Constants Electroweak Vevstructure Vev Wz Mass Hierarchy

The W and Z mass ordering

The W and Z bosons are the particles that carry the weak nuclear force, the interaction responsible for radioactive beta decay and the fusion that powers the Sun. The Z boson is heavier than the W boson: the measured masses are about 80.4 GeV and 91.2 GeV respectively, where one GeV is roughly the mass of a proton. That ordering, Z heavier than W, is a fixed fact of the standard model of particle physics.

In Recognition Science (RS), a framework that derives physical structure from a discrete record of recognition events, this mass ordering appears as a formal theorem. The theorem vev_wz_mass_hierarchy in the framework's machine-checked library of formal theorems states that there exist positive real numbers m_W, m_Z, and v such that m_Z is greater than m_W. The proof supplies the specific values 80.4, 91.2, and 246.0. The number 246.0 is the canonical value, in GeV, of the electroweak vacuum expectation value (VEV), the energy scale at which the electroweak symmetry breaks and the W and Z acquire their masses.

What the theorem establishes is narrow but real: within the framework, the existence of a positive Z mass exceeding a positive W mass is a derived statement, not an assumption. It is a consistency check that the framework's scale structure can accommodate the observed mass hierarchy. The theorem does not derive the masses from first principles. It takes the measured values as given inputs and proves that they satisfy the ordering. The framework's own documentation marks the full derivation of the VEV from its ledger rung structure as BLOCKED, meaning the precise numeric extraction of 246 GeV from the framework's phi-ladder remains an open target.

The broader context is the framework's claim about the hierarchy problem, the puzzle of why the electroweak scale (about 246 GeV) is so much smaller than the Planck scale (about 10^19 GeV). The framework's library contains a theorem stating that the ratio of these two scales is less than 10^-15, and it interprets this not as a problem requiring fine-tuning but as a natural consequence of discrete scale rungs. That dissolution claim is a separate structural statement, not part of the mass ordering theorem. The mass ordering theorem itself is a small, verified piece: it says the heavier Z and lighter W can coexist in the framework's numerical world.

/-- **THEOREM**: The VEV is related to the W and Z boson masses through
    the φ-ladder structure.

    m_W = v/2 × g (weak coupling)
    m_Z = v/2 × √(g² + g'²)

    In RS: g and g' are also φ-ladder quantities, making the entire
    electroweak scale a single φ-scaled hierarchy. -/
theorem vev_wz_mass_hierarchy :
    ∃ (m_W m_Z v : ℝ),
      m_W > 0 ∧ m_Z > 0 ∧ v > 0 ∧
      m_Z > m_W := by
  use 80.4, 91.2, 246.0
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  · norm_num

/-- **THEOREM**: The VEV is related to the W and Z boson masses through
    the φ-ladder structure.

    m_W = v/2 × g (weak coupling)
    m_Z = v/2 × √(g² + g'²)

    In RS: g and g' are also φ-ladder quantities, making the entire
    electroweak scale a single φ-scaled hierarchy. -/
theorem vev_wz_mass_hierarchy :
    ∃ (m_W m_Z v : ℝ),
      m_W > 0 ∧ m_Z > 0 ∧ v > 0 ∧
      m_Z > m_W := by
  use 80.4, 91.2, 246.0
  constructor
  · norm_num
  constructor
  · norm_num
  constructor
  · norm_num
  · norm_num

/-- The canonical RS VEV value in GeV. Equal to the standard EW scale. -/
noncomputable def vev_canonical : ℝ := 246

What this page does not claim

The theorem does not derive the numerical values of the W and Z masses from the framework's first principles. The theorem does not establish the physical mechanism by which the electroweak symmetry breaking occurs. The theorem does not prove that the framework's phi-ladder correctly predicts the measured VEV of 246 GeV.

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/Constants/ElectroweakVEVStructure.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

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