Constants Electroweak Vevstructure Vev Canonical Pos

The canonical VEV

The vacuum expectation value, or VEV, is the energy at which the Higgs field settles into its lowest state, breaking the electroweak symmetry and giving mass to the W and Z bosons. In the standard model this value is not predicted; it is an input parameter fixed by experiment at about 246 GeV. The declaration vev_canonical_pos in the Recognition Science framework's machine-checked library of formal theorems does something much narrower: it proves that this canonical value, defined as the number 246, is positive. The proof is a simple arithmetic check, but it anchors the VEV as a real, usable scale instead of a symbolic placeholder.

The framework defines the canonical RS VEV as the real number 246, matching the standard electroweak scale in GeV. A companion theorem proves this value sits inside the observed range (244, 248) GeV, and another proves it is positive. These are not empirical measurements; they are formal statements about a chosen constant. The framework's structural claim, expressed in the theorem vev_not_free_parameter, is that this scale is not a free input but is determined by the ledger structure of recognition events. The word ledger here means a discrete record of events whose recognition cost is forced by a proved uniqueness theorem.

The framework goes further in its documentation, proposing that the VEV sits on a ladder of powers of the golden ratio, with the ratio v/m_e of the VEV to the electron mass near φ^27. The theorem vev_electron_rung_27_order proves only that this ratio lies between 300,000 and 600,000, which is a far weaker statement than pinning it to a specific rung. The precise rung assignment, and the full derivation of the 246 GeV value from first principles, is marked as OPEN in the framework's own notes. The derivation strategy is a plan, not a result: it requires completing the electron mass derivation first.

What the declaration establishes, then, is a formal footing: the canonical VEV is a positive real number in the observed range, and the framework treats it as ledger-determined rather than free. What it does not establish is the numerical value itself. The framework's own documentation is explicit that the full numeric extraction of the laboratory VEV remains BLOCKED. The theorem vev_canonical_pos is a small but necessary brick in a larger structure whose completion is still a target, not an achievement.

/-- The VEV is positive. -/
theorem vev_canonical_pos : (0 : ℝ) < vev_canonical := by
  unfold vev_canonical; norm_num

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

/-- 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

What this page does not claim

The numerical value 246 GeV is not derived from the framework's principles; it is a defined constant. The precise φ-ladder rung for the VEV is not established by any theorem in this pack. The framework does not prove that the VEV-to-electron-mass ratio equals φ^27; it only proves a wide bounding interval.

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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