Algebraic Self-Duality, Triplicity, and Planck-Scale Boundary Structure in the Standard Model

This work develops a structural framework in which algebraic self-duality uniquely selects internal dimension three via the identity \Lambda^2 V \cong V. We prove that the only nontrivial complex solution is \dim V = 3, providing a mathematical basis for triplicity. We show that SU(3) uniquely realizes this self-duality at the representation level through the identity \Lambda^2(\mathbf{3}) = \bar{\mathbf{3}}. Embedding SU(2)_L \times U(1)_Y into SU(3) fixes hypercharge normalization and predicts \sin^2\theta_W = 1/4 at a high scale \Lambda \sim 10^{18}\,\mathrm{GeV}. Assuming classical scale invariance at this scale forbids a tree-level Higgs quartic, which is generated radiatively. Precision three-loop Standard Model renormalization-group analyses show compatibility between the geometric boundary condition and the observed near-critical Higgs behaviour. The proposal is formulated as a structural boundary-condition framework rather than a unification theory.