Dual-Mode Arithmetic Representation via Functorial Holography

This presents a framework that resolves the apparent incompatibility between algebraic and geometric representations of natural numbers. We demonstrate that while static two-dimensional coordinate systems are informationally incomplete for capturing the full arithmetic structure of prime factorization, a dynamic spectral representation—constructed as a holographic boundary dual—admits a functorial equivalence with the algebraic structure. Through categorical formalization, analytic number theory, and network embedding theory, we establish that prime factorization and enriched ordinal–rotational addressing are complementary epistemic interfaces to a single arithmetic reality. Critically, this framework preserves the computational hardness of integer factorization while providing mathematical unification, enabling arithmetic signal processing, and generalizing via Pontryagin–Bohr duality to broader mathematical contexts. The resonance conditions inherent in the spectral representation do not yield efficient factorization algorithms, as the inverse problem (phase retrieval) maintains equivalent computational complexity to traditional factorization. This work establishes a formal mathematical bridge between historical harmonic computing approaches (paramatrons), quantum computational models, and fundamental physics through the shared mathematical structure of prime-based spectral representations and their connection to gauge symmetries.