Unified Field Theory 𝛽 Complete Derivation of System and Field Equations and Quantitative Formulas for Parameters of Spatial Motion

Under Zhang Xiangqian's Unified Field Theory (UFT) paradigm of "Physics is Geometry," this paper systematically constructs and derives, for the first time, a unified mathematical framework connecting the microscopic quantum world with the macroscopic classical universe—theβ system and its dynamical field equations. The theory is grounded in the spacetime background inertia strength constantK=c^3/G (dimensionMT^(-1) ), which characterizes the intrinsic inertial property of spacetime resisting changes in motion patterns.We rigorously define, from first principles, the geometric amplification factorβ_dim=(L/l_p )^(1/n) describing the statistical amplification effect of spacetime quantum fluctuations. The continuous parametern (geometric perturbation dimension) dynamically characterizes the deviation of spacetime geometry from ideal three-dimensional Euclidean space. The theory demonstrates the macro-inaccessibility ofβ_dim=1 and reveals that its practical lower bound is determined by the system-specific value of n . By introducing the mass-enhancement efficiency factorβ_2 (with dimensionM^(-1) L ) and the environmental modulation factorβ_env , we establish the core relationshipβ_1=β_dim⋅β_env , where the global inertia ratioβ_1≡K/K_eff directly quantifies the modulation of spacetime inertia strength by matter distribution. Based on dimensional analysis and physical interpretation, we derive the dynamical equation□β_2=f(n,β_dim)⋅(4πc^2/K^2)ρ forβ_2 and naturally express it in the more general covariant spacetime response formS^μν=(8π/c^2)β_2 T^μν , revealing the essence of matter driving spacetime geometric response through the dynamic coupling fieldβ_2 . To solve this coupled system and connect it to observations, we designed a weighted residual-enhanced iterative coupling method, successfully achieving high-precision joint inversion of the key parameter set(n,β_2,几何常数g) .The theory first geometrizes the reduced Planck constantℏ as a function of spacetime properties:ℏ=KL^2/β_dim^2n , and derives the fine-structure constantα as a purely geometric quantityα=1/(4π^3+π^2+π) , achieving a paradigm shift from "input parameters" to "geometric outputs" for fundamental constants. Building upon this framework, this paper systematically derives universal quantitative formulas for all parameters of spiral motion—from hydrogen atoms to black holes—based on the UFT spacetime motion postulate (cylindrical helical motion at light speed). These include radius (R ), angular velocity (ω ), linear velocity component (h ), and frequency (f ). The derived formulas exhibit remarkable agreement with observational data across nearly 30 orders of magnitude.This work provides a rigorous, self-consistent, and computable mathematical framework for UFT, offering a novel geometrization program for the unified interpretation of quantum phenomena, gravitational anomalies (such as galactic rotation curves), black hole physics, and even cosmological structures. The sole author of this paper is Binggang Huang. The submitter, Lynn Beran, served as editor and submitter of the work and does not claim authorship. Editorial involvement was limited to organization, formatting, and language clarity; all theoretical content and conclusions remain the responsibility of the author.