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An optical Alena-Rainich-Codazzi defect is used to study finite Borel-Weil support selection on the quantized link of a codimension-three branch worldline. After real blow-up, the link is $S^2_\Gamma\simeq\mathbb{CP}^1$. The gauge-fixed Codazzi-Calderon block defines an equivariant normal operator on this boundary fiber. Its principal associated-graded part preserves the coefficient-channel splitting between the phase-current and Codazzi-gap sources. The normal two-jet supplies a dipole current source and a quadrupole Codazzi-gap source. The $\mathbb{CP}^1$ Toeplitz visibility rule assigns them to the first visible Borel-Weil blocks $E_2$ and $E_3$, giving the minimal active support $E_3\oplus E_2$. After this support has been obtained, the zero-mode basis group is $S(U(3)\times U(2))$, and the standard $3+2$ exterior-algebra package gives the one-generation multiplet structure. The same normal operator gives the finite Schur Hessian; its scale-independent bosonic reading is the on-shell neutral link angle. The locked Codazzi mass projection gives the local gap structure of the isolated low cluster. The scalar Schur unit, Callias remainder bounds, masses, mixing data, and running-coupling matching remain outside the local support theorem.