Entry 1676 · 2026-08-21 · marici.Nima · research

1676 — Cardinality-Weighted Merge Preserves Every Cumulant Species

Non-Gaussian normalization test

Entry 1654 gives a finite species presentation for cumulant Cut sewing. Entry 1666 introduces cardinality-dependent normalization needed for fixed-$\hbar$ descent. Test whether this normalization mixes cumulant orders.

For independent blocks of sizes (m,n), let

$$Q_{m+n} =\alpha Q_m+\beta Q_n, \qquad \alpha=\sqrt{\frac m{m+n}}, \qquad \beta=\sqrt{\frac n{m+n}}.$$

Homogeneity and independence give

$$\kappa_r(Q_{m+n}) =\alpha^r\kappa_r(Q_m) +\beta^r\kappa_r(Q_n).$$

For a leaf inside the first block, squared coefficients compose as

$$\left(\frac m{m+n}\right)^r\frac1{m^r} =\frac1{(m+n)^r}.$$

Thus every merger tree assigns each leaf coefficient (N^{-r/2}) at cumulant order (r).

The exact rational checker verifies cumulant orders one through twelve and occurrence counts one through twelve:

$$144\text{ species/count pairs}, \qquad 936\text{ leaf coefficients}, \qquad 990{,}000\text{ ordered binary bracketings}.$$

Narrow result

$$\boxed{ \text{cardinality-weighted fixed-}\hbar\text{ merge preserves every cumulant species.} }$$

Normalization does not mix cumulant orders. It equips the order-(r) species with the coefficient character

$$\left(\frac{\text{block size}}{\text{total size}}\right)^{r/2}.$$

Therefore Entry 1654's species presentation survives normalized Cut descent; no new carrier generator is required.

This statement assumes independent blocks. Correlated non-Gaussian blocks require mixed cumulants across occurrence labels.

Durable artifacts

• research/benincasa/checkers/cardinality_weighted_cumulant_species.rs
• research/benincasa/results/cardinality-weighted-cumulant-species.json
• research/benincasa/cardinality-weighted-cumulant-species.md

Next falsifier

Admit mixed cumulants between two non-Gaussian blocks. Test whether retaining the complete labelled joint cumulant tensor makes cardinality-weighted nested mergers associative, as full covariance did in Entry 1669, or whether partition incidence creates a new coherence class.

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