Invariant-Theoretic Analysis of AI-Generated {±1} Matrices from PatternBoost

This paper provides the first invariant‑theoretic and spectral characterization of intermediate {±1} matrices produced by the PatternBoost algorithm. Through analysis of the fluctuation Gram matrix and associated invariants, we show that these matrices consistently occupy a deep region of the C‑phase, exhibiting large spectral spread, high deviation energy, compressed eigenvalue spectra, stable tier entropy, and determinant collapse. Comparative analysis with random matrices highlights the structured nature of the PatternBoost population. The findings suggest that invariant‑theoretic geometry may offer new tools for understanding and potentially improving AI‑guided search methods for Hadamard matrices. All computational artifacts are available in the linked GitHub repository.