Geometric Sector Proximity as an Alternative to Gauge Unification

Distance Fields, Sector Geometry, and Emergent Unification

抄録

We propose a geometric reformulation of gauge-sector relationships based on a unified generator manifold, introducing inter-sector distances as fundamental physical quantities. In contrast to conventional approaches that assume unification through algebraic embedding into a larger symmetry group, we describe sector mixing in terms of geometric proximity, with interaction strength suppressed by inter-sector distance.
We show that the observed interaction hierarchy emerges naturally from a robust ordering of distances, $d_{12} \ll d_{13}, d_{23}$, corresponding to the relative proximity of U(1)-like, SU(2)-like, and SU(3)-like sectors. By promoting these distances to dynamical fields $d_{ij}(x)$, we obtain a distance-field framework in which gauge interactions, sector mixing, and spacetime geometry become interconnected.
Furthermore, we introduce a projection-based interaction principle, in which all interactions arise as projections of an underlying flow structure, with geometric separation controlling their strength. In this picture, unification is interpreted not as a fixed symmetry identity but as a geometric limit of small inter-sector distance, while gravity is understood as the geometry governing the dynamics of sector separation.
This suggests that unification and gravity are not independent structures, but arise from a common geometric framework governed by sector proximity and flow projection.