Phase-OS Isotropy Sweep & Multi-Scale κ Stability: Empirical Results from a Prime-Driven Graph Construction

This report documents the discovery of a candidate spectral-partition invariantκ ≈ 0.8509 in a prime-driven, τ-parameterized graph family. Across 22 ndependent runs spanning Pmax = 50K to 10M (a 200× scale range), the core-shareratio κ := K_core / |V_active| remains locked near 0.8509 while geometric observables(radial uniformity R, angular uniformity J) fluctuate freely (CV = 10–22% vs κ CV = 0.06%)Key results:- κ at τ* = 0.145: 1M → 0.8508631, 5M → 0.8510200, 10M → 0.8508507- Bootstrap 95% CIs of width ~3×10⁻⁵ (10,000 iterations)- 7-decimal reproducibility: wide-grid and ultrafine-grid return identical κ(τ*) on the same 1M instance- τ-stability: total κ band width Δκ ≈ 2.9×10⁻⁴ over τ ∈ [0.12, 0.18]- κ decoupled from geometry: Spearman ρ(κ, R) ≈ 0 at seed level- Coordination convergence: eCorePerN increases with Pmax (10.1 → 11.4 → 11.9), approaching 12 (icosahedral)- Not trivially constant: R and J vary 7–79× more than κ across τExperiment parameters:- 22 runs total (7 main + 15 pilot)- Main runs: seeds = 5,000, N = 500 primes, 10 roots, bootstrap = 10,000 iterations- System: 3× AMD Radeon AI Pro R9700 (96 GB VRAM), ~50 GPU-hours- Gauge artifact risk acknowledged; θ-sensitivity and null tests plannedThis is a data-priority preprint for timestamp and reproducibility purposes.A full analysis with null tests and artifact stress tests will follow.Author ORCID: https://orcid.org/0009-0009-9336-1043Keywords: Prime numbers, Spectral partition, Graph invariant, Isotropy, Bootstrap statistics,Multi-scale stability, Core-share ratio, Random matrix theory, Computational number theory, GPU computingRelated work: - Rotational Invariance of Prime-Indexed Quantum Graphs (DOI: 10.5281/zenodo.18627601)- RH-VERTEX-LOG: Universal Attractor Robustness Verification (DOI: 10.5281/zenodo.17553568)- RH-VERTEX-LOG: Complete Mathematical Framework — Phase 1–7 (DOI: 10.5281/zenodo.17467556)