Encoding the Four Fundamental Forces in a Single Tau Field at Atomic Scales

This paper presents a novel theoretical framework where the four fundamentalforces—strong nuclear, weak nuclear, electromagnetic, and gravitational—are en-coded within a single scalar field τ (r) at atomic scales. The theory is based on aforce balance equation FEM + Fτ = 0, where Fτ = −Gτ ∇τ represents the unifiedforce contribution. Calibrated from the hydrogen atom, the theory yields parame-ters Gτ = 0.006274 (in atomic units), λn = 0.8072a0, and λe = 0.5504a0. Predic-tions for atomic radii of 20 elements (Z=1-20) show an average error of 9.3%, with15/20 elements predicted within 10% error. While electromagnetic forces dominateat atomic scales, the τ field provides a unified mathematical framework encodinginformation from all fundamental interactions, offering a simplified approach toatomic structure prediction.