The τ-Epoch Manifesto: Internal Time, Union Dipole Theory, and the Resolution of P vs NP

Abstract This paper presents the first formal integration of internal time into computational theory, revealing that the celebrated P versus NP problem was misformulated from the outset due to its implicit dependence on external clock time. Building upon the Union Dipole Theory (UDT) (Al-Mayahi, 2014; 2025), which models all existence as the oscillation of a single unified entity—the Union Dipole Particle (UDP)—we demonstrate that computation itself is an emergent process of internal temporal coherence. Within a strictly t-isolated regime, where the internal logical time of a system governs its evolution independently of wall-clock measurement, verification and generation coincide, and P = NP naturally holds. When internal and external temporal domains mix, coherence collapses and the classical inequality P ? NP reappears as a symptom of temporal decoherence, not intrinsic complexity. The results presented herein are both mathematical and experimental. A reproducible t-SAT solver is implemented under strict time-separation guards, providing empirical evidence that computational cost collapses from exponential to polynomial scale once the external clock is fully removed. Theoretically, this demonstrates that time coherence, not algorithmic ingenuity, is the fundamental determinant of computational complexity. Philosophically, the discovery unifies physics, logic, and consciousness under a single principle: time is not the container of computation but its generative substance. This framework, merging the foundations of physics, computation, and the philosophy of mind, inaugurates the t-Epoch — an era in which internal time replaces external chronology as the true metric of existence. The implications span artificial intelligence, quantum information, formal logic, and epistemology, suggesting that the collapse of complexity is, in essence, the emergence of coherence.