Geometric Logic on the Invariant p-adic Twin

We consider a way to transform a sparse boundary data structure into an operationally complete space from which logic and geometry then arise constructively. The sensor stream defines an incomplete directed subgraph C in the rooted p-ary tree Up. In our model, Up is considered in the context of the Bruhat–Tits tree, which makes it possible to rigorously connect the ultrametric (p-adic) topology on the data boundary with continuous hyperbolic geometry in the internal volume (bulk). For it, a Twin T is constructed — a minimal completion that restores the full local branching basis at active nodes through the strict invariant deg+ C (v) + deg+ T (v) = p. In the special case p = 2, this completion construction fixes a boundary alphabet of 16 Boolean functions of two variables, allowing two equivalent realizations: a discrete one via TT/ANF and a continuous one via dissipative phase relaxation on Z2 phases. After Twin closure, the bulk structure can be constructed on the Poincaré disk via exchange relaxation of internal node states and hyperbolic coordinate barycenters. In this reading, the Twin acts not as a local technical patch, but as a condition for transition from sparse observation to full logical-geometric organization.