Kinetic Sectors and Nonequilibrium Dynamics in the $δ$-Ising Model

We introduce and analyze the $δ$-Ising model, a kinetic Ising ferromagnet in which single-spin flips are accepted when their energy change does not exceed a tunable threshold. Varying the field and threshold opens or closes local flip channels, dividing parameter space into kinetic sectors, each governed by a fixed Markov transition operator. We prove that the dynamics retains global spin-inversion symmetry exactly when up and down spins have equal numbers of open channels. In one such sector on a periodic chain of length $L$, unbiased random initial states therefore reach the field-aligned and antifield states with probability $1/2$ each. Closing a single channel leaves both states absorbing but makes the probability of ordering against the field decay as $(2/π)^L$. Parameter sweeps correspond to ordered products of sector operators, which separate reaching a state from holding it, so hysteresis appears when a state is reached and released at different sector boundaries. We also show how increasing the system size can induce a Mpemba-like reversal, allowing a state with more misaligned spins to reach the final ordered state sooner on average. On a periodic chain, this reversal arises because the mean absorption time grows logarithmically with system size for many short domains but linearly for a single macroscopic domain, and it disappears when the bulk flip channel is opened. These findings show how a simple, intuitive rule for Ising dynamics links local kinetic constraints to field response, memory, and relaxation within a unified framework.