Spectral Indistinguishability and Learning-Curve Identifiability in Quadratic Networks

Dekun Yang

Motivated by information-organization sculpting, we study spectral summaries and scalar learning curves in centered quadratic networks. We construct analytic positive input densities with Gaussian one- and two-coordinate marginals. Their complete fourth-moment operators are orthogonally intertwined by a map fixing the entire diagonal teacher algebra, giving equal anchored spectral measures. For selected common teachers with a nonzero cycle invariant, their factorized loss curves differ because the map does not preserve matrix multiplication. Cycle length $r$, which can be arbitrarily large, gives $O(\varepsilon^r)$ contrast uniformly in training time for each fixed construction. For a supplied pair, noisy scalar loss queries require order $\varepsilon^{-2r}$ observations, while bounded initialization interventions with scalar gradient readouts require order $\varepsilon^{-2}$. A shared-input score, equal to a paired-gradient difference, attains the optimal input-sample exponent without knowing the teacher. An explicit family of diverging auxiliary teacher scales instead admits a nonzero finite limiting contrast at coupling $1/\log L$. Conversely, for Gaussian inputs and a…