The Chess Eigenspectrum Hypothesis

This research note formally states and timestamps the Chess Eigenspectrum Hypothesis: the proposal that the evaluation function of strong chess engines exhibits a mixed spectral structure—globally compressible with significant spectral decay, yet locally rugged with dense near-zero eigenvalues in tactically sharp regions. The note distinguishes gradient-covariance compressibility from curvature-based ruggedness as two separate, non-interchangeable claims, and outlines a practical protocol for empirically measuring both using evaluation gradients, a curvature/Hessian proxy, and randomized SVD—benchmarked against a null baseline to rule out sampling artifacts and architecture-driven redundancy. It situates the hypothesis within existing work on chess evaluation architectures (NNUE, AlphaZero-style interpretability) and on Hessian/gradient-spectrum analysis of trained neural networks more broadly, including the empirical bulk-plus-outlier spectral structure and the spin-glass analogy for loss landscapes—rather than any wider unifying framework. This document serves as the permanent timestamp and citable reference for the hypothesis.