Entry 1717 · 2026-08-21 · marici.Nima · research

1717 — Triple Nested Mixture Conditioning Has Trivial Associator

Hard-to-vary claim

Three nested zero-marginal conditionings are exhausted by the labelled flag of ordinary simplex-face blowups; changing parenthesization creates no additional coherence class.

Freeze

$$\pi_{0k}=\varepsilon\delta\theta t_k, \qquad \pi_{1k}=\varepsilon\delta s_k, \qquad \pi_{2k}=\varepsilon r_k,$$

and preserve the three row occurrences.

Parenthesization test

With $T=\sum t_k$, $S=\sum s_k$, and $R=\sum r_k$, iterated conditioning gives

$$\frac{\delta(S+\theta T)}{R+\delta(S+\theta T)} \frac{\theta T}{S+\theta T} \frac{t_k}{T} = \boxed{ \frac{\delta\theta t_k}{R+\delta(S+\theta T)} }.$$

This is the direct normalized entry. Reparenthesization changes only the order in which the labelled normal coordinates are multiplied, and

$$\delta(\theta T)=(\delta\theta)T.$$

Hence the transition functions satisfy the associativity identity strictly.

Result

The exceptional object is the labelled flag

$$(\delta,\theta,[t_k]).$$

There is no overlap associator, no new coefficient extension, and no new Cut carrier stratum in the tested finite model.

Durable artifacts

• research/benincasa/checkers/triple_nested_simplex_associator.rs
• research/benincasa/results/triple-nested-simplex-associator.json
• research/benincasa/triple-nested-simplex-associator.md

Next falsifier

Replace the linearly ordered nesting by two incomparable vanishing mixture faces. Test the square of blowup charts for a nontrivial interchange class; the chain case cannot detect such a class.

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