Constants Electroweak Vevstructure Vev Not Free Parameter

The VEV's ledger rung

In the standard model of particle physics, the electroweak vacuum expectation value (VEV) is the energy scale at which the electromagnetic and weak forces separate. It is usually written as v and measured at about 246 GeV. It is not a derived quantity in the standard model; it is an input parameter, fixed by experiment. The VEV sets the masses of the W and Z bosons and, through the Higgs mechanism, the masses of the fundamental fermions.

Recognition Science (RS) is a framework that starts from a single postulate: reality keeps a discrete record of recognition events, and the cost of each recognition is forced, not chosen. From that cost function, a chain of theorems derives constants such as the golden ratio and the number of spatial dimensions. In this framework, mass scales are not continuous inputs. They sit on a ladder of discrete rungs, each separated by a factor of the golden ratio φ ≈ 1.618.

The declaration vev_not_free_parameter is a theorem in the framework's machine-checked library of formal theorems. It states that the electroweak scale is ledger-determined, meaning it belongs to the same fixed scale hierarchy as other masses rather than being an unconstrained input. The theorem does not compute the VEV from first principles. It establishes a structural claim: the VEV is not free. A companion theorem, vev_phi_window, pins the golden ratio to the interval between 1 and 2, and vev_implies_phi_ne_one shows the VEV structure rules out φ = 1.

The framework's own documentation is explicit about what is not yet done. The full numeric extraction of the laboratory VEV remains blocked. The derivation strategy lists four steps, and steps one and two are in progress; steps three and four wait on the closure of the electron mass derivation. The theorem vev_phi_ladder_position only proves that the ratio v/m_e is positive and that some power of φ is positive, not that the ratio equals φ^27. The value 246 GeV is hardcoded as a definition, and the theorem vev_in_range merely proves that 246 lies between 244 and 248.

What the declaration changes is the framing of a known problem. The hierarchy problem, the puzzle of why the electroweak scale is so much smaller than the Planck scale, dissolves in RS because there is no fundamental scale separation; all scales are rungs on the same ladder. The framework offers a structural answer, not a numerical one. The VEV is not free, but its precise value is still an open target.

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

/-- The VEV structural scale is pinned to the same phi interval. -/
theorem vev_phi_window : 1 < Constants.phi ∧ Constants.phi < 2 :=
  ⟨Constants.one_lt_phi, Constants.phi_lt_two⟩

/-- Electroweak-VEV structure implies `phi ≠ 1` via the inherited scale window. -/
theorem vev_implies_phi_ne_one (_h : vev_from_ledger) : Constants.phi ≠ 1 := by
  exact ne_of_gt Constants.one_lt_phi

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

/-- 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

What this page does not claim

The theorem does not compute the numerical value 246 GeV from first principles. The framework does not yet derive the electron mass, which is a prerequisite for the full VEV extraction. The theorem does not prove that v/m_e equals φ^27; it only proves the ratio is positive and bounded.

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)

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Derived articles

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