Entry 1686 · 2026-08-21 · marici.Nima · research

1686 — The Infinite Cubic Dyson-Schmidt Tower Has a Finite Shift Presentation

Finite-presentation falsifier

Entry 1685 proves unbounded Schmidt rank. Test whether this infinite coefficient object is nevertheless finitely presented and compatible with filtered Cut composition.

Let

$$A=q_1q_2q_3, \qquad U(z)=e^{zA} =\sum_{n\ge0}c_nA^n, \qquad c_n=\frac{z^n}{n!}.$$

The complete coefficient sequence obeys one first-order shift recurrence:

$$\boxed{ (n+1)c_{n+1}=zc_n. }$$

Equivalently, the operator family has the finite differential presentation

$$(\partial_z-A)U=0, \qquad U(0)=1.$$

Filtered composition

Composition of two copies uses binomial convolution:

$$\sum_{i+j=n}\frac1{i!j!} =\frac{2^n}{n!}.$$

For three copies, both parenthesizations give

$$\sum_{i+j+k=n}\frac1{i!j!k!} =\frac{3^n}{n!}.$$

Thus Dyson order is a canonical additive bond filtration and its convolution is associative.

For every finite truncation

$$U_{\le K}=\sum_{n=0}^Kc_nA^n,$$

the map

$$\rho\longmapsto U_{\le K}\rho U_{\le K}^{\dagger}$$

is one-Kraus and completely positive. Partial trace/Cut contraction preserves positivity. It is not generally trace-preserving because (U_{\le K}) is not exactly unitary.

The exact checker verifies 34 factorial recurrences, 61 binary convolutions, 61 ternary parenthesization identities, and 100 exact CP Gram factorizations.

Narrow result

$$\boxed{ \text{the infinite cubic Dyson--Schmidt tower is unbounded in rank but finitely presented by one shift generator.} }$$

This parallels Entry 1649's finite differential presentation of the infinite moment module. The two infinite coefficient realizations are structured, not arbitrary, and require no infinite family of carrier incidences.

Finite-order positivity is retained, but trace preservation requires the full unitary or a source-derived normalization/virtual comparison.

Durable artifacts

• research/benincasa/checkers/cubic_dyson_finite_presentation.rs
• research/benincasa/results/cubic-dyson-finite-presentation.json
• research/benincasa/cubic-dyson-finite-presentation.md

Next falsifier

Compare the moment-module presentation of Entry 1649 with the Dyson-Schmidt shift presentation. Construct the source-derived transform from bond order (n) to moment-degree growth and test whether it is a filtered chain map. Do not infer equivalence merely from both presentations being finite.

View source on GitHub