p-adic Sobolev Spaces and Perfectoid Geometry: Sobolev-Gauss-nilperfectoid charts and descent of the dagger-nilradical

Abstract: We introduce a class of geometric objects, Sobolev--Gauss (SG) nilperfectoid spaces, designed to connect p-adic harmonic analysis with perfectoid geometry through a common Banach--geometric framework. Motivated by the obstruction isolated by Ansch\"utz, Le~Bras, Bosco, Rodr\'iguez Camargo, and Scholze, especially Question~4. 5. 12 and Remark~4. 6. 8, we study the passage from local perfectoid charts to global perfectoidness for separable Gelfand-type rings in a Sobolev--Gauss setting. Our first result is that for SG--Gelfand rings the uniform completion acts as a geometric and categorical filter: the maximal power-multiplicative seminorm dominated by the Sobolev norm kills the -nilradical, and the resulting SG---reduction identifies canonically with the uniform envelope. On finite SG covers, the bounded Sobolev/Gauss calculus gives strict-kernel base change and Banach quasi-coherence for the -nilradical; combined with Kiehl-type acyclicity, this forces the vanishing of its higher cohomology, while after suitable pro-\'etale/arc refinements the higher cohomology of O^ becomes almost zero. From these inputs we derive an SG-perfectoid upgrade theorem: chartwise perfectoid data together with the required cohomological vanishing package force the global SG atlas to become perfectoid. On the resulting perfectoidized atlas, coefficient-side -flows and Mellin operators admit a compatible infinitesimal description, and their spectral decomposition is organized by a Tannakian trace formalism. In the rank-one local setting, after fixing a ramified sector, the associated Mellin spectral windows on the unramified circle are represented by finite-rank projection kernels whose bulk microscopic scaling limit is the sine kernel. Thus the rank-one local parameter space carries a natural determinantal spectral process arising directly from the Sobolev--Gauss perfectoid realization. Contact & Feedback: This upload is a research preprint and part of an ongoing independent research program. Comments, corrections, questions, and discussions are highly welcome. As I pursue this work independently alongside my regular professional commitments, my replies may take some time and are typically sent during weekends or holidays. Thank you for your understanding.