Foundation Wightman Axioms Status Wightman Status Cert

The status certificate

The Wightman axioms are a set of conditions, proposed by Arthur Wightman in the 1950s, that a quantum field theory should satisfy to be considered mathematically rigorous. They include requirements like Lorentz invariance, the existence of a vacuum state, and the condition that measurements at spacelike separated points commute. In the Recognition Science framework, a machine-checked library of formal theorems contains a declaration called WightmanStatusCert, which is a certificate: a packaged record of which of these axioms the framework can prove, and which it cannot.

The certificate establishes four specific facts. First, it counts five Wightman axioms in its formal list. Second, it proves the existence of a vacuum state, which in this framework is the state where the cost function J equals zero. Third, it proves the spectral condition, meaning that all states other than the vacuum have positive cost, a formal stand-in for positive energy. Fourth, it proves Lorentz invariance, expressed as the symmetry J(r) = J(1/r). Each of these is a proved theorem in the library, with no unproved assumptions.

What the certificate does not claim is just as important. It does not claim that all five Wightman axioms hold universally. The framework's own documentation identifies a remaining gap: the axiom of local commutativity (W4) is proved only in certain sectors, not in general, and the continuum limit is not yet established. The certificate is a status report, not a full proof of the Wightman axioms. It records what has been derived and marks what remains open, which is the honest state of the framework's progress.

/-- The vacuum state has J = 0 (W2: vacuum existence). -/
theorem vacuum_exists : Jcost 1 = 0 := Jcost_unit0

/-- Off-vacuum states have J > 0 (W1: spectral positivity). -/
theorem spectral_positivity {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- Lorentz invariance: J(r) = J(r⁻¹) (W0). -/
theorem lorentz_invariance {r : ℝ} (hr : 0 < r) :
    Jcost r = Jcost r⁻¹ := Jcost_symm hr

What this page does not claim

The certificate does not prove local commutativity (W4) in all sectors. The certificate does not establish the continuum limit. The certificate does not claim the Wightman axioms are the only way to axiomatize quantum field theory.

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/WightmanAxiomsStatus.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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