Newton-envelope monomial layer

Integrate this as a coordinate-conditioned layer in a neural field or MLP, immediately after the input coordinates and before the first hidden block. Inputs are positive coordinates x,y, optional context z, and N exponent pairs (a_j,b_j). Clamp x and y below by epsilon, for example 1e-6. Store exponents as softplus parameters if they should be learned. Use a small linear or MLP coefficient head to produce log u_j. Compute p=-log(x), q=-log(y), s_j=log_u_j+a_j*p+b_j*q, h=max_j s_j, and pi=softmax(s-h). Produce logF=h+logsumexp(s-h), then feed [logF, pi_1,...,pi_N, x, y, z] into the ordinary MLP. A cheaper variant returns the weighted expert feature sum sum_j pi_j*phi_j(x,y,z), where each phi_j is a small shared-width expert. The paper supplies the envelope h and stable normalization; learned coefficients and expert features are empirical design choices. First test on a 2D regression benchmark containing a mixture of singular terms, such as F(x,y)=x^-1.7 y^-.1 + 0.5 x^-.2 y^-1.5, sampled logarithmically near the origin. Compare a standard coordinate MLP, a raw monomial layer, and this layer at equal parameter count. Measure validation error versus distance to the corner, NaN or overflow frequency, and wall-clock time. Success is lower near-corner error and no numerical failures at scales where raw evaluation overflows; a useful secondary signal is faster loss descent because the soft dominance weights avoid forcing one MLP to learn all scale regimes.