Three Theorems from One Equation: su(3), su(2), and 3+1 Dimensions from -z = 1/z

We derive three uniqueness theorems from the equation -z = 1/z. (1) D=3 is the unique positive integer for which the centralizer of the charge operator and involution on C^ (2D) contains exactly one simple Lie algebra factor; that factor is su (3). (2) The quaternions are the unique normed division algebra where solutions to -z = 1/z close under the commutator to form a simple Lie algebra; that algebra is su (2), whose complexification is the Lorentz algebra so (3, 1). (3) For all D, the gauge-singlet sector is two-dimensional, giving a universal su (2) with weak-isospin-like properties. At D=3 these combine to give su (3) + su (2) + u (1) + u (1), the same Lie algebra type as the Standard Model gauge algebra plus one additional u (1). Machine-verified computational scripts are included as supplementary material. Manuscript preparation was assisted by Claude (Anthropic) ; the author takes full responsibility for all mathematical content.