Within the Recognition Science framework, the Global Co-Identity Constraint (GCIC) states that all stable recognition boundaries share a single, universe-wide phase parameter Θ ∈ [0, 1). In this paper, we derive GCIC from the canonical cost functional J(x) = 1/2(x + x^-1) - 1. The proof has three ingredients: (i) ratio-only dependence of J yields a continuous U(1) phase symmetry; (ii) strict convexity of J in log-coordinates yields a strictly positive edge penalty for nonzero phase mismatch; (iii) connectedness of Z3 propagates local equality to global equality. We formulate the derivation first on finite connected subgraphs (where the energy is well-defined and coercive), then pass to the infinite lattice by exhaustion. The resulting theorem is that any finite-volume ground state is phase-uniform, and every thermodynamic-limit ground state inherits a single global phase. We also show that the 8-tick neutrality constraint commutes with phase shifts, and we derive the quadratic stiffness of small fluctuations around the uniform phase vacuum.