Certified Nonblocking Geometry for Successor-Factor Polynomial Carriers

We study a successor-factor blocking problem in a fixed polynomial-carrier model in which admissibility is controlled by low-degree carrier geometry together with high-degree trace constraints. The main results are sectoral rather than global. First, the connected straddling real-root sector is strictly nonblocking simultaneously for the four successor indices under consideration. Second, the complete detached compact same-side continuum cover is closed by combining a projective mode-$24$ right-component argument with fixed-simplex certificates on the reflected left component. Third, the non-tangent R–S roof is closed by a projective Wronskian atlas whose charts cover the complete physical roof domain. The computer-assisted parts are fixed finite residual certificates after analytic reduction; they do not replace the structural arguments that define and own the residual domains. We also record structural progress on the remaining B–S branch. A global quadratic wall-witness route fails in the actual model, the surviving compatibility condition reduces to an exact secant-barrier criterion, and the actual carrier geometry has a uniform far-root barrier s₂ ≥ 7. Combining denominator collapse with fraction-free quadratic remainders then removes the carrier roots entirely. A blow-up analysis at the unique degenerate corner proves a uniform fixed-cell collar in every corner direction. The only unresolved B–S piece is therefore a complementary compact polynomial sign gate. We do not certify that compact residual here and do not claim closure of the full same-side, connected real-root, or complex-root problems.