Entry 1715 · 2026-08-21 · marici.Nima · research

1715 — Correlated Mixtures Require Complete Joint Weight Tensors

Correlated-label falsifier

Entry 1714 proves closure for independent product mixtures. Admit arbitrary correlations among the finite mixture labels.

Two blocks

For two label sets, the complete coefficient is a joint nonnegative table

$$\pi_{jk}.$$

The generator is the corresponding weighted sum over the existing labelled pairs. One-site marginals do not determine it: diagonal and anti-diagonal binary tables can have identical marginals but different collective displacement moments.

Hidden common-cause labels add no observable datum after pushforward; summing over the hidden label produces the same pi_jk and the same generator.

Three blocks

Pair tables are not sufficient. The even- and odd-parity distributions on three binary labels have identical pair marginals but distinct joint support. Therefore the required coefficient is the complete tensor

$$\boxed{ \pi_{jkl}. }$$

This is the finite-mixture analogue of Entry 1677's complete joint cumulant tensor.

Zero marginal

If a row vanishes as

$$\pi_{jk}=\varepsilon r_k,$$

then conditioning retains

$$\boxed{ \frac{r_k}{\sum_lr_l}. }$$

The exceptional datum is the projective row direction [r]. It is path-dependent after forgetting the approach direction, but canonical on the simplex-face Rees chart.

Narrow result

$$\boxed{ \text{correlated finite mixtures close under Cut only after retaining the complete joint weight tensor and its simplex-face Rees data.} }$$

No new Cut carrier stratum is needed. The enlargement is entirely in the sector-specific coefficient object.

Durable artifacts

• research/benincasa/checkers/correlated_mixture_joint_weights.rs
• research/benincasa/results/correlated-mixture-joint-weights.json
• research/benincasa/correlated-mixture-joint-weights.md

Next falsifier

Test nested conditioning on two simultaneously vanishing marginals. Determine whether projective row directions glue by ordinary iterated simplex blowups or whether their overlap carries a nontrivial extension class. Preserve all joint label occurrences.

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