Mean-Polynomial Positivity Head

Integrate the construction as a nonnegative energy or regularization head on top of an MLP representation h=f_theta(x), rather than modifying the main prediction logits. Choose J=8--32 atoms. For each atom j, create a small feature map P_j(h) producing m nonnegative arguments, set Y^(j)=softplus(P_j(h)), normalize weights with w^(j)=softmax(beta_j), and constrain the atom coefficient with a_j=softplus(alpha_j). Start with integer orders (q,p)=(2,1); later add (4,2) or (3,1) as an ablation. Compute h=f_theta(x), then for each j compute Y=softplus(P_j(h)), mq=(sum(w*Yq)/sum(w))(1/q), mp=(sum(w*Yp)/sum(w))(1/p), and atom=a_j*(mql-mpl), where l=lcm(q,p). Sum atoms into E and optimize loss=task_loss+lambda*E. The positivity guarantee comes from the power-mean inequality; no positivity estimate is needed during training. Empirically monitor the smallest E on random and adversarially scaled inputs, gradient norms, approximation error, and runtime. First fit a two-dimensional nonnegative target polynomial, including a Robinson-like quartic, on a dense grid. Compare against an SOS Gram parameterization and a sum of nonnegative circuit polynomials with matched parameter count. Then insert each penalty into constrained regression or energy-based classification. Success means lower target approximation error at equal parameter count, no sampled negative energies, fewer gradient spikes, or fewer constraint violations. Record wall-clock overhead and memory because repeated fractional powers may erase the benefit.