triangle_sum
Unitarity triangle angles α, β, γ in the CKM matrix sum to exactly 180 degrees from their assigned numerical values. Flavor physicists checking CP-violation consistency would cite this when validating Recognition Science quark-mixing derivations. The proof is a direct term reduction that unfolds the three constant definitions and applies numerical normalization to the arithmetic sum.
claim$α + β + γ = 180^∘$, where α, β, γ are the angles of the unitarity triangle formed by the rows of the CKM matrix.
background
The module constructs the CKM matrix from φ-quantized mixing angles that arise from the eight-tick phase structure. Unitarity of the 3×3 matrix produces a closed triangle in the complex plane whose interior angles must sum to 180°. Upstream results supply the active-edge count A (IntegrationGap and Masses.Anchor) and the actualization operator A (Modal.Actualization) that underwrite the coherence assumptions used to assign the angle values.
proof idea
The term proof unfolds unitarityAngle_alpha, unitarityAngle_beta, and unitarityAngle_gamma, then applies norm_num to verify the numerical identity 85 + 22 + 73 = 180.
why it matters in Recognition Science
The declaration confirms geometric closure of the CKM unitarity triangle inside the Recognition Science treatment of quark flavor mixing. It directly supports the module target of deriving the matrix from golden-ratio geometry and aligns with the T7 eight-tick octave in the unified forcing chain. The embedded comment sketches φ-based predictions for λ, A, and η/ρ, marking an open verification path against measured CKM parameters.
scope and limits
- Does not derive the numerical angle assignments from the J-cost function or Recognition Composition Law.
- Does not compute the CKM matrix elements or the CP-violating phase.
- Does not connect the angles to the mass ladder or Berry creation threshold.
- Does not prove full unitarity of the CKM matrix beyond the triangle sum.
formal statement (Lean)
196theorem triangle_sum :
197 unitarityAngle_alpha + unitarityAngle_beta + unitarityAngle_gamma = 180 := by
proof body
Term-mode proof.
198 unfold unitarityAngle_alpha unitarityAngle_beta unitarityAngle_gamma
199 norm_num
200
201/-! ## φ-Predictions for CKM -/
202
203/-- RS predictions for CKM parameters:
204
205 1. λ ≈ (φ - 1)² / φ ≈ 0.236 (vs observed 0.227)
206 2. A ≈ related to φ
207 3. η/ρ ≈ φ (possible?)
208 4. Unitarity triangle angles ≈ φ-related
209
210 These would be profound if verified! -/