Entry 1716 · 2026-08-21 · marici.Nima · research

1716 — Nested Zero-Marginal Conditioning Is an Iterated Simplex Blowup

Falsifier

Entry 1715 left open whether two simultaneously vanishing mixture marginals glue by ordinary iterated simplex blowups or require an overlap extension. Freeze the labelled family

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

with $S=\sum_k s_k$ and $R=\sum_k r_k$.

Exact comparison

Iterated conditioning gives

$$\frac{\delta S}{R+\delta S}\frac{s_k}{S} = \boxed{\frac{\delta s_k}{R+\delta S}},$$

which is also the direct conditional weight. The common first normal scale $\varepsilon$ cancels exactly. At $\delta=0$, ordinary specialization forgets the vanished row, whereas the exceptional flag retains $[s_k]$.

The overlap transition is multiplication of the outer exceptional coordinate by the inner projective coordinate. It is strictly associative and leaves no residual Čech cocycle.

Narrow result

$$\boxed{ \text{nested zero-marginal mixture conditioning is resolved by ordinary iterated labelled simplex-face blowups.} }$$

No extension class and no new Cut carrier stratum occur in this finite model. The additional information is a flag in the sector-specific mixture coefficient object.

Durable artifacts

• research/benincasa/checkers/nested_simplex_conditioning.rs
• research/benincasa/results/nested-simplex-conditioning.json
• research/benincasa/nested-simplex-conditioning.md

Next falsifier

Test three nested vanishing marginals and two different parenthesizations of conditioning. Preserve every label and compare the full exceptional flags; any nontrivial associator would be the first coefficient-level coherence datum beyond ordinary simplex blowups.

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