The Spectral Information State

We introduce the spectral information state as a canonical mode-level state object for spectral inference. After eigendecomposition, each mode \(k\) carries an estimated coefficient \(\hat{A}_k\) and a residual uncertainty \(\sigma_k^2\) induced by finite sample size, noise level, and eigenvalue strength. We define

\[ \psi_k = (\hat{A}_k, \sigma_k^2) \]

and argue that this pair is the correct reusable local state for inference in the spectral basis. In Gaussian linear models it is a complete sufficient statistic for the mode. More generally, it is the minimal practical state needed to support the main inferential readouts associated with that mode: confidence intervals, posterior means and variances, Bayes factors, minimum-description-length penalties, effective sample size, information gain, and decision weights.

The paper makes four claims. First, the spectral information state is not Bayesian or frequentist in itself; it is the common substrate on which both frameworks act. Second, many quantities that are usually computed by separate workflows are readouts of the same state once the eigendecomposition is fixed. Third, the state evolves with sample size as a progressive spectral collapse: high-information modes resolve first, boundary modes later, and noise modes remain unresolved. Fourth, the state connects naturally to the Universal Spectral Representation Theorem, which supplies an objective decay prior and an explicit complexity frontier.

The goal of the paper is foundational rather than polemical. We do not claim that the spectral information state replaces all of statistics, or that every problem admits a faithful spectral lens. We claim that whenever a spectral lens is meaningful, the pair \((\hat{A}_k, \sigma_k^2)\) is the right primitive state object for organizing inference, complexity, and decision. The companion bridge paper The Duality of Bayesian and Frequentist Statistics is then read as a corollary of this deeper object: once both traditions act on the same state, much of their practical disagreement collapses to a thin finite-sample boundary layer.