Foundation Wightman Axioms Status

Wightman Axioms in Recognition Science

The Wightman axioms are a set of rules, proposed by Arthur Wightman in the 1950s, that a quantum field theory should satisfy to be considered physically reasonable. They include requirements like Lorentz invariance (the laws of physics look the same in all inertial frames), the spectral condition (energy is bounded below), and the existence of a vacuum state (a state of lowest energy). A theory that satisfies all these axioms is a well-behaved quantum field theory.

In Recognition Science, the framework models physical structure from a single starting point: reality keeps a ledger, a discrete record of recognition events, and the cost of recognition is forced. This cost function, J(x) = (x + 1/x)/2 - 1, is proved to be the unique function satisfying five plain conditions. The framework's machine-checked library of formal theorems shows that this single cost function gives rise to the Wightman axioms.

Specifically, the framework proves that five of the six Wightman axioms hold on its Hilbert space. The vacuum state, where J = 0, exists at x = 1. The spectral condition is satisfied because off-vacuum states have J > 0. Lorentz invariance holds because the cost function is symmetric: J(x) = J(1/x). Completeness of states and local commutativity are also stated to hold, though local commutativity is sector-dependent and not yet universally proved.

The framework establishes that the count of these five axioms equals the framework's configDim D = 5, a number that appears elsewhere in the framework's structure. The remaining gap is W4, local commutativity, in its full universality, and the continuum limit. The framework's library verifies these results with no unproved assumptions (0 sorry, 0 axiom).

In plain language, this means the framework's core idea, a forced cost of recognition, is strong enough to reproduce the foundational rules of quantum field theory. It suggests that the Wightman axioms are not arbitrary, but follow from a deeper principle. The framework does not claim to have proved the full Wightman axioms in all generality; the sector-dependence of W4 and the continuum limit remain open targets.

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

/-- 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 full Wightman axioms are proved in all generality; W4 sector-dependence and the continuum limit remain open. The framework derives the specific particle content or interactions of the Standard Model from these axioms. The Wightman axioms are the only possible set of axioms for a 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)

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