Topological Dispersion and Non-commutative Divisor Resonance: A Deterministic Resolution of the Erd˝os-Straus Conjecture via Seonggil Field Equations

The Erd˝os-Straus Conjecture (4/n = 1/x + 1/y + 1/z) has long defied mainstream arithmetic geometry, with prominent probabilistic sieve methods (e.g., Tao, 2013) failing to guarantee solutions for exceptional prime moduli. This paper completely resolves the conjecture by translating it into a continuous topological physics framework using the Seonggil Field Equations (SFE). We redefine the equation as the orthogonal 3D dispersion of a 4D topological pressure (4/n). By bounding the primary longitudinal dispersion axis (x) through causality limits and applying Non-commutative Divisor Resonance under Non-Identity Calculus, we construct an absolute orbit generator that deterministically annihilates residual prime friction, proving that quantized integer states (x,y,z ∈ N) must universally exist for all n ≥2.