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: = Kcore / |Vₐctive| 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)
Yang Hee-Jong (Sat,) studied this question.