Standard Model Ckmmatrix

The CKM matrix in Recognition Science

The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a 3 by 3 unitary matrix that describes how quarks change flavor in weak interactions. Its four physical parameters, three mixing angles and one complex phase, set the probabilities for transitions between the up-type quarks (up, charm, top) and down-type quarks (down, strange, bottom). The matrix also encodes CP violation, the asymmetry between matter and antimatter behavior, through a single measurable quantity called the Jarlskog invariant.

In the Wolfenstein parameterization, the matrix is built from four real numbers: lambda, A, rho, and eta. The measured values from the Particle Data Group give lambda around 0.225, A around 0.82, rho around 0.14, and eta around 0.35. These produce the familiar hierarchy of magnitudes, with diagonal elements near 1 and off-diagonal elements shrinking by powers of lambda.

In Recognition Science, the framework models the CKM matrix as emerging from phi-quantized mixing angles related to an eight-tick phase structure. The framework's library defines the Wolfenstein parameters and the matrix elements as constants, and it establishes several interval results about them. For example, it establishes that rho lies between 0.10 and 0.20, that eta lies between 0.28 and 0.40, and that the Jarlskog invariant is positive but below 1e-4.

The framework also defines a set of hypotheses connecting the Cabibbo angle to the golden ratio phi, which solves r² = r + 1 and equals about 1.618. One hypothesis sets the sine of the Cabibbo angle to (phi - 1)² / phi, about 0.236. Another sets it to 1 / (2 * phi), about 0.309. A third uses sin(pi / (4 * phi)), about 0.368. These are definitional choices, not derived results, and the framework does not establish which one matches nature.

The library establishes that the three unitarity angles, alpha, beta, and gamma, sum to 180 degrees, which is a consistency condition for any unitary matrix. It also establishes that rho² + eta² is less than 1, which places the apex of the unitarity triangle inside the unit circle. These are formal consequences of the chosen constants, not measurements.

The framework's library lists four predictions for the CKM parameters, including the claim that lambda is approximately (phi - 1)² / phi. These are hypotheses with named falsifiers: if future measurements of the Cabibbo angle exclude 0.236 at high precision, that hypothesis fails. The framework does not derive the CKM matrix from first principles; it proposes phi-based values and checks them against the measured intervals.

/-- The Wolfenstein parameter λ = sin(θ_c) ≈ 0.227. -/
noncomputable def wolfenstein_lambda : ℝ := cabibboAngle

/-- The CKM matrix in Wolfenstein parametrization (to O(λ³)):

     ⎛ 1 - λ²/2      λ           Aλ³(ρ - iη) ⎞
V =  ⎜   -λ        1 - λ²/2         Aλ²       ⎟
     ⎝ Aλ³(1-ρ-iη)   -Aλ²            1         ⎠
-/
noncomputable def V_ud : ℂ := 1 - wolfenstein_lambda^2 / 2

/-- ρ̄ is in the RS-predicted interval (0.10, 0.20).
    From unitarity triangle with δ = π/2: the real part ρ̄ ≈ 0.13. -/
theorem rho_bar_interval : (0.10 : ℝ) < wolfenstein_rho ∧ wolfenstein_rho < 0.20 := by
  unfold wolfenstein_rho; constructor <;> norm_num

/-- η̄ is in the RS-predicted interval (0.28, 0.40).
    Derived from: J_CP = A²λ⁶η̄ and A = 9/11, λ ∈ (0.234, 0.238), J_CP ≈ 3.05×10⁻⁵. -/
theorem eta_bar_interval : (0.28 : ℝ) < wolfenstein_eta ∧ wolfenstein_eta < 0.40 := by
  unfold wolfenstein_eta; constructor <;> norm_num

/-- **THEOREM**: CP violation is small but nonzero. -/
theorem cp_violation_small :
    jarlskogInvariant > 0 ∧ jarlskogInvariant < 1e-4 := by
  unfold jarlskogInvariant
  constructor <;> norm_num

/-- **THEOREM**: Unitarity triangle angles sum to 180°. -/
theorem triangle_sum :
    unitarityAngle_alpha + unitarityAngle_beta + unitarityAngle_gamma = 180 := by
  unfold unitarityAngle_alpha unitarityAngle_beta unitarityAngle_gamma
  norm_num

/-- The unitarity constraint ρ̄² + η̄² < 1 holds (required for V unitary). -/
theorem unitarity_triangle_valid :
    wolfenstein_rho^2 + wolfenstein_eta^2 < 1 := by
  unfold wolfenstein_rho wolfenstein_eta; norm_num

What this page does not claim

The framework does not derive the CKM matrix from first principles; it proposes phi-based values. The phi-based hypotheses are definitional choices, not established results. The framework does not claim that the golden ratio uniquely determines all CKM parameters.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/StandardModel/CKMMatrix.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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