Within the unified framework of PFUSRC, this paper establishes the ontological foundation of convex optimization and duality theory based on the 45° triple coaxial bicone and Pythagorean Frustum topology. It is rigorously proved that core structures—including convex functions, concave functions, non-convex non-concave transition functions, strong duality theorem, optimality conditions and KKT conditions—are not artificially defined formal mathematical constructs, but necessary geometric projections of the bicone topology. The lower cone corresponds to convex functions and primal minimization; the upper cone corresponds to concave functions and dual maximization; the middle frustum corresponds to duality gap closure and the establishment of strong duality. Based on PFUSRC₀0 original topology, PFUSRC₄8 group isomorphism and structural invariance, this paper realizes the rigorous unification of algebraic structure, geometric topology and convex analysis, and proves that all principles of convex optimization originate from the inherent cosmic ontology.
Zhenmin Wang (Thu,) studied this question.