k-Uniform Sorting Numbers and Prym–Brill–Noether Loci

Jensen [arXiv:2607.01173] computes the tropical Prym–Brill–Noether locus V^r(Γ,π) of the k-uniform double cover of a loop of loops, proves dim V^r(Γ,π) ≤ g−1−⌈k(r+1)/2⌉ for k 2. We answer both questions in the negative. For g−1 ≥ N(k,r), the dimension equals g−1−N(k,r), where N(k,r) is the least number of k-uniform blocks whose Demazure product is the longest element w_0 of S_(r+1). By exhaustive computation N(3,5)=10>9, and at (k,r)=(4,5) the irredundant words have lengths 12 and 13, so V^r(Γ,π) is not pure for g−1 ≥ 13. We compute N(k,r) for all 2 ≤ k ≤ r ≤ 10 and in a larger range for odd k, always finding N(k,r)=(k+1)r/2 for odd k ≤ r, and conjecture this in general. We prove the upper bound N(k,r) ≤ (k+1)r/2 for every odd k ≥ 3 and r ≥ 2: by a 0-1 reduction the schedule c_t ≡ 2t (mod k) becomes a deterministic exclusion process whose hop times satisfy a constant-weight last-passage recursion, solved in closed form. For period-k schedules, which expand to repetitions of Coxeter elements, we prove that for r ≥ k no odd-k schedule sorts in fewer than (k+1)r/2 blocks; in the block-time formulation the obstruction is the parity of a winding number. For schedules of period 2k, and of period nk with n ≤ k+1 in which every class fires equally often, we prove the conjectured growth rate (k+1)/2 via maximum cycle means. We also record a structural conjecture on the graded orbit of the block monoid, with computational evidence.