Statistical Estimation of Extremal Characteristic-Function Functionals: Growing L^p Profiles, Random Frequency Banks, and Walsh Calibration

Let \(Δ_P\) be the difference between the joint characteristic function of a random vector and the product of its marginal characteristic functions, and let \(D_∞(P)=_t|Δ_P(t)|\). For generic even empirical samples with scalar blocks and absolutely continuous marginal laws, unrestricted empirical maximization reaches the universal population ceiling. A supremum over a suitably growing frequency window is already known to be strongly consistent. We prove that no estimator of \(D_∞\) is uniformly consistent over all laws supported on \([0,1]^2\), even though pointwise estimation remains possible. We then study two alternative regularized observation models. First, for a fixed full-support frequency law \(W\), the exact integral \(L^p(W)\) estimator is strongly consistent for \(D_∞\) whenever \(p=p_n→∞\) and \(p_n=o(n)\). Second, an independent random bank of \(K\) frequencies gives a computable finite-sample bracket whose error separates coefficient estimation, failure to hit a near-maximal region, and the \(L^p\) averaging penalty. Growing banks are strongly consistent under \(log K_n=o(n)\) and \(p_n/log K_n→∞\). A local growth condition yields explicit rates, and for Gaussian characteristic-function differences we compute the exact Morse–Bott near-peak exponent and normal Hessian determinant. As a finite spectral application, we give simultaneous Walsh confidence calibration, exact signed-box and known-marginal envelopes, phase and missing-mask impossibility results, and sharp-order recovery of a rational parity parameter. Exact Fourier queries, empirical coefficients, deterministic code distances, and iid sample complexity are kept distinct throughout.