Residue Recurrence and Scaling Properties in the Complex Embedding of Prime Numbers

This paper presents an exploratory analytic framework for examining the distribution of prime numbers through complex exponential embeddings and their associated Residue Recurrence Classes (RRCs). Each prime $p$ is mapped to a unit-magnitude complex oscillator $\omega \left(p\right)={\text{e}}^{ip}$ , allowing the study of angular dynamics modulo $2\pi$ within a multiplicative and rotational setting. This representation generalises classical modular analyses and reveals local angular recurrences and clustering within the prime sequence. Residue Recurrence Classes are defined as collections of primes whose angular residues approximate those generated by the integer or fractional roots of smaller primes. Empirical computations suggest that such classes contain numerous primes and exhibit coherent alignments on the unit circle, indicating possible small-scale regularities within the global modular uniformity of primes. We further introduce a quasi-harmonic prime function, formulated as a root-weighted cosine sum over prime angular residues, and analyse the cumulative complex prime centre summation. The latter exhibits an approximate power-law relation between its real and imaginary components, consistent with self-similar or fractal-like scaling. These observations are empirical in nature and do not contradict the asymptotic equidistribution of primes modulo $2\pi$ . Rather, they suggest that within finite intervals, the prime sequence may display local harmonic organisation and interference effects that warrant further formal investigation within analytic number theory.