phi_prediction_theta13
The definition supplies a Recognition Science prediction for the reactor neutrino mixing angle as the golden ratio divided by 100. A neutrino phenomenologist would cite it when testing φ-quantized angles against reactor data such as Daya Bay. The construction is a direct one-line assignment of the constant phi from the foundation.
claimThe predicted value for the squared sine of the reactor mixing angle is given by $sin^2 θ_{13} ≈ φ/100 ≈ 0.01618$, where $φ = (1 + √5)/2$ denotes the golden ratio.
background
In the PMNS matrix module the neutrino mixing angles are treated as φ-quantized quantities. The golden ratio φ is the self-similar fixed point forced at step T6 of the unified forcing chain. The module contrasts this approach with tribimaximal mixing, which predicts a vanishing θ₁₃, and introduces φ-corrections to account for the observed nonzero value near 8.6°.
proof idea
One-line definition that directly assigns the value of phi divided by 100.
why it matters in Recognition Science
The definition supplies Hypothesis 3 inside SM-014, the module that derives the PMNS matrix from Recognition Science. It supplies the φ-based input for θ₁₃ that feeds the larger claim of golden-ratio geometry for neutrino mixing angles. No downstream theorems yet reference it, leaving open the quantitative match to the observed 0.022 value.
scope and limits
- Does not derive the full set of PMNS matrix elements.
- Does not assert exact numerical agreement with the measured sin²θ₁₃ ≈ 0.022.
- Does not include higher-order φ-corrections or renormalization effects.
formal statement (Lean)
104noncomputable def phi_prediction_theta13 : ℝ := phi / 100
proof body
Definition body.
105
106/-- **Hypothesis 4: Tribimaximal mixing (TBM) + corrections**
107
108 TBM predicts:
109 - sin²θ₁₂ = 1/3 = 0.333
110 - sin²θ₂₃ = 1/2 = 0.5
111 - sin²θ₁₃ = 0 (wrong!)
112
113 Reality deviates from TBM by φ-corrections. -/