Self-Reconstructing Codazzi Defects, $\mathbb{CP}^1$ Quantization, and the Minimal Standard-Model Carrier

Abstract

A local reconstruction problem for a primitive optical Codazzi defect in a four-dimensional Lorentzian branch is analyzed. After resolution of the worldline core, the link is identified with $\mathbb{CP}^1$, and the primitive transverse class fixes $L_Γ\simeq\mathcal O(1)$. The resulting Borel-Weil tower and the $\mathbb{CP}^1$ Toeplitz visibility cutoff are used to turn a filtered transverse source into a finite internal carrier. For scalar-sector sources of transverse order at most two, two separated non-scalar channels are isolated: a $V_1$ phase-current channel and a $V_2$ trace-free Codazzi-gap channel. Separated central implementability and Toeplitz visibility select $E_3\oplus E_2$ as the minimal carrier, with mixed low blocks carrying only half-integer $SU(2)$ types. The finite determinant-obstruction count has the same local kernel as the top-form condition and gives the compact split basis group $S(U(3)\times U(2))$. The determinant-compatible even exterior package realizes a local one-generation Standard-Model module. The determinant global form further gives a $\mathbb Z_3$ family-response torsor. This torsor fixes a mod-three family factor, while exact family multiplicity and numerical vacuum, mass, mixing, and CP data are formulated as outputs of the subsequent Dirac-Callias/Riesz/Schur-Berry completion. An Alena-type residual collar is exhibited as a sufficient two-channel source.