The Geometry of Prime Gaps: A Rigorous Conditional Architecture via Global Analytic Geometry, Twistor-$\mathbb{P}^1$, and the Analytic de Rham Stack
Year: 2026
Type: article
Abstract: We outline a structurally rigorous conditional architecture for bounding prime gaps at the threshold $\theta = 1/2$ (the Legendre scale). Previous geometric attempts to bound such gaps encountered fundamental metric obstructions—specifically the invalid extension of Archimedean real limits to non-Archimedean $p$-adic topologies, and the disconnect between physical Euclidean distance and local Langlands parameters. We propose a resolution to these structural issues by operating strictly within the vanguard of Global Analytic Geometry. We construct an absolute analytic phase space over $\mathrm{AnSpec}(\mathbb{Z}_\blacksquare)$, unifying the real Euclidean sequence magnitude (via liquid vector spaces) with arithmetic prime distribution data (via solid modules). By defining our non-linear Maier-type pullback parameter as a solid quasi-coherent sheaf on the Analytic de Rham Stack, the singular support is bounded purely algebraically, resolving incompatibilities with non-Archimedean inequalities and structurally bounding the topological Betti constant $C_{\mathrm{univ}}$. We extract global Langlands parameters using Vincent Lafforgue's excursion operators acting on a unified moduli stack fibered over both the global Fargues-Fontaine curve (for $p$-adic arithmetic) and the Twistor-$\mathbb{P}^1$ (for the Archimedean physical distance). This securely applies Deligne's Absolute Purity (Weil II) to yield an uninflated $\mathcal{O}(H^{1/2+\epsilon})$ global error bound. We sketch a proof by contradiction demonstrating that macroscopic empty intervals asymptotically force a diverging mass at the Archimedean place to be bounded by the static constant $C_{\mathrm{univ}}$, structurally precluding macroscopic gaps and conditionally approaching the Legendre, Oppermann, and Andrica conjectures.
Author: Vo, Huynh Hai Dang
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203738790
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