Extended Spectral Field (ESF) via Analytic Filters: A Mellin–Fourier Framework for Prime Spectra and Energy Valuations

This preprint introduces the Extended Spectral Field (ESF), a field-theoretic packaging of Mellin spectra for multiplicative measures on (0,∞)(0,∞), realized as a subfield of Mer(U)Mer(U) on a vertical strip. A central tool is an exact log-Gaussian analytic filter: its Mellin transform is closed-form and yields Gaussian localization on the critical line. Using this filter, the paper develops: Mellin-domain diagonalization of smoothed observation operators, a weighted Plancherel “gate” for fixed filter parameters (h,ξ)(h,ξ), a smoothed explicit formula schema for the von Mangoldt measure μ_Λ, where the pole term cancels exactly against Lebesgue background subtraction, leaving a zero sum plus a tail remainder, and an effective variance-to-energy comparison linking the framework to multiplicative short-interval variance (Selberg-type integrals). A Riemann Hypothesis connection is stated explicitly as a conjectural program (valuation non-positivity), with open steps clearly identified. AI acknowledgement: An AI assistant (Chatgpt 5.2 thinking) was used for exposition/structure refinement and LaTeX polishing. The author is solely responsible for all mathematical content and any remaining errors. arXiv submission (primary: math.NT) is planned; feedback is welcome.