Koide Common Reducts and Proper Extensions: Kocik Circle Geometry, Diagonal Jordan Peirce Data, and Support-Completed C₃ Spectral Sectors

A rank-three quadratic object is isolated whose null cone is precisely Koide's charged-lepton mass relation after the substitution rᵢ = √mᵢ. The object is the triple (ℝ³, N, τ) with Euclidean norm N and trace functional τ(r) = r₁ + r₂ + r₃, together with the Lorentz formQ_K(r) = 2/3 τ(r)² − N(r).We prove that three constructions reduce functorially to this same object: Kocik's equiangular generalized Descartes circle equation at parameter p = 2/3 with one zero bend, the diagonal Peirce subalgebra of the exceptional Jordan algebra J₃(𝕆), and a support-completed C₃ compact-momentum sector with sharp spectral floor 1/(9ρ²). The main result is a common-reduct/proper-extension theorem. The common-reduct part is an isomorphism theorem in an explicitly defined category KQ. The proper-extension part is a collection of invariant-theoretic non-isomorphism theorems: the Kocik extension has a pointed hyperplane invariant, the full octonionic Jordan algebra has Peirce profile (8,8,8), the support-completed C₃ system has an infinite compact-momentum spectrum and a support idempotent not reconstructible from the rank-three reduct, and the four-dimensional Descartes and Koide--Kocik forms are real-congruent but rationally projectively non-equivalent. All claims are formulated as explicit morphisms, invariants, or obstruction theorems.