\sum_{i<j}w_iw_jQ_iQ_j. }$$

The pure terms are exactly the cardinality-rescaled local cubic evolutions. The mixed term is the labelled binary cocycle derived in Entries 1694--1695.

The merged variables remain Gaussian before the shear, with

$$A=\sum_iw_i^2a_i, \qquad B=\sum_iw_i^2b_i.$$

Substitution into Entry 1708 therefore gives the complete merged generator

$$K_{A,B,t}(x,y) =\frac B2y^2+tAy-\frac12\log(1+2Aty) +\frac{Ax^2}{2(1+2Aty)}.$$

Every mixed cumulant is supplied by the multinomial expansion of powers of A; no coefficientwise fitting is used.

Narrow result

$$\boxed{ \text{cardinality-weighted local evolution plus the labelled mixed cubic cocycle reconstructs the complete independent-block cumulant generator.} }$$

Thus the infinite coefficient object has an exact Cut coproduct generated by the already admitted pair incidence. No higher carrier operation is required.

Durable artifacts

• research/benincasa/checkers/cubic_generator_independent_cut.rs
• research/benincasa/results/cubic-generator-independent-cut.json
• research/benincasa/cubic-generator-independent-cut.md

Next falsifier

Admit a full correlated Gaussian covariance. Replace A by w^T A w and derive the generator including Q--P covariance. Test whether the same labelled quadratic cocycle suffices or whether conditioning introduces a Rees extension at singular covariance.

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