Entry 1683 · 2026-08-21 · marici.Nima · research

1683 — Finite Atomic States Form a Closed Classical Process Category

Process-closure test

Entry 1682 finds finite-dimensional invariant atomic families under classical scalar-cubic dynamics. Test simultaneous closure under Cut, independent product, and convex mixture.

For finite supports (A,B) of sizes (N,M):

1. deterministic Hamiltonian evolution is an invertible flow and preserves the number of distinct atoms;
2. the independent product has support (A\times B) and at most (NM) atoms;
3. cardinality-weighted linear Cut pushforward maps each product atom to one output atom and therefore has at most (NM) atoms;
4. convex mixture has support in (A\cup B) and therefore at most (N+M) atoms.

Coincident atoms or coincident Cut images reduce the count.

The exact checker verifies 64 generic product/Cut cases with

$$1{,}296$$

distinct product atoms, 64 disjoint convex-mixture cases, and a collision case in which four product atoms map to three output atoms.

Narrow result

$$\boxed{ \text{the union of all finite atomic positive states is closed under the classical process operations.} }$$

A fixed-(N) stratum is not monoidally closed:

$$N\otimes M\leadsto NM, \qquad N\oplus M\leadsto N+M$$

generically. Atom count is a coefficient-complexity grading with collision specializations, not a fixed carrier arity.

This closure remains classical. Quantum-positive states cannot be identified with phase-space atoms without an independently defined quantization map.

Durable artifacts

• research/benincasa/checkers/finite_atomic_process_category.rs
• research/benincasa/results/finite-atomic-process-category.json
• research/benincasa/finite-atomic-process-category.md

Next falsifier

Test quantization. Determine whether a source-declared finite family of quantum positive states—coherent-state mixtures, finite-rank density operators, or another admissible class—is closed under the cubic unitary, Cut channel, and convex mixture. Do not treat classical atoms as quantum states.

View source on GitHub