Tau Unified Field Theory: A Gradient-Based Framework for Unifying the Four Fundamental Forces

This paper presents the Tau Unified Field Theory (TUF), a novel framework thatunifies gravity, electromagnetism, the strong nuclear force, and the weak nuclear forcethrough a single scalar field τ (tau). In TUF, gravity emerges as the gradient of theτ field: ⃗Fg = −G∇τ, while the other three forces are encoded in the structure andparameters of the field. The strong force is represented by the nuclear τ field withdecay length λn = 0.8 a0 (∼0.42 Å), corresponding to hadronic scales. The weakforce is encoded in the finite interaction range of the τ field, and electromagnetism isgoverned by the fine-structure constant α = 1/137.036.Numerical simulations of the first 20 elements (H to Ca) demonstrate that 19/20elements achieve force balance within ±10% accuracy (|FEM/Fg − 1| < 0.1), with hydrogen showing perfect balance at the Bohr radius. Key parameters are determinedfrom first principles: gravitational coupling G = 0.014775, electron anti-tau factorτe = −0.3, and electronic decay length λe = 1.5 a0. TUF provides a mathematically consistent framework for four-force unification at atomic scales and suggests thatgravity may be an emergent phenomenon arising from information gradients.Keywords: unified field theory, emergent gravity, tau field, fine-structure constant, fundamental forces, atomic physics