 Exact linear optimization from a feasibility oracle

A polynomial-time algorithm for integer linear feasibility gives exact rational linear programming optimization. Rational vertex bounds give a polynomial-bit bound on every finite optimal value and its denominator. An objective threshold below that bound detects unboundedness after feasibility is established. Binary search with the additional inequality $c^{T}x\le t$ localizes the finite optimum to an interval containing a unique rational of bounded denominator; continued fractions recover it exactly. A feasible optimizer can then be found by successive coordinate minimizations on a bounded optimal face. These produce faces of one fixed rational polytope, so one uniform vertex-denominator bound works throughout, avoiding exponential growth of precision requirements.