What question did this study set out to answer?

This work aims to analyze the geometric properties of prime-indexed vertex-log embeddings and identify critical parameters affecting rotational invariance.

February 16, 2026Open Access

Rotational Invariance of Prime-Indexed Quantum Graphs at a Unique Threshold τ ∗

Key Points

This work aims to analyze the geometric properties of prime-indexed vertex-log embeddings and identify critical parameters affecting rotational invariance.
Examined vertex-log embeddings of the first 200 primes in four-dimensional space
Identified critical spectral parameter τ* at approximately 0.145
Conducted 3000 random SO(3) rotations to assess stability and distortion
Measured average distortion and stability index at threshold τ*
Identified τ* as approximately 0.145 with a stability index S at τ* of about 0.98
Observed minimal distortion D(τ*) at around 0.02, unique within the tested range of [0.10, 0.20]
Demonstrated rotational invariance of the threshold graph configuration at τ*

Abstract

This study examines the geometry of a vertex–log embedding of the first N=200primes into four-dimensional space and finds that the critical spectral parameter τ* ≈ 0. 145, previously observed near a Poisson→GOE-type transition inprime gap statistics, acts as a geometric fixed point. For the threshold graph defined by |log pᵢ − log pⱼ| < τ, the resultingconfiguration becomes nearly rotationally invariant at τ*, with averagedistortion D (0. 145) ≈ 0. 02 over 3000 random SO (3) rotations and stability indexS ≈ 0. 98. Neighbouring values of τ do not exhibit comparable stability. Key results: - Critical parameter: τ* = 0. 145 ± 0. 005- Distortion minimum: D (τ*) ≈ 0. 02 (unique in tested range 0. 10, 0. 20) - Stability index: S (τ*) ≈ 0. 98- 3000 Haar-measure SO (3) rotations tested - Reproducible with standard numerical libraries (NumPy, SciPy, NetworkX) This work complements the spectral analysis reported in the RH-VERTEX-LOG framework (DOI: 10. 5281/zenodo. 17553568) by providing independent geometricevidence for the distinguished role of τ* in prime number structure. No claims are made regarding classical conjectures in analytic number theory. Keywords: Threshold graphs, Rotational invariance, Spectral transitions, Random matrix theory, Geometric fixed points, SO (3) stabilityRelated work: - 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) Author ORCID: https: //orcid. org/0009-0009-9336-1043

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