Polynomial-time optimization and exact separation for rational mixed-integer bipartite covers

Polynomial-time exact rational linear optimization, strong separation and membership for mixed-integer bipartite covers with arbitrary rational edge demands, unrestricted variable signs and any selected integer coordinates. All graph, numerator, denominator, objective and rational query data are measured in total binary input size; dimensions and denominators vary. The actual convex hull is proved closed, with a polynomially computable facet-encoding bound, so exact boundary membership is included. This resolves the finite rational interpretation of Conforti's Problem 1, Oberwolfach Report 51/2008, pp.2904–2905. The separate subtree-hull Conjecture 2 remains unaddressed. This is an explicit assembled application of classical tools. Mixed proximity is credited to Cook–Gerards–Schrijver–Tardos (1986); hybrid partial minimization to Takamatsu–Hara–Murota (2004); scaling and ring-family lineage to Murota (2002/2003); exact submodular minimization and optimization–separation to Schrijver and GLS. The direct signed-threshold proof supplies attainment; integer labels have denominator-independent O(n) ranges. An exact LP completion oracle has uniform polynomial value encodings. A bounded priority audit located no matching full prior solution in inspected bodies, without certifying exhaustive originality or current worldwide openness. No strongly polynomial total algorithm, compact formulation, arbitrary-real oracle result or full polynomial optimizer implementation is claimed. The deposit includes the proof, TeX source, scope/priority supplement, authored independent adversarial reviews and portable exact arithmetic verification (Python 3.10 or later; standard library only). Small-case enumeration and actual defective-source controls corroborate identities and boundary examples; the written proof and established algorithms supply universal and complexity claims. AI tools were used extensively in solving, verifying, source auditing and preparation. This is an unrefereed preprint with AI adversarial reviews; conventional human peer review, journal acceptance and formal proof-assistant certification are not claimed.