Quasi-analytic response head

Use this as the final scalar or low-dimensional conditioning head of a neural operator, world model, or MLP that predicts a response y from an input context c and a continuous parameter z, such as time horizon, physical coefficient, control frequency, or data-corruption level. Encode c with the usual backbone, map it to Chebyshev coefficients a_0,...,a_K, and evaluate the output with Clenshaw's recurrence rather than a second unconstrained MLP. Normalize z to [-1,1]. Pseudocode: compute h=backbone(c); a=head(h); y=sum_k a[k]*T_k(z); loss=L_data(y,y_true)+lambda*R_an, where R_an penalizes coefficients above A exp(-rho k). Estimate A and rho from the training split or cross-validation; they are not supplied by the theorem. For a stronger test, use automatic differentiation at random z values to estimate derivative bounds and add penalties for violations of |f^(n)| <= C R^n M_n for n=1,...,N. The paper contributes the positive-measure-to-global-identification mechanism and the beta-function motivation; coefficient decay and finite-order penalties are engineering relaxations. First experiment: fit beta-like synthetic curves generated from analytic functions, and compare an unconstrained MLP, Fourier head, and Chebyshev head on one-dimensional parameter continuation. Train only on several disjoint intervals whose union has positive measure and evaluate dense held-out points. The falsifiable prediction is a sharp regularization transition: once rho is large enough that high-order coefficients decay exponentially, held-out error should fall together with anchor error; below this threshold an unconstrained model can interpolate anchors while retaining large off-anchor error. In a noiseless experiment, two constrained heads trained to agree on the same positive-measure anchor set should have sup-norm disagreement outside that set decreasing approximately exponentially with K, whereas unconstrained heads need not agree. With noisy labels, the exact rigidity claim should fail and the experiment should reveal an optimal finite rho rather than monotonic improvement.