The Geometry of Prime Vacuums: Legendre's Conjecture in Function Fields via Monodromy and Chebotarev Density

We prove the function field analogue of Legendre's Conjecture. For a squarefree monic polynomial f (t) in Fqt of degree d ≥ 1 with char (Fq) > 2d, the short interval f (t) ² + s (t): deg s ≤ d contains at least one irreducible polynomial over Fq when q is sufficiently large. The number of such irreducibles is asymptotically q^d+1/ (2d). The proof proceeds by constructing the universal polynomial family over the (d+1) -dimensional parameter space, establishing that its geometric monodromy group is the full symmetric group S₂₃ (via irreducibility, primitivity, and simple branching), and then applying the effective Chebotarev density theorem for higher-dimensional varieties. The topological error term is controlled by Katz's Betti number bounds, with the discriminant degree computed via the Sylvester resultant. The Grothendieck–Ogg–Shafarevich formula is used to analyze one-dimensional slices under tame ramification. This paper is part of the Titan Project, a programme investigating geometric and cohomological obstructions to classical prime number conjectures. It builds on prior work on conductor rigidity for Frey curves (Zenodo: 18682375), Sato–Tate equidistribution for the n²+1 family (Zenodo: 18683712), and conductor rigidity for primes in arithmetic progressions (Zenodo: 18684151).