Extensions and Exact Certificates: all-stage local existence, an explicit node family, and a limitation of the eigenpolynomial construction

For the collocation integration matrix $A$ on distinct positive nodes, the target is one positive diagonal $D$ for which $I-D^{-1}A$ is nilpotent. We give a self-contained all-stage existence theorem for nodes sufficiently close to a common positive point. The proof regularizes a determinant at coincident nodes, constructs an algebraic confluent seed by a triangular recursion, and applies the implicit function theorem with an explicitly nonsingular Jacobian. We also characterize the node sets admitting a scalar diagonal: they are scaled zeros of the ordinary Laguerre polynomial. Exact integer interval calculations prove twelve additional point or uniform-neighborhood assertions in dimensions four through seven. A further exact sign certificate shows that, at eleven stages near confluence, no eigenpolynomial with all nonnegative coefficients can produce the required diagonal, although a positive diagonal does exist. Complete proofs for every two- and three-node set are included. The arbitrary-node, arbitrary-stage assertion is not proved or disproved.