phi_prediction_theta12
The definition supplies a Recognition Science baseline for the solar neutrino mixing parameter as sin²θ₁₂ = 1/(1 + φ²) ≈ 0.276. Neutrino modelers deriving the PMNS matrix from φ-quantized geometry would cite it as the starting hypothesis before corrections. It is introduced by direct algebraic assignment of the golden-ratio expression.
claimThe predicted squared sine of the solar neutrino mixing angle satisfies $sin^2 θ_{12} = 1/(1 + φ^2)$ where φ denotes the golden ratio.
background
The PMNSMatrix module derives the Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix from Recognition Science, positing that the large mixing angles θ₁₂ ≈ 34°, θ₂₃ ≈ 45°, θ₁₃ ≈ 8.6° arise from φ-quantized geometry rather than the small angles of the CKM matrix. The module targets a full φ-angle construction and flags the maximal θ₂₃ as evidence of an underlying symmetry, with a potential PRD paper on golden-ratio neutrino mixing noted in the module documentation.
proof idea
The declaration is a direct one-line definition that assigns the algebraic expression 1/(1 + phi^2) to the real-valued constant. No lemmas or tactics are invoked; the upstream constants A, correction factor, and actualization operator supply the φ-ladder context but are not applied inside this definition.
why it matters in Recognition Science
This supplies the first explicit φ-mixing hypothesis in the PMNS derivation, providing the baseline sin²θ₁₂ that subsequent angle predictions such as phi_prediction_theta13 would extend. It instantiates the T5 J-uniqueness and T6 self-similar fixed-point landmarks by using φ as the scale for mixing, while the module documentation positions it as the opening step toward a complete RS-derived PMNS matrix. The 10% offset from observation remains an open correction question.
scope and limits
- Does not derive the expression from the Recognition Composition Law or forcing chain axioms.
- Does not incorporate the finite-N channel-capacity correction or projection-defect bounds from upstream results.
- Does not compute the full PMNS matrix or the other mixing angles.
- Does not address the observed 0.307 value or supply a deviation mechanism.
formal statement (Lean)
89noncomputable def phi_prediction_theta12 : ℝ := 1 / (1 + phi^2)
proof body
Definition body.
90
91/-- **Hypothesis 2: Maximal θ₂₃ from symmetry**
92
93 sin²θ₂₃ = 1/2 (maximal mixing)
94
95 Observed ≈ 0.545, close to maximal but slightly off.
96 A small φ-correction could explain the deviation. -/