Fock-Coercive Magnitude Loss for Complex Features

1. Integration point: replace the final complex feature head of a phase-retrieval or magnitude-only sensing network with a holomorphic polynomial head. For a scalar complex coordinate z, use F_theta(z)=sum_{k=0}^K a_k(theta) z^k, with complex coefficients produced by the preceding real network; for vector features, use one such head per coordinate. Sample fixed quadrature points z_i in the complex plane from a square or disk and assign Gaussian weights w_i proportional to exp(-pi|z_i|^2), normalized to sum to one. 2. Pseudocode: compute F_i=F_theta(z_i), F_0=F_theta(0), c_hat=sum_i w_i*abs(F_i)^2; L_mag=sum_i w_i*(abs(F_i)^2-c_hat)^2; L_square=sum_i w_i*abs(F_i2-F_02)**2; optimize L_task+alpha*L_mag+lambda*L_square. Initially set lambda=0 to test the theorem-inspired magnitude loss alone, then compare lambda>0 as an explicit finite-sample stabilizer. 3. The paper supplies the continuous coercivity relation; the Gaussian quadrature, finite polynomial truncation, and the constant multiplying the inequality are engineering approximations. Estimate the empirical coercivity ratio R=L_square/(L_mag+1e-8) on held-out random polynomials and verify it remains bounded for the chosen degree and grid. 4. First experiment: train a small complex MLP or complex polynomial decoder on synthetic one-dimensional phase retrieval, where inputs are y_j=|A_jx|^2 and the target is x up to global phase. Compare an ordinary intensity-MSE decoder with the Fock-weighted loss and with the additional square term at equal parameter count and FLOPs. Track reconstruction error after optimal global-phase alignment, intensity error, the held-out ratio R, and sensitivity to measurement noise. Success means lower aligned reconstruction error and substantially lower variance across random initializations at matched training intensity loss; a useful stability signal is bounded square-feature error as noise increases.