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.
Dongjo Kim (Mon,) studied this question.