The Causal Origin of Time: A Constraint Derived from the Realizability of Physical Law

We impose an exceptionally mild but unavoidable requirement: > the extremum of a variational principle must be attainable. This attainability constraint immediately enforces the existence of a zero information-time cost propagation mode, $$dt_{\mathrm{info}} = 0 .$$ The constraint functions as a powerful axiom filter: without introducing any additional physical postulates, it logically excludes Euclidean geometry and forces an indefinite (or degenerate) metric signature. --- ## How Minimal Action Generates Causality Once attainability is enforced, causal structure is no longer optional. **Geometric constraint.** Zero information-time cost modes correspond to null intervals, $$ds_{\mathrm{info}}^{2} = 0 .$$ For nontrivial displacements, the existence of solutions to the null condition forces the local metric to possess Lorentzian signature. **Emergence of causality.** The light cone thus appears naturally as the degeneracy boundary of the variational geometry. The invariant speed $c$ is defined by the slope of this boundary, rather than assumed a priori. Special Relativity emerges as the minimal geodesic dynamics compatible with this constraint, not as an independent postulate.