Loewner-Safe Monotone-Convex Gate

Integrate the gate after constructing a Min- or Max-cone affinity matrix K and before row normalization or value aggregation. Parameterize c_0,c_1,c_r,beta_r as softplus(raw_parameter), while tau_r is learned freely or initialized to empirical quantiles of K. For every batch, compute K from nonnegative sequence increments, then compute G=f_theta(K) elementwise and use G V or row-normalized G V in the attention or graph-message-passing update. A compact pseudocode loop is: x0=0; d=softplus(a); x=cumsum(d); K_ij=x_min(i,j); G=c0+c1*K; for r in 1..R, G += c_r*softplus(beta_r*(K-tau_r)); output=normalize_rows(G+epsilon)V. The displayed derivative formulas are the exact reason the gate is monotone and convex. The paper supplies the Loewner-order guarantee only for comparable Min/Max matrices; floating-point violations, row normalization, and task gains must be measured. Run a cheap ablation on a small Transformer for ordered synthetic sequences or a GNN on graphs with cumulative-distance affinities. Compare identity, unconstrained MLP, monotone-only, and monotone-convex gates at equal parameter count. Record validation loss, gradient norms, minimum eigenvalues, and the fraction of sampled ordered matrix pairs for which G(A)<=G(B) fails. Success is near-zero order violations, fewer unstable runs, and lower validation loss than identity gating without a material throughput cost.