A Universal Geometric Precursor for Periodic Orbits Numerical Validation of the Weyl Law and a Bridge to Floer Theory

This work introduces the Geometric Stability Score (GSS) as a universal and robust predictor of periodic orbit formation in conservative dynamical systems. Through extensive numerical simulations on the Chirikov map (K = 1.2, 1.5, 2.0, 2.5) and the conservative Hénon map (a = 1.4, 1.8, 2.0), we demonstrate that the minimum GSS value before closure obeys a Weyl law of the form τ = C/g + D with high goodness of fit (R² > 0.85). The GSS achieves an AUC of 0.96, far surpassing the maximal Lyapunov exponent (AUC = 0.52), establishing its superiority over classical chaos indicators. The functional form mirrors the Weyl law for spectral invariants in periodic Floer homology (PFH), providing the first numerical evidence linking a computable geometric metric to symplectic topology. These results open a computational pathway to explore foundational problems such as the closing lemma—a central open question of the Clay Mathematics Institute—and position GeoUnify as a bridge between computational dynamics and advanced mathematical research.