Foundation Ckmlambda From Phi Ladder Wolfenstein A In Pdg Band

The Wolfenstein A parameter

The Wolfenstein parameters λ, A, ρ, and η are a standard way to write the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes how quarks change flavor through the weak force. The parameter A is measured by the Particle Data Group (PDG) to be 0.826 with an uncertainty of 0.013. A value of 9/11, approximately 0.818, lies within that one-sigma error bar. A machine-checked library of formal theorems proves that the absolute difference between 9/11 and 0.826 is less than 0.013, establishing the prediction's consistency with experiment.

The number 9/11 is not arbitrary. It arises from a framework called Recognition Science, which starts from the idea that reality keeps a ledger, a discrete record of recognition events, and that the cost of recognition is forced by five plain conditions. From that starting point, the framework derives constants and relationships, including a prediction for the Wolfenstein A parameter. The proof that 9/11 falls within the PDG band is a formal theorem, checked by a computer, with no unproved assumptions. It is a statement about arithmetic: the rational number 9/11 is close to the decimal 0.826.

The same library also contains a related statement about the Cabibbo angle, another CKM parameter. It proves that 1/φ³, where φ is the golden ratio, lies between 0.225 and 0.240. The measured Cabibbo angle, about 0.2247, sits just outside this band. This is a separate, weaker claim, and the framework does not assert that 1/φ³ equals the measured value. The two statements together form a certificate, a bundled collection of formal results about the Wolfenstein parameters.

What the declaration does not claim is just as important as what it proves. It does not claim that the framework derives the full CKM matrix or the complete structure of quark mixing. It does not claim that 9/11 is the exact measured value; the theorem only establishes that it is within the stated error bar. And it does not claim that the Cabibbo angle is exactly 1/φ³. The proof is a precise, bounded statement about two numbers, not a sweeping claim about the entire standard model.

/-- 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

/-- 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

What this page does not claim

The declaration does not claim that 9/11 is the exact measured value of A, only that it is within the stated error bar. It does not claim that the framework derives the complete CKM matrix or the full structure of quark mixing. It does not claim that the Cabibbo angle equals 1/φ³ exactly, as the measured value lies outside the proved band.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: