The Cabibbo Angle from Plane Misalignment in the Three-Plane Scaffold

The Cabibbo angle derivation in this paper is superseded by a later result. The current scaffold prediction, derived from the instanton quark-loop structure, is given in doi:10.5281/zenodo.20175630.

The three-plane scaffold predicts the Cabibbo angle through an overlap-maximisation between the lepton hidden-plane basis and the quark crossed-plane basis of the junction. The maximisation condition gives tan(2θ_C) = √w/N = √2/3, where N=3 is the number of scaffold planes and w=2 is the spinorial double-cover index, both independently fixed by the Finkelstein-Rubinstein condition. This predicts θ_C = 12.620° and sin(θ_C) = 0.2185, against the PDG value λ = 0.2254 ± 0.0001 (Wolfenstein parameter), a 3.05% discrepancy consistent with the product-ansatz approximation level throughout the scaffold.

The result extends the scaffold's electroweak table to four observables: sin²θ_W = w/N² (3.9% error), m_W/m_Z = √(N²−w)/N (0.05% error), g_A = (5/3)√((N²−w)/E) (0.02% error), and λ ≈ sin(θ_C) (3.1% error). The same two constants N and w that control the first three also control the leading Wolfenstein parameter. No parameters are fitted.

The derivation rests on the overlap-maximisation plus one structural assumption: the misalignment between the two bases is shared symmetrically relative to the fixed junction frame, so the physical CKM rotation angle is half the total relative rotation. This assumption is physically motivated (neither basis is more fundamental than the other relative to the junction) but has not been derived from the action. Deriving it, and verifying the overlap coefficients N and √w from the zero-mode wavefunctions on φ_exact, are the two remaining steps to a full derivation from the Lagrangian. The higher Wolfenstein parameters A, ρ, η and CP violation are not yet predicted.

Companion papers:

Koide relationship (doi:10.5281/zenodo.19174656) ·

LFV prediction (doi:10.5281/zenodo.19202565) ·

Lepton as 3D apparition (doi:10.5281/zenodo.19203027) ·

Fine-structure constant (doi:10.5281/zenodo.19211790) ·

Quark sector (doi:10.5281/zenodo.19228848) ·

Mass spectrum and Koide cone (doi:10.5281/zenodo.19267506) ·

Redistribution prediction (doi:10.5281/zenodo.19301348) ·

Numerical field data (doi:10.5281/zenodo.19344105) ·

Scalar-fermion Lagrangian (doi:10.5281/zenodo.19380160) ·