Why it is a question
THEORY-158 holds Baez and Fong’s Theorem 1 on a finite state space, where four conditions are equivalent: [O, H] = 0; every polynomial in O has a constant mean; O and O² have constant means; and O is constant on each connected component of the transition graph. The record re-derived the cycle. Until 2026-10-19 THEORY-158’s promote_when also allowed “a checked extension of the component form to general state spaces”. At the owner’s direction, from the promotion audit’s batch 008 (ADR-047), the THEORY was narrowed to the finite case, and the general-space clause moved here.
It is open because, on σ-finite measure spaces, the source gives only the moment form. Theorem 3 covers a single stochastic operator and is proved by Chebyshev’s inequality. Theorem 2 covers Markov semigroups and is stated as a consequence. Neither has been re-derived here, and there is no component form.
What would answer it
A checked derivation of the moment form for Markov semigroups on general state spaces, under stated conditions, or a checked extension of the component form to general state spaces, with the notion of “component” that it needs.