Standard Model Ckmexact

The CKM parameter A

The CKM matrix is the standard model's bookkeeping for how quarks change flavor. When a quark decays, it can turn into a lighter quark of a different type, and the probability of each transition is set by one entry of this 3 by 3 matrix. The matrix is named after Cabibbo, Kobayashi, and Maskawa, who built it in the 1970s to explain why some decays are common and others are rare. Its four independent parameters are usually written as λ, A, ρ, and η, and the whole matrix is often expressed in the Wolfenstein parametrization, named after Lincoln Wolfenstein.

The parameter A controls the overall strength of the transitions between the second and third generations of quarks, such as a charm quark decaying into a strange quark. The Particle Data Group's 2024 review gives A = 0.826 ± 0.013, a value extracted from many measurements of B meson decays. The number is not predicted by the standard model itself; it is an input that experiments must supply. A good theory of flavor would derive A from something deeper, and that is exactly what the Recognition Science framework attempts.

In Recognition Science, the framework models the three generations of quarks as the three axes of a three-dimensional cube, a graph with 8 vertices and 12 edges. Each axis gets a flip count from a Gray code, a binary counting scheme where consecutive values differ by one bit. The framework assigns flip counts (4, 2, 2) to the three axes, and it assigns generation torsion values (0, 11, 17) to the three generations. The ratio of the torsion differences, (17 − 11)/(11 − 0) = 6/11, gives a structural value for A. A correction factor from the cube's face fluxes, 6/4 = 3/2, multiplies this to yield A = 9/11 ≈ 0.818. The machine-checked library of formal theorems proves this calculation with no axioms beyond the standard logical ones, and the result lands within 0.6 standard deviations of the measured value.

The number 44 appears as the product of the first axis flip count and the first torsion gap, 4 × 11, and it also appears in the framework's expressions for the fine-structure constant and the baryon-to-photon ratio. The framework interprets this as a common origin in the cube's chirality, the handedness of its structure. The CKM derivation is one of the framework's few concrete predictions for standard model parameters, and it is the one with the cleanest exact result: 9/11, not a fitted number.

What this establishes is a testable claim. The framework says A is not arbitrary; it is forced by the geometry of a cube to be exactly 9/11. The measurement agrees within the stated uncertainty, but the derivation is not a theorem about the physical world. It is a theorem about a mathematical model, and whether that model corresponds to nature is an empirical question that more precise measurements of A will answer.

theorem flip_axis0 : flipCount ⟨0, by norm_num⟩ = 4 := rfl

theorem flip_axis1 : flipCount ⟨1, by norm_num⟩ = 2 := rfl

theorem flip_axis2 : flipCount ⟨2, by norm_num⟩ = 2 := rfl

def tau : Fin 3 → ℕ
  | ⟨0, _⟩ => 0
  | ⟨1, _⟩ => 11
  | ⟨2, _⟩ => 17

theorem A_structural_eq : A_structural = 6 / 11 := by
  simp only [A_structural, deltaTau23, deltaTau12, tau]; norm_num

/-- 9/11 = (6/11) × (3/2) = (Δτ₂₃/Δτ₁₂) × (faceFlux₁₂/faceFlux₂₃).
    Every factor traces to D = 3 via the forcing chain. -/
theorem nine_elevenths_forced : A_corrected = (6 : ℝ) / 11 * (3 / 2) := by
  rw [A_corrected, A_structural_eq, berry_correction_eq]

theorem forty_four_connection :
    flipCount ⟨0, by norm_num⟩ * deltaTau12 = 44 := rfl

def ckmExactCert : CKMExactCert := {
  a_structural   := A_structural_eq
  a_corrected    := A_corrected_exact
  berry_factor   := berry_correction_eq
  berry_sq       := berry_sq_eq
  pdg_1sigma     := A_in_pdg_1sigma
  pdg_distance   := A_distance_from_pdg
  gap_closed     := gap_nearly_closed
  forty_four     := forty_four_connection
  flip_asymmetry := gray_asymmetry
  axis_symmetry  := gray_axis12_symmetric
  lam_interval   := lambda_RS_interval
  face_12        := faceFlux_12
  face_23        := faceFlux_23
}

What this page does not claim

The framework derives the full CKM matrix, including the phase that explains CP violation. A = 9/11 is a proven property of the physical standard model. The cube geometry is derived from the standard model; it is a model choice.

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

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