Legendre's Conjecture in Function Fields: the Classical Range, Full Monodromy, and the Open Range q <= d-2

For a monic polynomial f in F_q[t] of degree d, the Legendre interval I_f = {f^2 + s : deg s <= d} is the function field analogue of [n^2, (n+1)^2]. Version 7 revises v6 (doi:10.5281/zenodo.23179113) and v5 (doi:10.5281/zenodo.18705744). 1. The classical range. The explicit Hayes-Weil estimate gives |sum_{P in I_f} Lambda(P) - q^{d+1}| <= (d-2)(q^d - q) in every characteristic. With a bound on prime powers, it shows that every Legendre interval contains an irreducible polynomial whenever q >= d-1. This supersedes all explicit thresholds of v5 and v6, which overlooked the classical estimate. The genuine analogue of Legendre's conjecture (q fixed, d -> infinity) lies in the remaining range q <= d-2. 2. The open range. Every Legendre interval was examined for q = 2 (d <= 20), q = 3 (d <= 10), q = 4 (d <= 10) and q = 5 (d <= 8). None is empty. Hence the function field analogue of Legendre's conjecture holds for every q when d <= 8. 3. Fluctuations. For odd q, the variance of the prime count across intervals is already close to the Keating-Rudnick value (d-2)q^{d+1}, proved only for q -> infinity. The smallest normalized count tends to 1 as d grows. We conjecture that no Legendre interval is ever empty and give a Gaussian heuristic. The paper also contains a self-contained proof that the geometric monodromy group is S_2d for p > 2d, an exact twisted-variety identity, closed formulas for d <= 3, and a list of corrections to v5 and v6. Code, data, figures and a referee report on v5: https://github.com/Ruqing1963/legendre-function-field