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 := V_LCC|/|V_active| ≈ 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| < 10⁻¹⁵).• Q2 (Input replacement): κ does not distinguish primes from uniform-random, log-uniform, or ±2-perturbed inputs (two-sample z-test, all p > 0.4).• Q4 (Reference sweep): A connectivity threshold at τ_ref,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 (P_max = 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)