Foundation Wightman Axioms Status Lorentz Invariance

Lorentz invariance

In physics, Lorentz invariance is the principle that the laws of nature look the same for observers moving at constant velocity relative to each other. It is the mathematical backbone of special relativity, and it underpins the standard model of particle physics. In the Recognition Science framework, the term is used for a much narrower, purely algebraic statement about a function called the cost, a measure of the price of recognizing one state from another.

The framework's machine-checked library of formal theorems proves a statement it names lorentz_invariance. The statement is this: for any positive real number r, the cost J(r) equals the cost J(1/r). In symbols, J(r) = J(r⁻¹). The theorem is proved directly from the cost function's reciprocal symmetry, a property that is part of its defining conditions. The proof is a short, direct derivation, and the library reports no unproved assumptions in its development.

This algebraic identity is the entirety of what the declaration establishes. It does not construct a Lorentz transformation, it does not prove that physical laws are invariant under boosts, and it does not derive the spacetime structure of special relativity. The framework's own documentation marks the physical interpretation as an open gap: the step from this cost symmetry to a full physical Lorentz invariance is not made in the library. The declaration is a statement about a real-valued function, not about the geometry of spacetime.

What the declaration does do is place a small, precise stone in a larger structure. The framework identifies the cost's reciprocal symmetry as the seed of what it calls the Wightman axiom W0, the first of five axioms it claims hold on its Hilbert space. The other axioms, such as the spectral condition and vacuum existence, are each tied to separate properties of the same cost function. The Lorentz statement is thus one of several formal results that the framework assembles into a status report on how much of the Wightman axiomatic framework it can reproduce.

The honest summary is that this declaration proves a specific algebraic symmetry, and nothing more. A reader who wants the physical content of Lorentz invariance, the transformations of coordinates and fields, must look to conventional physics. The framework's contribution is a formal analogy, a small echo of a great principle in a different mathematical setting.

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

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

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 declaration does not prove that physical laws are invariant under Lorentz transformations. The declaration does not construct a Lorentz transformation or derive the spacetime structure of special relativity. The declaration does not establish the full set of Wightman axioms, since W4 local commutativity is not universally proved.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: