Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori

We prove a canonical (or fixed-population) local equilibrium theorem for the symmetric simple exclusion process on the discrete torus $\mathbb T_N^D$, $D\ge2$, at particle densities bounded away from $0$ and $1$, uniformly over all deterministic initial configurations with the prescribed particle number. At times \[ t_N(s)=\frac{\log(N^D)+s}{2γ_N}, \qquad γ_N=2-2\cos\left(\frac{2π}{N}\right), \] the Radon--Nikodym density of the process relative to equilibrium converges in $L^2$ to a canonical exponential tilt generated by the unique small mean-zero calibration field whose one-site marginals match the evolving one-particle heat profile. Consequently, whenever the profile coordinate converges, the corresponding total variation profile is a Gaussian shift. More precisely, for a deterministic initial sequence $(S_N)$, if the covariance-normalized squared amplitude $\mathfrak q_N^{S_N}(s)$ converges to a scalar $\mathfrak q$ at a fixed $s$, then the distance to stationarity converges to $ 2Φ\!\left({\sqrt{\mathfrak q}}/2\right)-1 $, where $Φ$ is the standard normal distribution function. The profile coordinate is asymptotically determined by the one-particle eigenspace corresponding to the smallest nonzero eigenvalue. The proof combines a calibrated canonical comparison, a fixed-degree comparison between independent and exclusion dynamics, and all-degree control obtained from pair energy estimates and preservation of the Strong--Rayleigh property.