deltaCP_pmns_in_third_quadrant
plain-language theorem explainer
The torsion correction to the PMNS CP-violating phase lies strictly between π and 3π/2. Neutrino phenomenologists citing Recognition Science derivations of mixing angles reference this bound to locate the phase in the third quadrant, matching the observed value near 197°. The proof unfolds the explicit correction formula and discharges both inequalities by linear arithmetic on the positivity of π.
Claim. Let δ denote the torsion correction to the CP-violating phase in the PMNS neutrino mixing matrix. Then π < δ < 3π/2.
background
In the Recognition Science treatment of the Standard Model the PMNS matrix is built from φ-quantized angles, producing the observed large neutrino mixing angles θ₁₂ ≈ 34°, maximal θ₂₃ ≈ 45°, and θ₁₃ ≈ 8.6°. The torsion correction is defined explicitly as π + (6/11)(π/4), arising from the ratio of angle defects scaled by π/4 together with a sign flip from sub-leading terms. The module SM-014 frames this construction as deriving the Pontecorvo-Maki-Nakagawa-Sakata matrix from golden-ratio geometry.
proof idea
The proof unfolds the definition of the torsion correction, splits the conjunction, and applies linarith to each conjunct using the positivity of π.
why it matters
This bound is invoked directly by the downstream theorem that places the full PMNS CP phase in (π, 2π), aligning with observation. It fills a concrete step in the SM-014 derivation of neutrino mixing angles from φ-geometry within the Recognition framework, where φ-connections account for the pattern of large mixing angles.
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