The claim
§4 cites Baez and Fong (LIT-773) correctly: for finite continuous-time Markov processes, a diagonal observable commutes with the generator exactly when its expectation and second moment are conserved for every initial distribution (THEORY-158). It then states its own discrete-time criterion, Kf = f, under which f(S_n) is a martingale and E f(S_n) is invariant, and bounds the drift of the mean by Σε_i when ‖K_i f − f‖ ≤ ε_i.
What it does not say
The manuscript says plainly that Noether’s hypotheses do not hold automatically in communicative systems, and that its bounds are expectation-preservation estimates only.
- That the observable’s distribution is conserved. Only its mean is. In a walk on {0, …, N} absorbed at the ends, f(x) = x is harmonic while the distribution of f(S_n) spreads to the endpoints (CLAIM-094).
- That the criterion is a Noether theorem. Mean conservation does not make an observable a symmetry. The Noether-type equivalence, for finite continuous-time processes, needs the second moment as well and is Baez and Fong’s (THEORY-158), cited here and not leaned on.
Where it came from
The owner’s U37 objection: “symmetry is a kind of invariance, no? isn’t that part of the point of Noether Theory?” (CLAIM-040). A105 conceded and found Baez and Fong’s Noether theorem for Markov processes: “a genuine conservation-theoretic connection for the continuous-time Markov special case”. It also gave the discrete harmonic condition Kf = f, its approximate form (‖Kf − f‖∞ ≤ ε gives drift at most nε, or Σε_i), and a candidate conserved quantity: “Whether the speaker and audience occupy positions of solidarity or normative asymmetry”. A105’s stronger condition, K(h∘f) = h∘f for all bounded h, which conserves the whole distribution, became the manuscript’s “stronger conditions” without the formula. The limit on borrowing Noether is CLAIM-093.
Note of 2026-10-10: harmonic observables on irreducible chains
A203 §26 restates the criterion and adds a limit: “For finite irreducible Markov chains, harmonic functions are constant.” That is right, by the maximum principle. A harmonic f attains its maximum somewhere, and Kf = f makes it attain the maximum at every state reachable from there, which on an irreducible chain is every state. So a non-trivial exactly conserved observable needs reducible dynamics with several closed classes, a restricted state space, or time-dependent kernels. A203 names the same three. Several closed classes are themselves a hypothesis about conventions: separate attractors from which reconstruction does not escape (TERM-027, CLAIM-059). The approximate form survives without them: ‖Kf − f‖∞ ≤ ε bounds the drift of the mean by nε, as the manuscript already says.
Note of 2026-10-19: narrowed to what is shown
The two steps the title now states are one line each. If K_i f = f for every step, E[f(S_(i+1)) | S_i] = (K_(i+1) f)(S_i) = f(S_i), so f(S_n) is a martingale and E f(S_n) = E f(S_0). If ‖K_(i+1) f − f‖∞ ≤ ε_(i+1), then E f(S_(i+1)) − E f(S_i) = E[(K_(i+1) f − f)(S_i)], whose absolute value is at most ε_(i+1), so the mean drifts by at most Σ ε_i. The note of 2026-10-10 proves the irreducible case. What the old title added, conservation of the observable and a Noether-type criterion, is CLAIM-094’s objection, granted and Active, and it now stands in “What it does not say”.