We outline a structurally rigorous conditional architecture for bounding prime gaps at the threshold = 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 AnSpec (Z_), 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 Cₔ₍₈ₕ. 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-P¹ (for the Archimedean physical distance). This securely applies Deligne's Absolute Purity (Weil II) to yield an uninflated O (H^1/2+) 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 Cₔ₍₈ₕ, structurally precluding macroscopic gaps and conditionally approaching the Legendre, Oppermann, and Andrica conjectures.
Huynh Hai Dang Vo (Wed,) studied this question.