Foundation Ckmlambda From Phi Ladder Cabibbo Phi

A number from the golden ratio

The Cabibbo angle is a number in particle physics that measures how strongly quarks of different generations mix when they decay through the weak force. Its sine, often written as λ in the Wolfenstein parameterisation, is measured at about 0.2247. The recognition science framework, which builds physical constants from a forced cost function, defines a candidate for this quantity as the reciprocal of the golden ratio cubed: cabibboPhi = 1/φ³. Since φ³ equals 2φ + 1, which is about 4.236, the candidate is about 0.236.

The framework's machine-checked library of formal theorems proves that this candidate lies strictly between 0.225 and 0.240. That interval contains the measured value 0.2247, but it is wider than the measurement's own uncertainty, so the theorem does not single out the measured number. The declaration is a definition, not a derivation: it chooses a specific expression involving the golden ratio and then proves a band around it. The choice of 1/φ³ itself is not forced by the framework's axioms; it is a proposed identification.

In Recognition Science, the same file proves a separate result for the Wolfenstein parameter A, which controls the overall strength of the mixing. The framework sets A equal to 9/11, about 0.818, and proves this lies within one standard deviation of the measured 0.826 ± 0.013. This is a theorem with a machine-checked proof. The Cabibbo angle candidate, by contrast, only earns the band statement, not a point prediction.

What the declaration does not claim is as important as what it proves. It does not claim that the Cabibbo angle is exactly 1/φ³, nor that the framework derives the measured value from first principles. The band theorem is a consistency check: the golden ratio produces a number in the right neighborhood. The exact value of the Cabibbo angle, and the question of whether the framework can pin it down precisely, remain open targets.

/-- 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

/-- Cabibbo angle proxy: 1/φ³. -/
noncomputable def cabibboPhi : ℝ := (phi ^ 3)⁻¹

What this page does not claim

The exact Cabibbo angle is not derived; only a containing interval is proved. The expression 1/φ³ is not a forced consequence of the framework's axioms. The measured value 0.2247 is not reproduced to its own precision.

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: