The claim
Baez and Fong’s theorem needs the second moment conserved as well as the mean; their three-state counterexample conserves the mean for every initial state while the second moment moves, and the observable does not commute with the generator (NOTE-598 checked it by hand). The manuscript’s discrete-time criterion Kf = f is a mean condition, so it is a conservation criterion but not the Noether-type one its section heading claims.
What it does not say
It does not say Kf = f is useless: a harmonic observable is a martingale, and the manuscript’s drift bound is correct. Whether Kf = f together with K(f²) = f² gives the discrete-time analogue of the theorem is the record’s extrapolation; Baez and Fong treat continuous time only, and only diagonal observables, so symmetries that permute states are out of scope.
Why the Markov theorem is the one that applies
Baez’s account of Noether’s theorem (LIT-819, NOTE-630, THEORY-160) puts the equivalence of symmetry and conservation in an antisymmetric bracket between observables, with reversible one-parameter groups. A Markov kernel has neither. The manuscript’s setting is therefore the Baez–Fong case, where the mean alone is not enough, and not the Hamiltonian case where it would be. This is the record’s connection; neither paper draws it.
Note of 2026-10-13: the converse case, in learning dynamics
This claim says that conserving an observable’s mean for every initial state does not make it a symmetry. The law of balance (THEORY-211) gives the converse case. Under SGD on a loss with a rescaling symmetry, the symmetry holds exactly for every minibatch, yet the quantity it conserves under gradient flow, ‖u‖² − ‖w‖², is not conserved even in mean: it is driven toward a balanced value. A symmetry of a stochastic process need not give even a mean-conserved observable. No typed edge: the claim is about the other direction.