Foundation Qrft Higgs Potential From Recognition Vacuum Higgs Nonneg

The nonnegative potential

The Higgs potential, the energy function that gives mass to elementary particles, is usually written as V(H) = −μ²|H|² + λ|H|⁴. For the standard model, this potential has its minimum when the field magnitude |H| equals v/√2, about 174 GeV, the electroweak vacuum expectation value. The potential itself is a quartic curve that dips to zero at that minimum and rises on both sides. A reader might ask: can this potential ever be negative? In the standard treatment, the quartic form is chosen so that it is bounded below, but the question of nonnegativity is a property of the specific parameters.

In Recognition Science, the framework models the same potential as a cost function on the ratio r = |H| / (v/√2), the field strength measured against its vacuum value. The framework's cost, a measure of how far a configuration sits from a preferred state, takes the form J(r) = ½(r + r⁻¹) − 1. This function is zero at r = 1, which is the vacuum, and it grows as r moves away from 1 in either direction. The framework proves, in a machine-checked library of formal theorems, that this cost is never negative for any positive field ratio. That is the content of the declaration higgs_nonneg: for every positive r, the potential J(r) is greater than or equal to zero.

The theorem is not a numerical approximation; it is an exact statement. The proof relies on the fact that for any positive r, the arithmetic mean ½(r + r⁻¹) is at least 1, so subtracting 1 leaves a nonnegative result. This is a classical inequality, but the framework encodes it as a formal theorem about the Higgs potential. The declaration also comes with three companion results: the vacuum has zero potential, the potential is symmetric under r → 1/r, and the vacuum is the unique minimum. Together, these four properties form a certificate that the potential has the shape a physical vacuum needs.

What the declaration does not claim is just as important. It does not prove that the standard model's actual potential is nonnegative for all field values; that is a property of the standard model's parameters, not a theorem about them. It does not derive the Higgs mass or the vacuum expectation value from first principles; those are inputs to the framework, not outputs. And it does not claim that the framework's cost function is the only possible form for the Higgs potential; it is a modeling choice, not a uniqueness proof. The theorem is a statement about the framework's own defined potential, and it is true within that definition.

For a reader, the practical takeaway is simple: in this framework, the vacuum is a stable, lowest-energy state, and the potential never dips below it. That stability is a necessary feature for a physical vacuum, and the framework provides a formal guarantee of it. The declaration is a small piece of a larger structure, but it shows how a classical physics concept can be recast as a cost-minimization problem and then verified with machine precision.

/-- The potential is non-negative (all field values above vacuum). -/
theorem higgs_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ higgsPotential r :=
  Jcost_nonneg hr

What this page does not claim

The theorem does not prove the standard model's potential is nonnegative for all field values; it only applies to the framework's defined cost function. The theorem does not derive the Higgs mass or the vacuum expectation value from first principles. The theorem does not show the cost function is the only possible form for the Higgs potential.

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/Foundation/QRFT/HiggsPotentialFromRecognitionVacuum.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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