Beyond the Spectral Horizon: Predicting and Accelerating Scientific Discoverability

Building on the recently proposed zeta law of discoverability, we develop new mathematical extensions

that yield a spectral theory of learning curves for finite-data machine learning. For a binary classifier

scored by the area under the ROC curve, AUC(N), we expand the discriminative signal in the eigenbasis

of the data covariance. Finite-sample perturbation theory implies that only eigenmodes exceeding a

random-matrix detection threshold are recoverable, defining a spectral horizon K(N). The accumulated

Mahalanobis information is proportional to the recoverable mutual information and, for power-law spectra,

reduces to a truncated Riemann zeta sum, giving closed-form scaling laws for AUC(N). The theory

predicts crossover points between competing models, staircase learning curves, spectral phase transitions,

percolation-like transitions in discoverability, and when additional data, richer representations, or

new sensing modalities yield the greatest gains.

We then develop a stochastic extension in which AUC(N) evolves as a martingale. Malliavin calculus

yields a confidence cone whose width is determined by the sensitivity of future performance to perturbations

of the observed data stream. A Clark–Ocone representation reveals a duality between forward

accumulation across spectral modes and backward accumulation across observations, unifying spectral

and Malliavin cones within a common variational framework.

Finally, we derive double-spectrum (double-zeta) laws coupling multiple covariance operators, providing

a unified framework for multimodal learning, heterogeneous populations, acquisition-site effects,

and domain adaptation. More broadly, the theory reframes learning as the controlled emergence of recoverable

information, making the future rate of scientific discovery itself a mathematical object that can

be predicted, accelerated, and optimized.