Entry 1685 · 2026-08-21 · marici.Nima · research

1685 — The Source Cubic Unitary Generates Unbounded Schmidt Rank

Entanglement-rank falsifier

Entry 1684 shows that finite global density rank does not control reduced rank. Test whether the actual scalar-cubic interaction preserves a finite Schmidt-rank sector before partial trace.

Across the observed/internal bipartition

$$q_1\mid(q_2,q_3),$$

the interaction unitary is

$$U(t)=e^{-igtq_1q_2q_3}.$$

Its exact expansion is

$$\boxed{ U(t) = \sum_{n=0}^{\infty} \frac{(-igt)^n}{n!} q_1^n\otimes(q_2q_3)^n. }$$

Acting on a generic product vector

$$f(q_1)g(q_2,q_3),$$

the two families

$$q_1^nf(q_1) \quad\text{and}\quad (q_2q_3)^ng(q_2,q_3)$$

are linearly independent. Hence the order-(K) Dyson truncation has coefficient matrix with (K+1) nonzero diagonal entries and Schmidt rank

$$\boxed{K+1.}$$

The exact checker verifies 257 truncations through (K=256), 33,153 labelled Schmidt terms, and 97 explicit matrix-rank calculations.

Narrow result

$$\boxed{ \text{the source cubic unitary has no uniform finite Schmidt-rank closure.} }$$

At generic nonzero coupling and time, the complete analytic expansion has infinite Schmidt rank. Therefore neither finite Schmidt rank nor finite reduced density rank can serve as the closed quantum process coefficient family.

Exceptional wavefunctions annihilated by polynomial relations may have lower rank; the claim is generic, not universal at every state.

Durable artifacts

• research/benincasa/checkers/cubic_unitary_schmidt_rank.rs
• research/benincasa/results/cubic-unitary-schmidt-rank.json
• research/benincasa/cubic-unitary-schmidt-rank.md

Next falsifier

Determine whether the resulting infinite Schmidt spectrum has a finite differential or tensor-network presentation compatible with the moment-module presentation of Entry 1649. Test whether Dyson order supplies a canonical bond filtration whose Cut contraction is associative and positive.

View source on GitHub