Closure Geometry and Unified Dynamics: A Ten-Dimensional Action for Mass, Scalar Relaxation, Quantization, and Curvature

This paper presents a mathematically explicit unification framework in which proto-spinor alignment, inertial mass, scalar relaxation, discrete spectral structure, gravitational curvature, and complex quantum phase structure arise as structural consequences of a single closure geometry defined on an admissible modal configuration space. The construction begins with a ten-dimensional configuration manifold equipped with a closure-strain field and a local closure cost functional satisfying three key conditions: the existence of an alignment minimum, a positive-definite quadratic normal form on anchored directions near alignment, and divergence as a nil-boundary coordinate approaches the admissibility limit. Synchronization constraints define a distribution on an induced Lorentzian encoding whose integrability depends on the closure field. Within this framework, alignment is identified with the global minimum of the closure cost, inertial mass with anchored quadratic displacement from alignment, the Higgs scalar with the unique radial relaxation mode in closure space, and discrete spectra with nil-boundary survivorship under finite coherence capacity. Gravitational curvature appears as the integrability obstruction of synchronization constraints under nonuniform closure strain, while complex quantum phases arise as holonomy of a principal U(1) circle bundle in the coherent sector. These mechanisms are simultaneously realized by a single covariant ten-dimensional action functional containing an Einstein–Hilbert term, a closure-strain sigma model, a nil-boundary barrier term, and a minimal curvature–strain coupling. Under coherent truncation and admissible projection, the action reduces to a four-dimensional Einstein–Hilbert action coupled to a Standard-Model-like gauge and scalar sector, with controlled higher-derivative corrections. The paper does not assume spontaneous symmetry breaking, primitive metric quantization, or primitive Hilbert-space structure. It is a structural and action-level unification result rather than a complete numerical or phenomenological completion.