  Progressive quenching (PQ) is a stochastic process during which one fixes,
one after another, the degrees of freedom of a globally coupled Ising spin
system while letting it thermalize through a heat bath. It has previously been
shown that during PQ, the mean equilibrium spin value follows a martingale
process and this process can characterize the memory of the system. In the
present study, we find that the aforementioned martingale implies a local
invariance of the path weight for the total quenched magnetization, the
Markovian process whose increment is the spin that is fixed last. Consequently,
PQ lets the probability distribution for the total quenched magnetization
evolve while keeping the Boltzmann-like factor, or a canonical structure, under
constraint, which consists of a path-independent potential and a path-counting
entropy. Moreover, when the PQ starts from full equilibrium, the probability
distribution at each stage of PQ is found to be the limit distribution of what
we call recycled quenching, the process in which a randomly chosen quenched
spin is unquenched after a single step of PQ. The local invariance is a
consequence of the martingale property, and not an application of known
theorems for the martingale process.
