Cocircuit-Brion Factorization and Forced Root Skeletons in Fully Mixed Flagged Determinant Gates

This paper proves a cocircuit-Brion skeleton theorem for a family of fully mixed flagged determinant gates arising from three-row Schur/Jacobi-Trudi coefficient packages. For integers n ≥ 2 and k ≥ 4, the fully mixed gate A_k(n,m) is a polynomial in the flag-size variable m. The main theorem identifies an explicit forced-root skeleton S_{k,n}(m) dividing A_k(n,m), equivalently establishing three nested vanishing layers: value, first derivative, and second derivative. The proof is organized around three exact finite-algebraic mechanisms. First, the fully mixed flagging is encoded by a flag-simplex series. Second, the low boundary kernel factors as a fourth-order cocircuit finite difference. Third, the finite chain satisfies a Brion-type endpoint split, reducing the boundary-reduced operator to low and high endpoint packages. These identities convert the observed root skeleton into a forced boundary-cancellation phenomenon. The lower boundary is governed by ghost rank and proper-face clearing, with the adjacent X/Y face removed by triangular exactness. The upper boundary is governed by overflow matching and Brion gluing of the unique XZ boundary atom. Together these mechanisms prove the divisibility S_{k,n}(m) | A_k(n,m). This work is Paper III in a sequence developing boundary-absorbed Schur-minor and flagged determinant-gate mechanisms. It moves from reciprocal-pair coefficientwise positivity to forced-root skeletons. The result is a finite algebraic divisibility theorem. It does not claim a proof of the Riemann Hypothesis, a general Jensen-polynomial hyperbolicity theorem, or a Laguerre-Pólya-class criterion.