Entry 1682 · 2026-08-21 · marici.Nima · research

1682 — Finite Atomic Measures Give Invariant Nonlinear Classical Realizations

Finite-family falsifier

Entry 1681 proves that scalar-cubic evolution is not autonomous on the quadratic moment cone. Test whether this forces every exact positive process realization to be infinite-dimensional.

For

$$\dot q=p, \qquad \dot p=-q^2,$$

take an (N)-atomic positive measure

$$\mu=\sum_{s=1}^Nw_s\delta_{(q_s,p_s)}, \qquad w_s>0, \qquad \sum_sw_s=1.$$

Hamiltonian pushforward transports each atom independently:

$$\dot q_s=p_s, \qquad \dot p_s=-q_s^2, \qquad \dot w_s=0.$$

Therefore the family remains (N)-atomic and positive. Its moments

$$m_{ab}=\sum_sw_sq_s^ap_s^b$$

satisfy exactly

$$\boxed{ \dot m_{ab} =a,m_{a-1,b+1}-b,m_{a+2,b-1}. }$$

The exact checker verifies twenty three-atom families, 1,320 moment-hierarchy identities through degree ten, and 800 positive polynomial-square evaluations.

For fixed (N), the parameter space has dimension

$$2N+(N-1)=3N-1.$$

In particular, the tested three-atom realization is eight-dimensional.

Narrow result

$$\boxed{ \text{the classical scalar-cubic process has finite-dimensional invariant positive families.} }$$

This does not contradict Entries 1648 and 1681. The moment hierarchy is an infinite linear coefficient representation; an (N)-atomic family is a finite nonlinear subvariety whose infinitely many moments satisfy algebraic rank relations.

This result is classical. Delta measures on phase space do not automatically define positive quantum states and generally violate quantum uncertainty.

Durable artifacts

• research/benincasa/checkers/finite_atomic_classical_realization.rs
• research/benincasa/results/finite-atomic-classical-realization.json
• research/benincasa/finite-atomic-classical-realization.md

Next falsifier

Test whether the finite atomic realization is compatible with cardinality-weighted Cut merge. Determine how atom counts multiply under independent product and how coincident pushed-forward atoms reduce the count. Classify whether the resulting finite-state category is closed under Cut, dynamics, and convex mixture simultaneously.

View source on GitHub