Labeling without Constructing: The DD Origin of the Weak Mixing Angle
A Conditional Theorem under the EW Matching Axiom

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Abstract

This paper establishes a conditional theorem on the weak mixing angle within the Self-as-an-End (SAE) framework. One generation of fermions comprises exactly 16 = 2⁴ Weyl components, reflecting a four-layer binary structure where each layer corresponds to a DD level: chirality (4DD), isospin (2DD), and color/lepton identity (3DD). The first DD level (1DD), being the unique abelian layer, contributes no multiplicity but only labels — it "labels without constructing." The sole fully gauge-trivial component is the right-handed neutrino (ν_R, y_SAE = 0), which serves as the group-theoretic identity of the four-bit encoding. Under a new Electroweak Matching Axiom, the DD eigenvalue of the weak mixing angle equals the ratio of 1DD-nontrivially-labeled states (15 = 2⁴ − 1) to the DD Splitting structural capacity (65):

sin²θ_W|_{DD eigen} = (2⁴ − 1) / (C(12,2) − 1) = 15/65 = 3/13 ≈ 0.2308

The experimental value ŝ_Z²(M_Z) = 0.23122 ± 0.00006 (PDG 2025) differs from the DD eigenvalue by +4.51 × 10⁻⁴ (relative deviation −0.195%, approximately 7.5σ), consistent in magnitude with one-loop electroweak radiative corrections. This constitutes a conditional theorem whose conditions are the Paper II one-generation hypercharge table, the Prequel's structural capacity 65, and the EW Matching Axiom above. The axiom belongs to the same class of structure-to-EFT bridge axioms as the Prequel's κ = 65/4 postulate.

Keywords: Self-as-an-End, weak mixing angle, Weinberg angle, DD hierarchy, fermion spectrum, electroweak mixing, DD eigenvalue