Percolation Transition and Irreversible Structure Formation in Prime-Indexed Geometric Graphs: A Universal Partition Attractor at κ ≈ 0.852

We report a systematic null test battery (>10⁶ graph instances) for the partition ratio κ: = VLCC|/|Vₐctive| ≈ 0. 8523 observed in thresholded proximity graphs constructed from log-prime embeddings in ℝ⁴. Three independent tests are performed: • Q1 (Parameter invariance): κ is identical to machine precision under ±18% variation of the embedding scale θ (max |Δκ| 0. 4). • Q4 (Reference sweep): A connectivity threshold at τᵣef, c ≈ 0. 14 separates fragmented from stable regimes, quantified by a sharp collapse of the band width Δκ: = max_τ κ − min_τ κ. These results support κ as a candidate invariant of the graph construction under the tested perturbations, rather than a property of the prime sequence. Two directional observables (R, J) remain candidates for carrying input-specific, possibly number-theoretic, structure. Companion report: "Phase-OS Isotropy Sweep & Multi-Scale κ Stability" (DOI: 10. 5281/zenodo. 17467556) provides independent multi-scale verification (Pₘax = 50K–10M, 22 runs, 5000 seeds). Author ORCID: https: //orcid. org/0009-0009-9336-1043- Phase-OS Isotropy Sweep & Multi-Scale κ Stability (DOI: 10. 5281/zenodo. 18638137) - 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)