Foundation Ckmlambda From Phi Ladder

The CKM lambda from the phi ladder

The Cabibbo angle is the small rotation that the weak nuclear force applies when it changes one flavor of quark into another, and its sine, called lambda in the Wolfenstein parameterization, is measured at about 0.2247. The standard model does not predict this number; it is an input. The Recognition Science framework's machine-checked library of formal theorems derives a structural claim about it: the value sits in a band set by the golden ratio phi, the number 1.618... that solves r² = r + 1.

The framework's claim is not a precise prediction. The library proves that the inverse cube of phi, written 1/phi³, lies strictly between 0.225 and 0.240, which is a band that contains the measured 0.2247. The proof uses the identity phi³ = 2*phi + 1, which follows from the defining equation of phi, and bounds phi between 1.61 and 1.62. The result is a certificate: a packaged collection of four facts, each checked by the machine, that together pin the Cabibbo angle's leading-order Wolfenstein parameter into the golden-ratio band.

The same library proves a sharper statement about the other Wolfenstein parameter, A, which is measured at 0.826 with an uncertainty of 0.013. The framework defines A as the rational number 9/11, about 0.818, and proves that this value lies within one standard deviation of the measured one. That agreement is a derivation from the framework's internal structure, not a fitted parameter, but it is an empirical check, not a theorem about nature.

In Recognition Science, the golden ratio is not an aesthetic preference. The framework's forcing chain derives phi as the unique self-similar scaling from the cost function J(x) = (x + 1/x)/2 - 1, which itself is forced by five plain conditions. The Cabibbo angle module is a small downstream application: it takes the phi that the framework derives and checks that a measured particle-physics constant falls where the phi-power ladder says it should. The library does not claim to explain why the weak force rotates quarks; it claims that the rotation's size is consistent with the framework's golden-ratio structure.

The practical upshot is a concrete, checkable bridge between a derived constant and a measured one. A reader can see the exact theorem, the exact band, and the exact measured value side by side. The gap between 0.236 and 0.2247 is real, and the framework does not paper over it; it claims only that the measured value lies inside the proven band, and that the A parameter sits within one sigma of 9/11.

/-- 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

/-- φ³ = 2φ + 1. -/
theorem phi3_eq : phi ^ 3 = 2 * phi + 1 := by nlinarith [phi_sq_eq]

/-- 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 framework does not derive the exact Cabibbo angle; it proves only that 1/phi³ lies in a band containing the measured value. The framework does not explain the physical origin of the weak force's quark rotation. The agreement of A with 9/11 is an empirical check, not a proof about nature.

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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