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

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 p_i − log p_j| < 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) stability, graph cosmologyRelated 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