# Verification and scope

The note gives a complete proof of a standalone finite-state theorem:
one-hidden-unit binary RBMs are invariant under independent bit-flip heat
flow, while for n >= 4, two-hidden-unit RBMs generically leave even their
algebraic closure at all sufficiently small positive times. Rates are fixed
and positive; in the four-visible case any nonzero nonnegative rate vector
is allowed.

The exact certificate uses 16 visible configurations, 14 parameter
derivatives, one scale column, and one heat column. Its determinant is
11241871441920. Verification uses independent integer and rational
elimination, four modular checks, hidden-state enumeration at 100 additional
parameter tuples, all 24 coordinate permutations, and 120 exact rational
mixture-channel/reconstruction examples. Ordinary and optimized Python
runs agree.

The generic-exit argument separately treats possible singularities of the
full model. The explicit rational mass vector is not asserted to be a smooth
point or an individual immediate-exit example. Conditioning, not visible
marginalization, supplies the extension to larger n. The generic exceptional
polynomial and exit times are existential.

The model geometry, mixture representation, Maas metric, and pullback
projection construction are classical and cited. The specific source
webpage was unavailable; its typographical reconstruction remains unverified.
The official AIM workshop context was checked. The broad PDE question is
not claimed solved, and no frozen-record closure count is added.

The manuscript has a detailed originating-researcher self-audit. It is
unrefereed and not formally verified. No independent reviewer or proof
assistant has certified it. AI assistance is disclosed in manuscript prose.
The bounded primary-source search does not certify absolute novelty or
priority. Full-support and regular-image hypotheses are explicit throughout.