rho_bar_pos
plain-language theorem explainer
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.
Claim. In 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
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.
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