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.
Chihiro Yokota (Tue,) studied this question.