Exchange-based two-qubit logic admits a regime of zero-distance control in projective Hilbert space. For the exchange family U_Li(rho) = exp(ipirho*H_XY) interpolating between identity and iSWAP, the Fubini-Study (QFIM) component satisfies the exact identity g_rr = pi^2 Var_psi(H_XY) and is invariant along the exchange orbit. Hence metric degeneracy occurs precisely on generator eigenstates. Crucially, the antisymmetric sector H_- is a subspace-level degeneracy: every state is an eigenstate, so g_rr = 0 and the rho-direction lies in a symmetry-protected kernel of the metric tensor. In contrast, the symmetric sector exhibits only state-dependent degeneracy. Metric collapse further enforces exact gradient death for any cost Hamiltonian. Under symmetry-breaking perturbations that mix exchange sectors, the collapsed direction reopens continuously with g_rr proportional to epsilon^2 and ds proportional to epsilon d rho. KeywordsQuantum Fisher information; Fubini-Study metric; exchange symmetry; metric degeneracy; gradient death; symmetry breaking; geometric teleportation