What question did this study set out to answer?

The aim is to examine the stability of the partition ratio in thresholded proximity graphs formed from log-prime embeddings.

February 19, 2026Open Access

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

Key Points

The aim is to examine the stability of the partition ratio in thresholded proximity graphs formed from log-prime embeddings.
Conducted a null test battery over >10⁶ graph instances.
Assessed parameter invariance of partition ratio at embedding scale variation.
Tested input replacement to evaluate prime vs. random inputs.
Analyzed connectivity thresholds to measure stability in fragmented vs. stable regimes.
Observed a stable partition ratio κ approximately equal to 0.852.
Found no significant distinction between primes and various random inputs.
Determined a sharp transition at a connectivity threshold that signals stability.

Abstract

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)

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