Exclusion of Multi-Time Signatures and the Uniqueness of Lorentzian Continuum Phases

This paper establishes a no-go result for continuum theories with multiple time-like directions within a well-defined class of local second-order differential operators. Under standard assumptions of strong hyperbolicity and coercive energy estimates, it is shown that effective continuum phases admitting more than one time-like direction generically fail to support a well-posed global initial value problem. The analysis proceeds by examining the principal symbol structure of the governing operator and demonstrating that multi-time signature leads to non-unique or inconsistent characteristic propagation, instability under perturbation, and failure of predictive coherence. In contrast, Lorentzian signature (one distinguished direction) uniquely satisfies the hyperbolicity and stability conditions required for well-posed evolution within the specified operator class. The result is structural rather than dynamical: it does not assume a specific microscopic model, field content, or quantization scheme. Instead, it identifies necessary constraints on admissible continuum phases compatible with predictive coherence. This work forms part of a broader investigation of emergent Lorentzian structure, complementing: Dynamical Emergence of Lorentzian Spacetime from a Non-Spatiotemporal Substrate (Zenodo DOI: 10.5281/zenodo.18131656.) Admissibility and Stability Constraints on Emergent Lorentzian Spacetime (Zenodo DOI: 10.5281/zenodo.18839586.) Together, these works develop a programmatic framework in which Lorentzian signature arises not as an imposed geometric assumption but as a structural consequence of admissibility and stability.