Standard Model Wzmass Ratio

The W and Z boson mass ratio

The W and Z bosons are the particles that carry the weak nuclear force, one of the four fundamental forces of nature. The W boson has a measured mass of about 80.4 GeV, and the Z boson has a measured mass of about 91.2 GeV, where GeV is a unit of energy used for particle masses. Their ratio, m_W / m_Z, is measured to be 0.8815 with an uncertainty of 0.0002, a value that is precisely known from experiments at particle colliders.

This ratio is not just a random number. In the standard model of particle physics, the ratio equals the cosine of the Weinberg angle, a parameter that describes how the electromagnetic and weak forces mix. The measured value of the ratio, 0.8815, corresponds to a Weinberg angle whose sine squared is about 0.223. The Weinberg angle is a fundamental input to the standard model, not something the theory predicts on its own.

In Recognition Science, the framework models the electroweak mixing angle as being constrained by the golden ratio phi, the number approximately equal to 1.618. The framework's library defines several candidate formulas for the mass ratio, each built from phi in a different way. For example, one hypothesis is the square root of 1 minus 1 divided by phi squared, another is (phi + 1) divided by (phi + 2), and a third is the square root of 1 minus 1 divided by (2 times phi plus 1). The library also defines a prediction for sin squared of the Weinberg angle as 1 divided by (2 times phi plus 1), which gives about 0.236.

The framework's library proves only narrow facts about these definitions, such as that the mass ratio lies between 0.87 and 0.89, which is consistent with the measured value. It does not prove that any single phi-based formula is the correct one. The list of predictions in the library states that sin squared of the Weinberg angle is related to 1 divided by (2 times phi plus 1), that the mass ratio is about 0.88, and that the angle is constrained by an eight-tick geometry, a concept from the framework's recognition cycle.

The measured value of 0.8815 is close to the phi-based predictions, but the match is not exact. The prediction of 0.236 for sin squared of the Weinberg angle differs from the measured 0.223 by more than the experimental uncertainty. The library itself records this as a hypothesis in progress, with a falsifier structure that names the mass ratio measurement and the sin squared measurement as the checks. Until a phi-based formula matches the measured value within its error bars, the connection between the golden ratio and the W and Z boson masses remains an open prediction, not a derived result.

/-- W boson mass (GeV). -/
noncomputable def m_W : ℝ := 80.377

/-- Z boson mass (GeV). -/
noncomputable def m_Z : ℝ := 91.1876

def experimentalStatus : List WZFalsifier := [
  ⟨"m_W / m_Z measurement", "0.8815 ± 0.0002, precisely known"⟩,
  ⟨"sin²(θ_W) measurement", "0.2229 ± 0.0003"⟩,
  ⟨"φ-connection", "In progress - promising"⟩
]

/-- **THEOREM**: Mass ratio is approximately 0.88. -/
theorem mass_ratio_value : massRatio > 0.87 ∧ massRatio < 0.89 := by
  unfold massRatio m_W m_Z
  constructor <;> norm_num

/-- In RS, the Weinberg angle emerges from 8-tick phase geometry:

    1. The 8 phases form a group: Z₈
    2. The electroweak group SU(2) × U(1) embeds in this
    3. The embedding angle is constrained by φ
    4. This gives sin²(θ_W) related to 1/(2φ + 1)

    Specifically: sin²(θ_W) ≈ 1/(2φ + 1) = 1/4.236 ≈ 0.236

    Compare to observed: 0.223. Error: ~6% -/
noncomputable def sin2ThetaW_predicted : ℝ := 1 / (2 * phi + 1)

def experimentalStatus : List WZFalsifier := [
  ⟨"m_W / m_Z measurement", "0.8815 ± 0.0002, precisely known"⟩,
  ⟨"sin²(θ_W) measurement", "0.2229 ± 0.0003"⟩,
  ⟨"φ-connection", "In progress - promising"⟩
]

What this page does not claim

No phi-based formula is proved to equal the measured mass ratio. The Weinberg angle is not derived from the framework; it is defined from the measured masses. The eight-tick geometry is mentioned as a prediction, not demonstrated as a derivation.

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/StandardModel/WZMassRatio.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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