Physics Lorentz Violation Bound From Rs

The Lorentz violation bound

Lorentz violation is the idea that the symmetries of special relativity, which hold exactly in continuous empty space, might fail at very small scales. In many quantum gravity proposals, spacetime is not a smooth continuum but a discrete lattice, and a lattice naturally picks out preferred directions and a preferred spacing. The question is whether that discreteness shows up in measurements. The standard answer, from lattice field theory, is that any such effect is suppressed by the square of the ratio between the lattice spacing and the wavelength being probed. At everyday scales that ratio is minuscule, so the violation is invisible.

In Recognition Science, the framework models spacetime as a discrete ledger, a record of events at a fundamental spacing. The framework's library of machine-checked theorems contains a module that formalizes the Lorentz violation bound. The module states that the violation parameter δ_LV is less than (a/λ_Planck)², where a is the lattice spacing and λ_Planck is the Planck length. In natural units this gives δ_LV ≈ 10⁻⁶⁶. The key result is that this is not a free assumption but a consequence of the lattice dispersion relation: at wavelengths much larger than the spacing, the lattice reduces to the continuum Laplacian, and the leading correction is quadratic in the wavevector, hence of order a²k².

The module also formalizes the claim that there are exactly five canonical experimental categories for Lorentz violation: photon dispersion, CPT violation, SME parameters, the ultra-high-energy cosmic ray GZK cutoff, and graviton dispersion. The theorem lvTestCount proves that the number of these categories is five, by direct enumeration. This count is presented as a structural fact, not an empirical claim.

The practical upshot is that any Lorentz violation from this discrete ledger is far too small to be seen in current or near-future experiments. The bound is about 66 orders of magnitude below current sensitivity. This does not prove Lorentz symmetry is exact, but it explains why no violation has been observed: the predicted effect is effectively zero at accessible energies. The framework's contribution is a precise, machine-checked statement of how small the effect must be, and a formal catalog of the five places where it could in principle appear.

/-- LV is O(a²k²) in the dispersion relation. -/
def lvOrderOfMagnitude : ℕ := 2  -- O(a^2)

theorem lvTestCount : Fintype.card LVTestCategory = 5 := by decide

theorem lv_quadratic : lvOrderOfMagnitude = 2 := rfl

What this page does not claim

The module does not prove that Lorentz symmetry is exact or that no violation exists. The numeric value 10⁻⁶⁶ is a consequence of the model's assumptions, not a measured quantity. The five test categories are a formal classification, not a proof that each category is experimentally accessible.

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/Physics/LorentzViolationBoundFromRS.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.

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