Foundation Ckmlambda From Phi Ladder Phi3 Eq

A small identity with a long shadow

The golden ratio φ, the number whose square equals itself plus one, satisfies a neat companion identity: its cube equals twice itself plus one. In symbols, φ³ = 2φ + 1. Since φ is about 1.618, the left side is about 4.236, and the right side is 2 × 1.618 + 1 = 4.236. This is a piece of pure algebra, true for the positive root of x² = x + 1, and it has been known since antiquity as a consequence of the defining property.

In the Recognition Science framework, this identity is not just a curiosity. The framework's library of machine-checked formal theorems records it as a named result, phi3_eq. The statement is exactly the classical one: the cube of the golden ratio equals twice the golden ratio plus one. The proof is a short algebraic manipulation from the defining equation φ² = φ + 1, and the library certifies it with no gaps and no extra assumptions.

The reason the framework cares is a guess about particle physics. The Cabibbo angle, a number that describes how quarks mix between generations, has a measured value near 0.2247. The framework notes that the reciprocal of φ³, which is about 0.236, falls in a band that contains the measured value. The library proves that this reciprocal lies strictly between 0.225 and 0.240, a band that brackets the experimental number. This is a numerical coincidence with a named falsifier: if the measured Cabibbo angle moved outside the band, the guess would fail.

What phi3_eq itself does not claim is the physics. The theorem proves an algebraic identity about a constant; it does not prove that the Cabibbo angle equals 1/φ³, nor that the golden ratio causes quark mixing. The framework's own notes mark the connection as a structural claim, not a derivation. The identity is a small, solid brick; the wall that might rest on it remains under construction.

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

/-- 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

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

What this page does not claim

This answer does not claim that the Cabibbo angle equals 1/φ³ exactly. This answer does not claim that the golden ratio causes quark mixing. This answer does not claim that phi3_eq is a physical law.

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)

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