atmospheric_correction_eq
The equality establishes that the atmospheric radiative correction coefficient equals six times the fine-structure constant. Neutrino physicists deriving the atmospheric mixing angle prediction sin²θ₂₃ = ½ + 6α would cite this when incorporating geometric corrections from the 3-cube topology. The proof is a one-line term reduction that unfolds the two defining equations and closes by reflexivity.
claimThe atmospheric radiative correction coefficient satisfies atmospheric_correction = 6α, where α is the fine-structure constant.
background
This module derives the integer coefficients (6, 10, 3/2) in PMNS mixing angle predictions from the geometry of the 3-cube voxel ledger. Atmospheric mixing is governed by cube faces: each of the six faces represents a passive mode in the Z-projection, so the correction is proportional to face count. The fine-structure constant α is taken from Constants.Alpha as α = 1/α_inv, with the module operating under the Recognition Science setting of discrete φ-tiers and 3D spatial structure (T8). Upstream results supply the Physical bridge anchors and alpha definition used in the equality.
proof idea
The proof is a term-mode one-liner. It unfolds atmospheric_correction and atmospheric_coefficient, then applies reflexivity to obtain the definitional equality.
why it matters in Recognition Science
This supplies the coefficient 6 for the atmospheric PMNS prediction sin²θ₂₃ = ½ + 6α, derived from the six faces of the 3-cube. It fills the geometric step in the module's derivation of radiative corrections and connects to the eight-tick octave and D = 3 framework landmarks. The result sits inside the broader PMNS coefficient derivations; no downstream uses are recorded yet.
scope and limits
- Does not derive the numerical value of α itself.
- Does not address solar or Cabibbo sector corrections.
- Does not compute explicit mixing angle values.
- Does not extend beyond the PMNS neutrino framework.
formal statement (Lean)
96theorem atmospheric_correction_eq : atmospheric_correction = 6 * alpha := by
proof body
Term-mode proof.
97 unfold atmospheric_correction atmospheric_coefficient
98 rfl
99
100/-! ## Derivation of the Coefficient 10 (Solar) -/
101
102/-- The solar correction coefficient is edges minus the active pair.
103
104 **Physical interpretation**: The solar sector involves 2-step torsion
105 (φ⁻² weight), coupling via edges. The 12 edges minus the 2 active
106 modes (e and ν_e in the 1-2 sector) gives 10 passive contributions. -/