An 11-Dimensional Topological Continuum and Space Field Substructure: Geometrical Formulation of Gauge Interactions, Planar Vortices, and the Resolution of the Hierarchy Problem

Modern physics encounters severe mathematical divergences and conceptual ambiguities due to its reliance on continuous quantum field theories operating against an isotropic, multi-dimensional spacetime background. This paper establishes a non-singular, alternative framework by modeling the cosmos as an 11-dimensional topological continuum structured upon a pairwise symmetric decahedral lattice. Rejecting the traditional assumption of dimensional equivalence, we partition the manifold into a single, invariant, global temporal monotonicity parameter ($t$) acting as a thermodynamic pressure drive, and 10 internal spatial dimensions structured via interlocking 2D planar sub-manifolds. Within this multi-planar hyper-space, gauge fields are derived as intrinsic geometric field orientations rather than independent physical entities. Fundamental fermions are rigorously formulated as localized topological vortices (holonomy singularities) residing within specific interaction planes, while gauge bosons emerge as multi-component transverse wave modes generated by mechanical resonance between intersecting planes. This Space Field Theory configuration naturally accounts for the three generations of elementary particles and geometric fractional charges without requiring a fundamental scalar Higgs field. Furthermore, we show that while gauge forces are confined to highly stretched 2D planar surfaces ($S_2$), the gravitational field tension is diluted across the 3D interior spatial volume ($V_3$). This dimensional volumetric dilution provides a purely geometric resolution to the gauge hierarchy problem at the microscopic scale.