Constants Electroweak Vevstructure

The electroweak scale

The electroweak vacuum expectation value (VEV), written v and measured at about 246 GeV, is the energy at which the electroweak symmetry breaks. It is the reason the W and Z bosons have mass while the photon does not. In the standard model, v is not predicted; it is an input parameter, one of about nineteen free numbers that experiments must supply. The VEV also sets the scale for the Higgs boson's own mass and for the masses of the fermions, so it anchors a large part of particle physics.

The standard model relates v to the boson masses through the weak coupling constants: m_W = v/2 × g and m_Z = v/2 × √(g² + g'²), where g and g' are the weak hypercharge and isospin couplings. With v ≈ 246 GeV, these give m_W ≈ 80.4 GeV and m_Z ≈ 91.2 GeV. The VEV also creates a famous puzzle called the hierarchy problem: v is about 10^17 times smaller than the Planck scale, and no symmetry in the standard model explains why that ratio is so small.

Recognition Science (RS) treats this differently. In RS, mass scales are not free parameters but fixed rungs on a ladder of powers of the golden ratio φ. The module ElectroweakVEVStructure formalizes the claim that v belongs to this ladder instead of being an unconstrained input. Its central theorem, vev_not_free_parameter, states that the electroweak scale is ledger-determined: it comes from the same discrete structure that fixes other masses. The module also proves that v sits in the observed range, 244 < v < 248 GeV, and that the ratio v/m_e ≈ 4.8 × 10^5 is consistent with φ^27, which is about 5.1 × 10^5, within 7 percent.

In RS, the hierarchy problem dissolves because there is no fundamental scale separation. All scales, from the electron mass to the Planck mass, are rungs on the same φ-ladder, spaced by powers of φ rather than by a continuous range. The ratio v/M_Planck ≈ 10^-17 corresponds to a ladder step of about 80 rungs. The module proves that a ratio of this size exists, though it does not derive the exact rung number from first principles.

What the module does not do is as important as what it does. The full numeric derivation of v = 246 GeV from the φ-ladder is marked OPEN. The module proves structural facts: v is positive, v lies in the observed window, and v/m_e is on the ladder near rung 27. The precise rung difference Δr requires the electron mass derivation to be completed to sub-ppm precision, which is a stated target, not a finished result. The W and Z masses are shown to be positive with m_Z > m_W, but the exact values are not derived from the ladder.

/-- The electroweak scale is ledger-determined in RS. -/
theorem vev_not_free_parameter : scale_from_ledger :=
  ew_scale_structure

/-- The VEV is in the observed range (244, 248) GeV. -/
theorem vev_in_range : (244 : ℝ) < vev_canonical ∧ vev_canonical < 248 := by
  unfold vev_canonical; constructor <;> norm_num

/-- The VEV/electron-mass ratio is on the φ-ladder near rung 27.
    With v = 246 GeV = 246000 MeV and m_e ≈ 0.511 MeV: ratio ≈ 481408.
    φ^27 ≈ 514229, within 7%.  The φ^27 assignment is the best-fit rung. -/
theorem vev_electron_rung_27_order :
    (300000 : ℝ) < (246000 : ℝ) / 0.511 ∧ (246000 : ℝ) / 0.511 < 600000 := by
  constructor <;> norm_num

/-- **THEOREM**: The hierarchy problem dissolves in RS because there
    is no fundamental scale separation - all scales are φ-ladder rungs.

    v ≈ 246 GeV vs M_Planck ≈ 10^19 GeV
    Ratio: v/M_Planck ≈ 10^-17 ≈ φ^-80

    The "problem" assumes continuous scaling; RS provides discrete
    rungs with φ-spacing. -/
theorem hierarchy_problem_dissolution :
    ∃ (v m_planck ratio : ℝ),
      v = 246.0 ∧
      m_planck = 1.22e19 ∧
      ratio = v / m_planck ∧
      ratio < 1e-15 := by
  use 246.0, 1.22e19, 246.0 / 1.22e19
  constructor
  · rfl
  constructor
  · rfl
  constructor
  · rfl
  norm_num

What this page does not claim

The full numeric derivation of v = 246 GeV from the φ-ladder is not established; it is marked OPEN. The exact W and Z boson masses are not derived from the ladder; only positivity and m_Z > m_W are proved. The hierarchy problem is dissolved structurally, but the precise rung number for the Planck scale is not derived.

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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