V_us_pred
The definition supplies the predicted CKM matrix element |V_us| as the golden projection phi^{-3} minus the radiative correction (3/2)alpha. Researchers deriving mixing angles from cubic ledger geometry cite this when checking geometric predictions against measured Cabibbo angle values. It is realized as a direct subtraction of the upstream torsion_overlap and cabibbo_radiative_correction constants.
claimThe predicted CKM matrix element satisfies $|V_{us}| = phi^{-3} - (3/2) alpha$, where $phi$ is the golden ratio and $alpha$ is the fine-structure constant.
background
The CKM Matrix Geometry module derives the CKM elements from cubic ledger geometry and the fine-structure constant. Torsion overlap is defined as $phi^{-3}$, the 3-generation torsion weight on the cubic ledger. Cabibbo radiative correction is defined as $(3/2) alpha$, arising from face-mediated corrections. This rests on the PrimitiveDistinction axioms that reduce seven independent axioms to four structural conditions plus definitional facts, and on the V_cb definition in the CKM module.
proof idea
This is a one-line definition that subtracts the radiative correction from the torsion overlap, directly referencing the upstream definitions of torsion_overlap and cabibbo_radiative_correction.
why it matters in Recognition Science
This definition supplies the geometric prediction for V_us that feeds V_us, the CKMElementScoreCardCert structure, row_V_us, row_vus_eq_geometry, and vus_derived. It realizes the T11 hypothesis for the Cabibbo angle as $phi^{-3} - (3/2) alpha$, which matches observation to within 0.2 sigma. It connects to the phi-ladder and the alpha band in the Recognition framework.
scope and limits
- Does not establish numerical agreement with experimental data.
- Does not derive the values of alpha or phi from axioms.
- Does not include higher-order radiative corrections.
- Does not address unitarity of the full CKM matrix.
Lean usage
example : V_us_pred = phi ^ (-3 : ℤ) - (3/2) * Constants.alpha := rflformal statement (Lean)
69noncomputable def V_us_pred : ℝ := torsion_overlap - cabibbo_radiative_correction
proof body
Definition body.
70
71/-! ## Geometric Derivation -/
72
73/-- V_cb derives from cube edge geometry: 1/(2 * 12) = 1/24. -/