C0

The ambient geometry is the square whose vertices are the four configurations of two binary channels: $v_{00}$, $v_{01}$, $v_{10}$, $v_{11}$. Oriented edges (bottom, top, left, right) connect them, so the complex is the standard combinatorial square with a single 2-face.

In this discrete setting one works with integer chain groups. A $0$-chain assigns an integer coefficient to each vertex; a $1$-chain does the same for edges; the $2$-chain group collapses to a single copy of $\mathbb{Z}$ because there is one face. The sibling edge type and the boundary operators $d_1$, $d_2$ live in the same module and turn these groups into a short cochain complex.

The local $V_{tx}$ inductive is distinct from the cosmological diamond-vertex abbreviation of the same name; only the four-point square is in play here.

Pure definitional abbreviation: C0 is definitionally equal to the function type Vtx → ℤ. No lemmas, tactics, or proof obligations.

This type is the carrier for vertex-supported integer data in multi-distinction geometry, the discrete complex underlying primitive recognition calculus. Downstream maximal-forcing and RS-closure machinery repeatedly names a claim parameter C0 when extending claim universes, preserving admissibility, realization, and the Forced/Independent/Selected trichotomy, and when absorbing forced invariants. The chain-group notation therefore sits at the interface between the geometric square model and the claim-closure layer that certifies completeness of forced facts. It does not itself invoke T5–T8 or the Recognition Composition Law; it supplies the combinatorial substrate those later arguments may quantify over.