unitarityAngle_beta
The declaration assigns the numerical value 22 degrees to the CKM unitarity angle beta. Researchers modeling quark flavor mixing via Recognition Science would cite this constant when checking triangle closure. It is supplied as a direct constant definition that feeds the summation identity proved in the same module.
claimThe CKM unitarity triangle angle $β$ equals $22^∘$.
background
The StandardModel.CKMMatrix module targets derivation of the CKM matrix from φ-quantized mixing angles tied to the eight-tick phase structure. The three unitarity angles are introduced as numerical constants so that their sum enforces the required closure condition on the complex plane. This setup implements the module's core insight that the CKM matrix emerges from RS with its four physical parameters.
proof idea
The declaration is a direct constant assignment of the real number 22. No lemmas or tactics are applied; the definition simply supplies the value for downstream use.
why it matters in Recognition Science
It supplies the beta component required by the triangle_sum theorem, which proves the three unitarity angles sum to 180 degrees. The definition advances the CKM construction in the Recognition Science framework by instantiating one of the φ-angles linked to the eight-tick octave. It touches the open question of exact derivation from the phi-ladder.
scope and limits
- Does not derive the 22 degree value from the J-cost or Recognition Composition Law.
- Does not specify the corresponding CKM matrix element magnitudes.
- Does not address the CP-violating phase parameter.
Lean usage
theorem triangle_sum : unitarityAngle_alpha + unitarityAngle_beta + unitarityAngle_gamma = 180 := by unfold unitarityAngle_alpha unitarityAngle_beta unitarityAngle_gamma; norm_numformal statement (Lean)
192noncomputable def unitarityAngle_beta : ℝ := 22 -- degrees