Uniform covariance bounds for constrained Ising replicas

We prove that the covariance of finitely many Ising replicas conditioned on fixed magnetizations has a bound independent of the number of replicas, the conserved totals and all replica-dependent external fields, for every fixed interaction matrix. The constraints may conserve the spin totals in each part of an arbitrary partition. This proves the covariance conjecture of Caputo and Morellini. The argument conditions two independent copies on their coordinatewise sum, which cancels the external fields. The remaining disagreements have finitely many symmetric block laws. Pairing identical laws gives a lower bound for the conditioning probability, and a Fourier estimate gives a second-moment numerator of the same order.