Fractal Frequency Combs for Prime Gap Analysis: A Spectral Approach via √2-Emergent Potentials

We develop fractal frequency combs — bounded, log-periodic potentials encoding prime spacing and gap structure — as deterministic spectral tools for analyzing prime number distributions. The construction is non-circular, relying only on the Prime Number Theorem. The fractal scaling parameter α = √2 emerges variationally as the unique optimizer balancing spectral energy against resonance overlap, not assumed. The resulting kinetic operator Hₖin = −∇·A (x) ∇ + Vₐdd is rigorously self-adjoint with bounded potential, establishing a well-defined spectral theory for additive prime problems. Three applications are developed. Prime gap analysis: the spectral encoding reformulates Cramér-type gap distributions, with phase-alignment mechanisms rigorously connecting the comb potential to prime positions. Twin prime analysis: spectral resonance conditions encode twin prime occurrences, benchmarked on 10⁵ verified pairs. Benford's law emergence: the log-periodic structure with irrational log√2/log 10 produces exact Benford digit distributions via Weyl equidistribution, verified on 10⁶ integers with <2% maximum deviation across all digit frequencies. The framework is positioned as complementary to Random Matrix Theory, not alternative. Where RMT characterizes ensemble averages (statistical), fractal frequency combs reveal individual structural features (deterministic). Comparisons with Connes' noncommutative geometry (explicit potentials vs abstract spectral triples), Berry-Keating (variational √2 derivation vs conjectured connections), and Miller's RMT analysis (individual structure vs family averages) are provided. Limitations are stated explicitly: this is a framework for spectral reformulation, not complete proofs of Cramér, Goldbach, or twin prime conjectures. Numerical benchmarks extend to n ~ 10⁶; asymptotic behavior requires further analysis. Full spectral analysis of the continuous spectrum remains open. Computational algorithms are provided in appendix for independent reproduction.