From Prime Numbers to Quantum Turbulence: A Prime–Golden Ratio Law for Kelvin Waves in Superfluid Helium

We introduce a novel theoretical framework linking prime number distributions to Kelvin wave excitations in quantized vortices of superfluid helium-II. A prime–golden ratio scaling law is derived: fn=ln⁡(pn)φ,f_n = \frac{\ln(p_n)}{φ},fn=φln(pn), where pnp_npn is the n-th prime number and φφφ is the golden ratio.This model predicts that Kelvin wave spectra encode logarithmic prime-based frequency bands converging to universal attractors at approximately 141.7001 Hz and 888 Hz. We extend the framework by connecting Kelvin spectra to the distribution of non-trivial zeros of the Riemann zeta function, suggesting that superfluid helium may provide a condensed-matter testbed for number-theoretic hypotheses. Empirically falsifiable predictions include: observable prime–logarithmic frequency bands, golden ratio modulation in harmonic ratios, universal attractors at 141.7 Hz and 888 Hz, multiplicative fractal synchronization of coupled vortices. An experimental protocol is proposed using nanoparticle-trapped vortices and high-resolution spectral analysis (Δf < 0.01 Hz).Potential implications span quantum computing (prime-coded qubits), astrophysics (log-prime quasi-periodic oscillations in neutron stars), and neuroscience (EEG coherence near 141.7 Hz). This framework bridges number theory, condensed matter physics, and universal coherence, offering a paradigm shift in our understanding of quantum hydrodynamics and its connections to mathematics and cognition.