A note on complete gauge-fixing and the constraint algebra

The admissibility of a gauge-fixing is governed by the invertibility of $\Delta=\{\sigma^a,\gamma_b\}$ where $\sigma^a$ are gauge-fixing conditions and $\gamma_b$ are independent first-class constraints. We prove, via the Schur complement, that the determinant of the combined constraint matrix $\mathcal{M}=\{\Phi_A, \Phi_B\}$ built from all constraints and gauge-fixing conditions factorises as $\det\mathcal{M}\approx\pm(\det\Delta)^2\det C$, where $C$ is the second-class constraint matrix, providing an alternative criterion for admissibility. Since $\det C\neq0$ by definition, the second-class sector decouples entirely from the gauge-fixing sector. In the algebraic case, this factorisation identifies the Hamiltonian admissibility criterion of Henneaux and Teitelboim with the Lagrangian completeness criterion of Motohashi, Suyama, and Takahashi. We identify a metric ansatz as gauge-fixing at the action level and analyse completeness in the context of spherically symmetric spacetime. The factorisation ensures that completeness is robust to the second-class sector that arises in modified theories of gravity.