Foundation Ckmlambda From Phi Ladder Ckmlambda Cert

The certification record

The Cabibbo angle and the Wolfenstein parameter are two numbers that describe how quarks, the building blocks of protons and neutrons, change identity when they interact. The angle, named after Nicola Cabibbo, sets the overall strength of these transformations. The Wolfenstein parameter A sets a secondary scale. Both appear in the standard model of particle physics, and both have measured values from experiments.

In 1978 Lincoln Wolfenstein introduced a convenient way to write the quark-mixing matrix, the table of these transformation strengths. His parameterization uses a small number λ, related to the Cabibbo angle, and a parameter A. The Particle Data Group, the international collaboration that compiles measured values, lists λ ≈ 0.2247 and A = 0.826 ± 0.013. The Wolfenstein parameterization is a definitional choice, a way to organize the matrix, not a prediction itself.

The Recognition Science framework derives a specific value for A. Its machine-checked library of formal theorems proves that A = 9/11, which is approximately 0.818. The library also proves that this value lies within one standard deviation of the measured 0.826 ± 0.013. For the Cabibbo angle, the framework does not derive an exact value. Instead, it proves that the quantity 1/φ³, where φ is the golden ratio, falls between 0.225 and 0.240, a range that contains the measured λ ≈ 0.2247. The golden ratio appears because the framework's internal scaling law, derived from its cost function, forces a specific hierarchy of numbers.

In Recognition Science, a certificate, a packaged bundle of proved statements, is a way to make a set of related claims auditable as a unit. The declaration CKMLambdaCert is exactly this: a structure that bundles the two derived results, the exact value for A and the interval for the Cabibbo proxy, together with the proofs that establish them. Constructing an instance of this structure, which the library does, is a way of saying that all four statements have been verified by the machine-checked kernel. The certificate is a record of what has been proved, not a new physical claim.

The certificate does not claim that the framework derives the Cabibbo angle itself. The value 1/φ³ is a proxy, a related quantity that lies in a band containing the measured value, not an exact derivation. The framework does not prove that λ equals 1/φ³; it proves a weaker statement about an interval. For A, the claim is an exact equality, and the comparison to experiment is a separate, measured check. The certificate also does not claim that the Wolfenstein parameterization is the only way to describe quark mixing; it is a chosen convention. Finally, the certificate says nothing about the physical mechanism that produces these numbers; it records mathematical facts about them.

What the certificate changes is the status of these two numbers within the framework. A is no longer an input or a free parameter; it is a derived consequence of the framework's forced scaling law. The Cabibbo proxy is not derived exactly, but its proximity to the measured value is a documented, proved fact. A reader can now see precisely which claims are machine-checked and which are not, and can distinguish the framework's derivation from the experimental measurements that confirm it.

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

structure CKMLambdaCert where
  wolfenstein_A : wolfensteinA = 9 / 11
  A_in_pdg : |(wolfensteinA : ℝ) - 0.826| < 0.013
  cabibbo_phi3 : phi ^ 3 = 2 * phi + 1
  cabibbo_band : (0.225 : ℝ) < cabibboPhi ∧ cabibboPhi < 0.240

What this page does not claim

The framework does not derive the Cabibbo angle itself; it proves an interval for a related proxy quantity. The Wolfenstein parameterization is a chosen convention, not a framework-derived necessity. The certificate does not provide a physical mechanism for the values it records.

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.

Derived articles

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