Multi-output BBL mass constraint

Integrate the constraint after the final positive prediction heads of a multi-view or multi-sensor model. For branch i, produce f_i(u)=softplus(F_i(u))+epsilon from its input representation and g_i(v)=softplus(G_i(v))+epsilon from its output representation, where epsilon is 1e-6. Sample N tuples (x_1^(n),...,x_m^(n)) from the branch coordinate grids, set z^(n)=sum_i lambda_i x_i^(n), and evaluate r_i^(n)=g_i(z^(n))/f_i(x_i^(n)). Estimate masses with Monte Carlo: ||f_i||_1 approximately equals B^(-1) sum_b f_i(x_i^b), and ||g_i||_1 approximately equals B^(-1) sum_b g_i(v_i^b). Compute q=Q_d(p)=p/(1-dp), then evaluate the right-hand power mean. Use logarithmic formulas for p=0 and clamp values before negative powers. Since the theorem uses an essential infimum, use smin_tau(u)=-tau log sum_n exp(-u_n/tau), with tau initially 0.2 times the median u and annealed toward 0.02 times the median. Add eta times L_BBL to the task loss. Prevent collapse by normalizing each predicted field to a fixed mass, or by adding a mass-matching penalty against a target estimated from labels. Use a detached sample pool or periodically refreshed hard-min samples to prevent the network from exploiting favorable samples. The theorem determines the inequality, valid p range, and exponent q; lambda, p, eta, tau, and sampling are empirical choices. First test a two-view MNIST or CIFAR-10 classifier with two positive confidence heads, comparing ordinary training against the penalty at equal FLOPs. Measure clean accuracy, accuracy under one-view corruption, calibration error, cross-view disagreement, and the lower quantile of normalized ratios. Success is improved corruption calibration or lower disagreement at unchanged clean accuracy without predicted-mass collapse.