Certified Active-Tail Ising Layer

Integrate this into the dense interaction evaluation of a binary Hopfield layer, Ising optimizer, or discrete latent-variable module, rather than into the surrounding neural MLP. Let S be the symmetric interaction matrix, mu the bias, and v the current sign state. Start with H empty and Q={0,...,n-1}. Run the existing synchronous or simulated-bifurcation update on the full state for a short warmup of K steps. At each reduction check, mark a coordinate i as a candidate only if its sign has stayed unchanged for R consecutive steps and its continuous pre-sign amplitude is outside a saturation threshold tau. For each candidate compute fixed_field_i = mu_i + sum_{j in H} S[i,j]*v[j] and unresolved_bound_i = sum_{j in Q, j != i} abs(S[i,j]). Freeze i only when sign(v[i]) equals sign(fixed_field_i) and abs(fixed_field_i) > unresolved_bound_i + epsilon. Move accepted indices from Q to H. Rebuild or maintain the active block S_QQ, and update the active bias as mu_eff = mu_Q + S_QH v_H. When a new coordinate freezes, update mu_eff incrementally by mu_eff[j] += S[j,i]*v_i for every active j, then remove row and column i from the active matrix. The mathematical quantities are the exact block reduction and the interval bound; the stability window R, amplitude threshold tau, safety margin epsilon, and reduction frequency are empirical hyperparameters. For a first experiment, use a 1,024- or 4,096-node Hopfield or Max-Cut layer with a dense symmetric random or graph-derived S, compare against the same solver without reduction and against naive early freezing, and measure wall-clock time, matrix-multiply FLOPs, final energy, and agreement of the final state at equal random seed. Success means at least 2x lower interaction cost after polarization with no increase in final energy and zero violations of the certified sign condition.