Emergent Causality in Multitemporal Manifolds: A Reformulation through Semigroup Restrictions in the Lagrangian

Theoretical physics models incorporating multiple time dimensions (T > 1) have been historically dismissed due to severe pathologies: causality violation, Ostrogradsky ghosts, and dynamical instability. We argue these anomalies are not intrinsic to multitemporal dimensionality, but arise from a categorical error: treating time as an isotropic, reversible dimension.

We propose a geometric framework where unidirectionality is elevated from emergent property to definitional axiom by dynamically restricting temporal evolution to a positive semigroup (R+)^T via a logarithmic barrier term in the Lagrangian. Under this fundamental constraint, multiple temporal dimensions collapse into a scalar Effective Time (T_eff), recovering hyperbolicity of wave equations and ensuring stable causal structure.

We explicitly demonstrate how this formalism resolves known pathologies in Bars' 2T-physics without requiring massive gauge symmetries. Finally, we show that the observable speed limit (V_L) emerges as a property dependent on the internal architecture of the temporal flow vector.

This approach focuses on internal logical consistency for T > 1 frameworks, providing rigorous foundations for any theory postulating multiple times.