phi_prediction_theta12
plain-language theorem explainer
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.
Claim. The 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
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.
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