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 F_q[t] of degree d ≥ 1 with char(F_q) > 2d, the short interval {f(t)² + s(t) : deg s ≤ d} contains at least one irreducible polynomial over F_q when q is sufficiently large. The number of such irreducibles is asymptotically qᵈ⁺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 S2d (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).