The Spectral Resolution Theorem: Dimension-Free Compression of Analytic Objects

We prove that any analytic function on \(\mathbb{R}^n\) --- probability density, regression surface, time series, transfer operator, spatial field --- is representable by \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) spectral coefficients at accuracy \(\varepsilon\), where \(\rho > 1\) is the Bernstein ellipse parameter measuring analyticity. Critically, \(N\) does not depend on the ambient dimension \(n\). This breaks the curse of dimensionality for all analytic objects simultaneously.

The result unifies and extends several known dimension-reduction results: the Universal Risk Representation Theorem for portfolio distributions (Nagy, 2026a), the Eckart-Young theorem for optimal low-rank approximation, and the classical Bernstein ellipse theory for Fourier convergence. The spectral coefficients \(\{A_k\}\) simultaneously encode all moment tensors of all orders --- mean (\(A_0, A_1\)), covariance (\(A_1, A_2\)), skewness (\(A_3\)), kurtosis (\(A_4\)), and all higher-order structure --- into a single finite-dimensional vector. For a Gaussian distribution, \(A_k = 0\) for \(k \geq 3\); the spectral representation detects and quantifies non-Gaussianity through the higher-order coefficients.

We demonstrate seven applications spanning statistics, finance, machine learning, and experimental design, all derived from the same \(K^* = \Theta(\log(1/\varepsilon)/\log\rho)\) formula with zero tuning parameters. All core results are formalized in Lean 4 with zero sorry across 27 files.