Hermite's 1848 problem seeks a representation of reals whose eventual periodicity characterizes cubic irrationals; the complex signature (1,1) case (Karpenkov's Problem 4) is open. A deterministic sin^2-type algorithm needs certified evidence of periodicity with exact unit certificates.

A deterministic algorithm implements the analytic extension of the sin^2 formula, selecting the most negative score and resolving ties by a declared ordering, with transitions certified by exact comparisons in Q(alpha). Forward closures compute finite functional graphs whose cycles and basins are decided by exact finite computation. For the plastic instance x^3+x-1, the exact transition map's functional graph is machine-checked in Lean 4 using the kernel only.

On 205 complex cubic polynomials every run closes projectively with an exact unit certificate; an exhaustive box campaign closes 194/194; 457 deformed bases confirm terminal-cycle invariance. Four fields get certified finite transition graphs; the plastic case (2285 states) is Lean 4 kernel-verified. The general periodicity theorem remains conjectural.