Connectivity, not channel count, sets how fast a discrete glass model forgets repeated thermal cycling

<jats:p>Real thermal-cycling protocols–multi-scan calorimetry, industrial annealing and tempering–repeat a cooling-heating ramp many times, yet discrete-state glass models have so far only been driven through a single pass. We drive a minimal three-state master-equation model1 through many non-reset cooling-heating cycles and ask how many are needed before the response becomes reproducible. Because the dynamics are exactly linear, one cycle is an exact map Pn+1 = MPn, whose second-largest eigenvalue λ2 is the exact–not fitted–rate at which memory is erased, verified against direct simulation to within 10−5. Comparing topologies that share every barrier–a chain and a fully-connected triangle–we find that channel count does not control λ2: the triangle’s extra direct bond lowers the turnaround temperature needed for efficient forgetting by a consistent 15-26K, robust across a decade of cooling rate. Generalizing to a continuous connectivity parameter, a graph-theoretic invariant–the effective resistance between the two states a cycle must exchange population between–tracks λ2 monotonically, while the exact steady-state dissipation per cycle does not: it peaks at intermediate connectivity and falls for still stronger coupling, so the kinetic benefit of connectivity need not cost maximal dissipation. Connectivity, not channel count, thus sets how fast a glass forgets its thermal history–a benefit that is, in principle, separable from its thermodynamic cost.</jats:p>