A Proportional-Arithmetic Framework for Fourier Analysis on the Positive Real Line

This paper develops a Fourier framework internal to proportional arithmetic on the positive real line. We construct the corresponding complex scalar field, differential and integral operators, oscillatory kernel, Fourier transform, and proportional function spaces. A correspondence theorem proves that the representative of the proportional transform is the classical Fourier transform under the logarithmic identification. Consequently, inversion, Plancherel, convolution, Schwartz invariance, and Sobolev characterizations follow by transport. We establish the exact relation with Fourier analysis on the multiplicative group and with the Mellin transform on the imaginary axis. Model resolvent and heat equations illustrate the operational calculus, while a scale-localized profile shows how spectral modulus and phase encode log-scale width and preferred scale. The construction is therefore a systematic proportional-arithmetic realization of classical harmonic analysis, rather than an analytically independent Fourier theory.