Wave Arithmetic: A Smooth Integral Representation of Number Theory

We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this formulation, each number arises as the accumulated amplitude of a structured waveform -- an interference pattern encoded by carefully designed kernels. Arithmetic operations such as addition, multiplication, and exponentiation are realized as geometric and tensorial constructions over multidimensional integration domains. Rational numbers emerge through amplitude scaling, and negative values through sign inversion, preserving all classical arithmetic identities within a continuous and differentiable structure. This representation embeds number theory into the realm of smooth analysis, enabling new interpretations of primality, factorization, and divisibility as geometric and spectral phenomena. Beyond technical formulation, Wave Arithmetic proposes a paradigm shift: numbers as the collapsed states of harmonic processes -- analytic resonances rather than atomic symbols.