Foundation Wightman Axioms Status Spectral Positivity

The positive-energy condition

The spectral condition is one of the Wightman axioms, the standard list of requirements that a quantum field theory must satisfy to be considered physically reasonable. In plain terms, it says that the energy of any physical state, measured in any inertial frame, is never negative. This is a basic fact about how the universe works: you cannot extract an unlimited amount of energy from a system, and no state has less energy than the vacuum itself.

The Recognition Science framework contains a machine-checked library of formal theorems. Within that library, a declaration called spectral_positivity proves a specific instance of this condition. It shows that for any positive real number r that is not equal to 1, the value of the framework's cost function Jcost(r) is strictly greater than zero. The cost function is a measure of how expensive it is to recognize a state; the framework models the energy of a state as this cost. The theorem states that only the state with Jcost equal to 1 has zero energy, and every other state has positive energy.

This result is a theorem, meaning it is proved with no gaps and no unproven assumptions in the library's logic. It is derived from a more general theorem about the cost function, Jcost_pos_of_ne_one, which itself follows from the defining properties of the cost function. The proof is complete and checked by the machine.

However, the declaration does not claim to prove the full Wightman axiom W1 for a real quantum field theory. The library's own documentation states that the spectral condition is one of five axioms that hold on the framework's Hilbert space, but it also identifies a remaining gap: the local commutativity axiom (W4) is only proven to hold in certain sectors, and the continuum limit is not yet established. The spectral_positivity theorem is a statement about the abstract cost function, not a proof that a specific, physically realized quantum field theory satisfies all the Wightman axioms.

What this means for a reader is that the framework has a proven, rigorous core result about the positivity of its own energy measure. It is a necessary piece of a larger structure, but it is not the whole structure. The theorem is a foundation stone, not the completed building.

/-- 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

def wightmanStatusCert : WightmanStatusCert where
  five_axioms := wightmanAxiomCount
  vacuum := vacuum_exists
  spectral := spectral_positivity
  lorentz := lorentz_invariance

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

What this page does not claim

The declaration does not prove the full Wightman axiom W1 for a real, physically realized quantum field theory. The declaration does not establish that all five Wightman axioms hold universally in the framework. The declaration does not provide a physical model of particles or fields; it is a statement about an abstract 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/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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