deltaCP_pmns_torsion_correction
The definition supplies the explicit torsion correction to the Dirac CP phase in the PMNS neutrino mixing matrix. Neutrino modelers working with φ-quantized angles cite it when confirming the predicted δ_CP lies near 197°. The value follows from direct substitution of the 6/11 ratio (extracted from Δτ23/Δτ12) into the base expression π plus a scaled quarter-pi term.
claimThe torsion correction to the Dirac CP-violating phase in the PMNS matrix is given by $π + (6/11)·(π/4)$.
background
The PMNS matrix module derives neutrino flavor mixing from Recognition Science, where angles are φ-quantized and the atmospheric angle is near-maximal at 45°. The torsion correction is introduced via the ratio Δτ23/Δτ12 applied to a π/4 base, producing the 6/11 factor that places the phase in the third quadrant after a sign flip from sub-leading terms.
proof idea
This is a one-line definition that performs the arithmetic combination of π with the product of the 6/11 ratio and π/4. No external lemmas are invoked beyond the field operations on real numbers.
why it matters in Recognition Science
The definition is called by deltaCP_pmns_in_third_quadrant and deltaCP_pmns_range to establish the interval (π, 2π), which in turn supports the PMNSScoreCardCert structure that certifies predicted mixing parameters against data. It fills the SM-014 derivation of neutrino mixing from golden ratio geometry and connects to the eight-tick octave through the discrete torsion increments.
scope and limits
- Does not derive the 6/11 ratio from the φ-ladder.
- Does not compute the full PMNS matrix or Jarlskog invariant.
- Does not include experimental uncertainties on observed δ_CP.
Lean usage
theorem usage : Real.pi < deltaCP_pmns_torsion_correction := deltaCP_pmns_in_third_quadrant.1formal statement (Lean)
292noncomputable def deltaCP_pmns_torsion_correction : ℝ :=
proof body
Definition body.
293 Real.pi + (6 : ℝ) / 11 * (Real.pi / 4)
294
295/-- The torsion correction is in (π, 3π/2) — in the third quadrant where δ ≈ 197°. -/