Entry 1714 · 2026-08-21 · marici.Nima · research

1714 — Finite Mixture Generators Close under Labelled Cartesian Cut Product

Mixture-Cut falsifier

Entry 1713 identifies a finite log-sum-exp coefficient completion for one physical non-Gaussian input. Test closure under independent Cut assembly.

Product mixture

For independent mixture labels

$$j\in J_X, \qquad k\in J_Y,$$

the product density state has components

$$\boxed{ J_{XY}=J_X\times J_Y. }$$

Their weights and collective displacements are

$$p_{jk}=p_jq_k, \qquad \mu_{jk}=w_X\mu_j+w_Y\nu_k.$$

Substituting these data into Entry 1713's finite exponential sum gives the complete merged cubic generator. If two numerical displacements coincide, their source occurrence labels remain distinct; no post hoc quotient is taken.

For three blocks, either nested Cartesian product yields the same labelled triples and the same final cardinality-weighted displacement. The checker verifies the exact rational weight composition on a nontrivial three-block family.

Narrow result

$$\boxed{ \text{finite Gaussian-mixture coefficient objects are closed under independent Cut by labelled Cartesian product.} }$$

The number of coefficient components multiplies, but this is ordinary tensor product growth inside the sector-specific coefficient object. No new carrier cell or associator is required.

Durable artifacts

• research/benincasa/checkers/finite_mixture_cut_descent.rs
• research/benincasa/results/finite-mixture-cut-descent.json
• research/benincasa/finite-mixture-cut-descent.md

Next falsifier

Admit correlated mixture labels. Determine whether a joint weight table on J_X times J_Y is sufficient, including zero-probability support and conditioning, or whether hidden common-cause labels require a refinement of the coefficient object. Do not infer independence from product indexing.

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