Foundation Wightman Axioms Status Wightman Axiom Count

The axiom count

The Wightman axioms are the standard list of conditions that a quantum field theory should satisfy to be mathematically respectable. They were set out by Arthur Wightman in the 1950s. The list begins with Lorentz invariance, the requirement that the laws of physics look the same for observers moving at constant velocity relative to each other. It continues with the spectral condition, which says that energy is bounded below, and the existence of a vacuum state, the lowest-energy state of the theory. Completeness of states and local commutativity, the condition that measurements at spacelike separation cannot affect each other, complete the standard set. These five conditions are usually numbered W0 through W4.

A machine-checked library of formal theorems, the Recognition Science framework's library, contains a declaration called wightmanAxiomCount. The declaration, a theorem in the library's formal system, establishes one fact: the number of Wightman axioms in the framework's list is exactly five. The theorem is verified by direct computation over a finite list of named axioms, with no unproved assumptions and no gaps. The count is a statement about the framework's own list, which includes W0 through W4 and treats a fifth condition, Hermitian analyticity, as following from the others rather than as a separate axiom.

What the count does not claim is more important than what it does. The theorem does not prove that any physical theory satisfies the Wightman axioms. It does not prove that the axioms are consistent with each other, nor that they are complete. The framework's own documentation marks one of the axioms, local commutativity, as sector-dependent, meaning it has not yet been verified in full generality. The continuum limit, a technical step needed to connect discrete structures to continuous spacetime, is also listed as an open gap. The count is a bookkeeping fact about a list, not a verdict on the axioms' truth or applicability.

The count matters because it pins down the framework's own vocabulary. When the framework's library says "the five Wightman axioms," the number is not a casual reference; it is a verified statement with a machine-checked certificate. That precision lets a reader know exactly which conditions are being discussed and which are excluded. The remaining work, verifying local commutativity in all sectors and taking the continuum limit, is stated openly in the same module that contains the count. The count itself is settled; the physics it describes is not.

theorem wightmanAxiomCount : Fintype.card WightmanAxiom = 5 := by decide

theorem wightmanAxiomCount : Fintype.card WightmanAxiom = 5 := by decide

inductive WightmanAxiom where
  | W0_lorentz | W1_spectral | W2_vacuum | W3_completeness | W4_commutativity
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The count does not prove that any physical theory satisfies the Wightman axioms. The count does not establish that the axioms are consistent or complete. The count does not resolve the open status of local commutativity or the continuum limit.

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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