The Family: A Unified Framework for Nonassociative Algebras From T3 to T21 and Beyond 代数家族 T3

This paper introduces and systematically investigates the T₃ₖ₊₃ family of commutative nonassociative algebras, whose members are defined by a unified multiplication rule and a signature-(k+1, 2k+1) inner product. The family includes T3 (3-dimensional, neutrino physics), T6 (6-dimensional, universal framework), T9 (9-dimensional, supergravity R-symmetry), T12 (12-dimensional, F-theory/M-theory), T15 (15-dimensional, d=5, N=2 supergravity), T18 (18-dimensional, d=4, N=4 maximal supergravity), T21 (21-dimensional, M-theory AdS₄×S⁷), and higher-dimensional members. All members share: (i) commutativity, (ii) weak power-associativity, (iii) an exact Fundamental Theorem of Calculus in mid-point calculus, (iv) a rich automorphism group O(k+1, 2k+1), and (v) associator-induced non-perturbative effects. This paper conducts the most detailed study centered on T9 (k=2), including O(3,5) gauge field theory, topological solitons (Kink, Vortex, Monopole, Instanton), lattice Monte Carlo quantum effects, and supersymmetry breaking. Higher members T12, T15, T18, and T21 exhibit progressively deeper connections with string theory/M-theory.