The Topology of Obstruction__An Executable Proof of Phase Separation in a Finite Registry.

This work presents a structural analysis of interaction, phase separation, and admissible persistence in a finite discrete registry governed by closure and adjacency constraints. Rather than assuming a unified underlying geometry, the analysis demonstrates that the registry enforces a topological obstruction: lawful configurations form disconnected islands, while interaction is mediated exclusively through non-persistent forbidden states that route strain and terminate. Direct interaction between lawful states is structurally impossible. A complete census of admissible triadic incidence shows that the registry does not admit a global Fano plane or unified octonionic geometry. Instead, it bifurcates into two immiscible structural regimes: (1) a single high-connectivity, high-chirality vortex structure, and (2) multiple low-connectivity shell structures supporting standing-loop closure. Residue-phase enforcement within the registry deterministically fixes phase values, leading to a unique generative topology at phase . When joined against the Periodic Atlas, this topology corresponds uniquely to Carbon, establishing its structural primacy as a theorem of the registry rather than a contingent chemical fact. The paper is non-normative and introduces no new physical laws or parameters. All admissibility criteria are inherited from an external Canon. To ensure reproducibility, the work includes an embedded, self-contained verification suite that mechanically audits the broken-incidence geometry, residue-phase determinism, and vortex chirality directly from the canonical dataset. This document is intended as a structural and mathematical contribution, emphasizing finite topology, executability, and falsifiability over interpretive or phenomenological modeling.