T₃-Algebraic Unified Medium Dynamics From Bubble Nucleation to Electromagnetic Fields via Discrete Fractal Geometry T₃ 代数统一介质动力学 从气泡成核到电磁场:离散分形几何的路径

Abstract We construct a unified field-theoretic framework based on the three-dimensional fractal algebra T₃ with basis {1, e, i} and constraints e² = 1, i² = −1, e·i = 0. The radial component β drives bubble nucleation and gravitational compression, while the angular component γ generates vortex spin and magnetic circulation. A double-well potential U′(β) = β(1 − β/βc) realizes genuine threshold nucleation, and a soft activation function A(β) controls the angular mode coupling. The framework is embedded in the Discrete Exhaustive-Reservoir Energy Gradient (DEREG) geometry, where the discrete vector Helmholtz operator Dℓ yields Yukawa-type medium propagation with damping coefficient ξ = 1/φ fixed by the golden-ratio steady state. We demonstrate that (i) symmetric bubbles exhibit spontaneous spin decay, (ii) shear-induced vortices require external asymmetry, (iii) current magnetism arises from β̇ × γ statistical polarization, and (iv) the macroscopic electromagnetic fields E, B are statistical coarse-grained images of the underlying vacuum medium circulation, compatible with Maxwell's equations in the continuum limit.