This companion paper provides the rigorous mathematical constructions underlying Logic Realism Theory. We prove three key results: (1) the Hardy kernel construction derives inner product structure directly from the distinguishability metric D without presupposing Hilbert space or the Born rule; (2) LRT axioms imply all five Masanes-Müller axioms, including MM5 (entanglement connectivity) via the Lee-Selby theorem applied to CBP-enforced purification uniqueness; (3) complex quantum mechanics is the unique theory satisfying LRT axioms. The derivation chain from logical constraints (3FLL) to quantum mechanics is complete—no conditional hedges or irreducible gaps remain.