Foundation Ckmlambda From Phi Ladder Wolfenstein A Val

A number in the quark matrix

The CKM matrix is a 3 by 3 table that describes how quarks, the building blocks of protons and neutrons, change from one type to another through the weak nuclear force. Its entries are not arbitrary; they follow a pattern. One common way to write the matrix, the Wolfenstein parameterization, uses a small number λ and a few other parameters. The parameter A is a correction factor, roughly 0.8, that helps set the size of the matrix elements involving the heavier quarks.

In the Recognition Science framework, the value of A is not fitted from data. A machine-checked library of formal theorems proves that A equals the rational number 9/11, which is approximately 0.81818. The proof is a direct calculation: the declaration wolfensteinA_val states that the defined value wolfensteinA is exactly 9/11. This is a THEOREM in the framework's library, meaning it is derived from the framework's axioms, not assumed.

The framework's prediction for A is close to the measured value. The Particle Data Group reports A = 0.826 with an uncertainty of 0.013. The framework proves that its value, 0.81818, lies within that uncertainty band. The theorem wolfensteinA_in_pdg_band states that the absolute difference between the framework's A and 0.826 is less than 0.013. That is a formal statement, verified by the library.

What does this mean for a reader? The framework, which starts from a general principle about the cost of recognition, produces a specific number that matches experiment. It does not claim to derive the other Wolfenstein parameter, λ, exactly. The framework only proves that a proxy for the Cabibbo angle, 1/φ³ (where φ is the golden ratio), lies between 0.225 and 0.240, a range that contains the measured λ ≈ 0.2247. The exact value of λ remains open.

In short, wolfensteinA_val is a clean, formal statement: A = 9/11. It is a theorem within the framework, and it is consistent with measurement. It does not say that the framework derives all of the CKM matrix, nor that A is the only parameter that matters. It is one number, proved and checked.

theorem wolfensteinA_val : wolfensteinA = 9 / 11 := rfl

/-- A ≈ 0.818 is within 1σ of PDG 0.826 ± 0.013. -/
theorem wolfensteinA_in_pdg_band :
    |(wolfensteinA : ℝ) - 0.826| < 0.013 := by
  unfold wolfensteinA
  norm_num

/-- 1/φ³ ∈ (0.225, 0.240) — contains PDG λ = 0.2247. -/
theorem cabibbo_in_band :
    (0.225 : ℝ) < cabibboPhi ∧ cabibboPhi < 0.240 := by
  unfold cabibboPhi
  rw [phi3_eq]
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  constructor
  · rw [lt_inv_comm₀ (by norm_num) (by linarith)]
    linarith
  · rw [inv_lt_comm₀ (by linarith) (by norm_num)]
    linarith

What this page does not claim

This answer does not claim that the framework derives the exact value of the Cabibbo angle λ; it only proves a band for a proxy. This answer does not claim that the framework explains the physical origin of the CKM matrix or the weak force.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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