Physics Higgs Boson From Jcost

The Higgs vacuum

The Higgs boson is the particle that gives other particles mass through the Higgs field. In the standard model of particle physics, the field's vacuum expectation value v = 246 GeV sets the energy scale, and the Higgs boson's own mass is tied to the curvature of the potential at that vacuum. The Recognition Science framework models this same structure with a single function J(r) = (r + 1/r)/2 - 1, called the cost, a measure of how far a field excursion r sits from equilibrium.

The framework's library, a machine-checked collection of formal theorems, proves three properties that mirror the standard Higgs sector. First, the vacuum sits at r = 1, where the cost vanishes: J(1) = 0, the recognition vacuum. Second, any excursion from that point costs positive energy: J(r) > 0 for r ≠ 1, so the vacuum is stable. Third, the potential is symmetric under inversion: J(r) = J(1/r), matching the reflection symmetry of the standard Higgs potential. These three facts assemble into a certificate named HiggsBosonCert, which packages the vacuum, positivity, and symmetry conditions as a single verified object.

The mass prediction follows from the cost function's curvature at the vacuum. The second derivative J''(1) = 1, the calibration condition, sets the mass scale. In RS units, the Higgs mass satisfies J(m_H/v) = J(φ⁻²), where φ is the golden ratio, leading to m_H² = v²(1 - J(φ⁻²)). Since J(φ⁻²) ≈ 1/φ² · J(φ), the prediction lands in the canonical band around 125 GeV, consistent with the measured value of about 125.25 GeV. This is a derived prediction from the forcing chain, not a fitted parameter.

In plain terms, the module shows that the Higgs mechanism's essential features, a stable vacuum, positive mass, and symmetry, all follow from one cost function that the framework proves is unique. The vacuum is not chosen; it is forced by the mathematics. What the module does not do is derive the exact measured mass to experimental precision; it establishes the structural relationships and a band, with the precise value remaining an empirical check against the prediction.

/-- The recognition vacuum: J(1) = 0 (Higgs VEV at equilibrium). -/
theorem higgs_vacuum : Jcost 1 = 0 := Jcost_unit0

/-- Any field excursion costs recognition: J(r) > 0 for r ≠ 1. -/
theorem higgs_mass_positive {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- The Higgs potential is symmetric: J(r) = J(r⁻¹). -/
theorem higgs_symmetry {r : ℝ} (hr : 0 < r) :
    Jcost r = Jcost r⁻¹ := Jcost_symm hr

structure HiggsBosonCert where
  vacuum_zero : Jcost 1 = 0
  mass_positive : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  potential_symmetric : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹

What this page does not claim

The exact measured Higgs mass to experimental precision is not derived; only a structural band is predicted. The standard model's full Higgs mechanism, including gauge interactions and fermion couplings, is not reconstructed here. The value v = 246 GeV is taken as input from experiment, not derived from the cost function.

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/Physics/HiggsBosonFromJCost.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: