Spectral Partition Invariance κ = 0.8509: Dynamical Evidence and Falsiable Predictions for Topological Transitions

This record documents computational and analytic results related to the spectral partition ratio κ ≈ 0.8509 observed in proximity graphs constructed from prime-indexed phase embeddings. The study investigates whether a stable partition ratio emerges in large-scale graph structures when points are embedded in ℝ⁴ using a phase operator with threshold parameter τ*. Numerical experiments were carried out up to π(10⁹) ≈ 5.08×10⁷ primes using a stride-1 exact computation engine. Three independent lines of evidence are presented: Computational verification — Multi-stage exact calculations (10⁶–10⁹ scale) show convergence toward a stable spectral partition ratio κ ≈ 0.8509 at τ* ≈ 0.143–0.145, with increasing spectral alignment as system size grows. Dynamical invariance tests — Real-time simulations of proximity graphs under extreme perturbations (structural collapse, energy spikes, and core-shell inversion) indicate that the partition ratio remains stable within numerical tolerance. Analytic structure — A derivation based on the mod-30 Croft ring decomposition connects the invariant κ to prime residue structure with a single free parameter τ*. The results suggest that κ ≈ 0.8509 may represent a recurring stable partition ratio in proximity-graph spectral dynamics. The work proposes falsifiable tests for physical systems undergoing three-state topological transitions. This record includes the full manuscript (PDF + LaTeX source) and supporting data files (JSON convergence reports). Limitations: The physical predictions have not yet been experimentally tested, and the parameter τ* has not been derived from first principles. Source code will be made available upon publication of the accompanying manuscript.The study lies at the interface of analytic number theory and spectral graph methods.Keywords analytic number theory, prime numbers, spectral graph theory, proximity graphs, spectral partition, topological transitions, complex networks, phase operator, prime distributions