rho_bar_pos
Particle physicists checking quark mixing in the Recognition Science model cite the strict positivity of the Wolfenstein ρ-bar parameter. This matches PDG observations for the CKM matrix in the φ-angle framework. The result supports consistency checks for CP violation in the quark sector. The proof reduces to unfolding the parameter definition followed by a numerical normalization step.
claimIn the Wolfenstein parametrization of the CKM matrix, the parameter $¯ρ$ satisfies $¯ρ > 0$.
background
The module derives CKM matrix elements from φ-quantized mixing angles linked to the eight-tick phase structure. The CKM matrix is the 3×3 unitary matrix for quark flavor changes in weak interactions, parametrized by three angles and one CP phase; Wolfenstein parameters λ, A, ρ-bar, η-bar give an approximate form with observed magnitudes near 0.974, 0.227, 0.004 for the first row. Upstream results supply the active edge count A := 1 from IntegrationGap and Masses.Anchor, which enters the φ-power balance identity at D = 3.
proof idea
The proof is a one-line wrapper that unfolds the definition of the Wolfenstein ρ-bar parameter and applies norm_num to confirm the numerical value is positive.
why it matters in Recognition Science
This result establishes positivity of ρ-bar inside the CKM derivation from Recognition Science, directly supporting the module target and the paper proposition on CKM Matrix from Golden Ratio Geometry. It aligns with the eight-tick octave (T7) in the forcing chain and the overall φ-ladder structure for mixing angles. No downstream uses are recorded yet.
scope and limits
- Does not compute a numerical value for ρ-bar.
- Does not prove unitarity of the full CKM matrix.
- Does not derive other Wolfenstein parameters such as η-bar.
- Does not incorporate experimental error bounds beyond sign.
formal statement (Lean)
240theorem rho_bar_pos : (0 : ℝ) < wolfenstein_rho := by
proof body
Term-mode proof.
241 unfold wolfenstein_rho; norm_num
242
243/-- η̄ is in the RS-predicted interval (0.28, 0.40).
244 Derived from: J_CP = A²λ⁶η̄ and A = 9/11, λ ∈ (0.234, 0.238), J_CP ≈ 3.05×10⁻⁵. -/