An algebraic gate for emergent Lorentzian signature: the clock–Schur criterion

In approaches where the spacetime metric is not assumed but read out from a more primitive (“premetric”) response form, a basic question is when that form yields a Lorentzian geometry — exactly one timelike direction — rather than a Euclidean or pathological multi-time one. We give a sharp, elementary answer relative to a distinguished “clock” covector. Splitting the symmetric response matrix into a clock component and a spatial block B, the geometry is Lorentzian, with inertia (1, 0, n-1), if and only if B is positive definite and the clock–Schur complement sτ = a-b⊤B-1b is negative. The proof is a one-line congruence followed by Sylvester’s law of inertia, equivalent to Haynsworth inertia additivity. We then show the criterion is independent of the lift used to represent the spatial complement, that it defines an open region with explicit stability margins mspace = λmin(B) and mtime = -sτ , and that its three boundaries classify the ways a Lorentzian readout can fail: clock degeneracy (sτ → 0), Euclideanisation (sτ > 0), and spatial signature change (λmin(B) < 0). The result is mathematically modest — a packaging of standard linear algebra — but it isolates the emergence of time into a single sign test with quantitative robustness, and supplies a falsifiable language for any model in which Lorentzian structure is meant to be derived rather than postulated.