Third-Hurwitz Closure and the Fourth-Hurwitz Renewal Frontier: Cut–Cauer reduction, cumulative raw renewal, and the sharp fourth-Hurwitz frontier

This work studies the descent polynomials of naturally labelled, graded width-two posets encoded by squarecase words over the alphabet {M, L, R}. It develops exact transfer, Cauer, Bézout, lattice-path, heap, and Schur-complement methods for investigating their real-rootedness through Hurwitz determinants. The principal unconditional result is an all-length closure theorem for the third Hurwitz determinant: D3 is nonnegative for every squarecase word and either prospective side, and it is strictly positive for words of length at least two. The proof uses a normalized four-port recurrence, a sharp height-two renewal inequality, and exact Newton-binomial positivity certificates. The next determinant, D4, is reduced to several precise equivalent or sufficient targets. These include coupled two-port transition inequalities, raw suffix renewal, aggregate Schur curvature, and an interval-capacity flow problem. The paper proves the opposite-hard reset, both normalized D4 transitions through reset distance three, an exact cut-Cauer determinant decomposition, a Schur-variance identity, and a complete half-open Hall/min-cut characterization of the interval-flow certificate. An exact cumulative-polar certificate proves strict raw suffix renewal at every cut of the infinite families M^e L^(s+5) M^b L M^s, where e is 0 or 1 and s,b are arbitrary nonnegative integers, together with the reflected R-side families. The corresponding degree-nine Newton certificate contains 2,035 coefficients, none negative. A hostile exact interval-flow campaign reaches length 500 and 76 simultaneous negative-curvature cuts without finding an obstruction. The paper also records decisive limitations. Normalized suffix renewal, every fixed-buffer repair, and terminal-only absorption are disproved by exact counterexamples or infinite counterfamilies. All computational results use exact integer or rational arithmetic and are kept logically separate from the all-length proofs. Universal D4 positivity, higher Hurwitz closure, and real-rootedness for all squarecase posets remain open. The manuscript therefore presents a complete D3 theorem together with a sharply documented D4 frontier, rather than claiming full squarecase real-rootedness.