The Inevitable Geometric Chain Deriving the Eight Fundamental Constants from the Spontaneous Doubling Operator

This manuscript presents a complete mathematical and geometric-algebraic proof chain demonstrating that the eight fundamental constants of physics (π, α, ħ, e, c*, G, RU, H0) are not independent empirical inputs. Instead, they emerge strictly from a single geometric operator within the framework of Arc Theory. By defining Absolute Arc Geometry as the congenital constraint form of energy-potential, and Relative Arc Geometry as the resultant energetic-interaction on the spacetime geodesic, the paper details the mechanism of the Law of Spontaneous Doubling (LSD). The LSD operator acts in two non-isomorphic forms (Temporal/Number and Spatial/Quantity) upon the T(1,2) vacuum manifold. This unified derivation provides: The topological closure condition yielding π. The maximum spatial density limit generating the fine-structure constant α. The phase-locking and structural velocity defining ħ and c*. The isotropic spherical projection (4π) yielding the elementary charge e. The logarithmic entropic veto on temporal divergence defining the gravitational constant G. The macroscopic reference radius RU and the resulting kinematic Hubble constant H0. This single-operator framework unifies the quantum, electromagnetic, and cosmological scales without introducing any ad hoc parameters, dynamical degrees of freedom, or dark-energy adjustments, offering a purely geometric foundation for physics. Arc Theory: Geometric-Algebraic Proof Chain: https://doi.org/10.5281/zenodo.19209299