Physics Gauge Boson Masses From Rs

Gauge boson mass ratio

The gauge bosons W and Z carry the weak nuclear force, one of the four fundamental interactions. Their masses are not equal: the Z boson is heavier. The standard model of particle physics predicts the ratio m_Z/m_W through the weak mixing angle, but it does not explain why that angle has the value it does.

Recognition Science offers a derivation. The framework's central result forces a unique cost function for recognition events, and from that cost function a chain of theorems produces the golden ratio φ, about 1.618, as a fundamental scaling constant. The module GaugeBosonMassesFromRS takes this constant and defines two quantities. The mass ratio is m_Z/m_W = 6/(3+φ), which evaluates to about 1.298. The weak mixing angle appears as sin²θ_W = (3-φ)/6, about 0.230.

These are not free parameters fitted to data. They are definitions built from the golden ratio. What the framework proves, in a machine-checked library of formal theorems, is that these definitions land inside the measured range. The ratio is positive and greater than one, so the Z is heavier than the W. The sine-squared value lies strictly between 0.228 and 0.232, the band where experiments place it.

The physical bridge from recognition events to actual particle masses remains open. The framework establishes the arithmetic: if the golden ratio is the right scaling constant, then the gauge boson mass ratio follows. It does not yet connect that ratio to the full mechanism of mass generation in the standard model.

theorem massRatio_pos : 0 < massRatio :=
  div_pos (by norm_num) (by linarith [phi_gt_onePointSixOne])

theorem massRatio_gt_one : massRatio > 1 := by
  unfold massRatio
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hd : (0:ℝ) < 3 + phi := by linarith
  rw [gt_iff_lt, lt_div_iff₀ hd]
  -- Need 3+φ < 6, i.e., φ < 3
  linarith

theorem sin2thetaW_band :
    (0.228 : ℝ) < sin2thetaW_RS ∧ sin2thetaW_RS < 0.232 := by
  unfold sin2thetaW_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  constructor
  · have : (3 - phi) / 6 > (3 - 1.62) / 6 := by
      apply div_lt_div_of_pos_right _ (by norm_num)
      linarith
    linarith
  · have : (3 - phi) / 6 < (3 - 1.61) / 6 := by
      apply div_lt_div_of_pos_right _ (by norm_num)
      linarith
    linarith

What this page does not claim

The framework does not derive the individual masses of the W and Z bosons, only their ratio. The framework does not explain the physical origin of the weak mixing angle, only its value in terms of the golden ratio. The framework does not connect the golden ratio to the Higgs mechanism or any other mass generation process.

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/Physics/GaugeBosonMassesFromRS.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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