Complexity-Bounded o-minimal Observers and Wasserstein-Indistinguishability of High-Frequency Superpositions

Here we develop a measurement model in which "classicality" reflects a bound on the descriptive complexity available to an observer, rather than a separate dynamical regime. The observer is modeled by a compact class C(N,d) of probability measures on a bounded detector domain, whose densities have bounded representational format, e.g. piecewise polynomials of degree at most d on at most N cells (hence definable in an o-minimal structure). Given an incoming quantum probability law mu, the reported measurement outcome is defined as the Wasserstein-1 best approximation Phi(N,d)(mu) in argmin over nu in C(N,d) of W1(mu,nu). For the oscillatory family with density 1 + cos(2*pi*k*x) on [0,1], we prove a quantitative complexity-versus-frequency tradeoff: if k exceeds the observer's alternation budget O(Nd), then the optimal approximation cannot retain interference visibility and becomes Wasserstein-indistinguishable from the phase-averaged classical mixture at rate Theta(1/k). The proof uses a quantitative monotonicity budget for definable CDF deviations, showing that a tame CDF cannot track high-frequency oscillations beyond its format budget. We also justify using W1 (Kantorovich-Rubinstein) as the natural metric for complexity-bounded observation, since it corresponds to distinguishability by Lipschitz observables, whereas total variation corresponds to discontinuous probes requiring unbounded resolution.