The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N) , which losslessly compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in . Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over \ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.
Ioannis Papadakis (2026) studied this question.