The Arrow of Time as Signature Monotonicity

We derive an arrow of time on the Minimal Discrete Épure G■ as the monotonicity of signature under deployment. Encoding each axis as flagged (temporal, negative-signature) or free (spatial, positive-signature), the deployment D→2D induces a signature-transition matrix whose central property is monotonicity: the number q of flagged axes never decreases. We prove this three independent ways: (A) algebraically, by Sylvester's law of inertia (extending a quadratic form by direct sum preserves existing signs); (B) geometrically, the causal vertex Θ■=arctan 2 forms a barrier — de-flagging would require crossing the null cone (the limiting velocity); (C) dynamically, the cost-asymmetric Wilson gauge action exhibits the same barrier (vanishing kinetic term at the vertex). The result is discriminant: it holds for the Lorentzian saddle x²−4y² but fails for Euclidean signature (no time). It is a structural derivation; the bridge to a laboratory observable is named as an open problem. Verified numerically.