Solution to the Yang–Mills Mass Gap Problem Based on SEP in TRR–NOTIME Framework

This document presents a formal, self-consistent solution to the Yang–Mills Mass Gap problem based on the TRR–NOTIME framework, where time is eliminated as a fundamental physical quantity. Instead of temporal evolution, the framework introduces a layered structure of spatial projections defined by directional energy potential (SEP). Axioms and postulates are clearly formulated to replace the standard time-dependent Hilbert space with a timeless but layered directional space, retaining all key features necessary for a Yang–Mills-type gauge theory: symmetry, nonlinearity, confinement, and structure-preserving projections. The mass gap Δ>0 is proven formally as a direct consequence of the omnidirectional, non-zero vacuum field , whose projection onto any directional layer is strictly positive. This projection yields a bounded below spectrum of interaction amplitudes, establishing the existence of a spectral gap without introducing time, geometry, or additional fields. The document includes a section mapping TRR–NOTIME structures to Yang–Mills concepts, a philosophical commentary on the origin of interaction, and a formal spectral analysis consistent with the Clay Mathematics Institute’s criteria (with the exception of peer-reviewed publication, addressed via this open repository). An appendix is provided: TRR–NOTIME Interpretation of Riemann Zeros and the Universe Without Dark Energy, which extends the theoretical implications of the SEP framework toward cosmology and number theory, offering a structural reinterpretation of observational anomalies without hypothetical constructs.