How Pattern Algebra Derives the Planck Mass M_P Without Free Parameters

A complete first-principles derivation: axioms → ħ, c, G → M_P = √(ħc/G)with full non-circularity proof and SI unit equivalence Abstract: The Planck mass M_P = √(ħc/G) ≈ 2.176×10 ⁸ kg ≈ 1.2209×10⁻ ¹⁹ GeV/c² is thefundamental mass scale of nature — the point where quantum mechanics and gravity areequally strong. In standard physics, M_P is computed from the measured values of ħ, c, and G;no framework derives these three constants from first principles. This paper presents thecomplete Pattern Algebra (PA) derivation of M_P from axioms A1–A11 and the Hurwitz theorem,without circular reasoning and without free parameters.