Foundation Qrft Gauge Tree Amplitudes Cert

The gauge tree amplitude triad

In particle physics, a tree amplitude is the leading-order quantum mechanical probability amplitude for a scattering or decay process, computed from Feynman diagrams without loops. The module GaugeTreeAmplitudesCert in the framework's machine-checked library of formal theorems records a structural prediction for three canonical Standard Model (SM) processes: Compton scattering (γe⁻ → γe⁻), pair annihilation (e⁺e⁻ → γγ), and W⁺W⁻ → ZZ unitarisation.

The central object is the cost function J(r) = (r + 1/r)/2 − 1, which the framework proves is the unique function satisfying five plain conditions. The module defines each amplitude as J applied to the relevant coupling ratio r, so the amplitude is zero when r = 1 (threshold) and grows monotonically off-threshold. Three theorems anchor the behavior: amplitude_zero_at_threshold, amplitude_reciprocal_symm, and amplitude_nonneg. The structure GaugeTreeAmplitudesCert packages these as a certificate, with a theorem stating the process count is exactly 3.

In Recognition Science, this is a structural opening for the A1 layer. The claim is that RS-native amplitudes match SM leading-order results in the canonical sector, with zero free parameters. The full derivation requires the Wightman/OS continuum limit (S1 in progress). The module has 0 sorry and 0 axiom in the Lean kernel.

/-- Per-process amplitude (J-cost on the relevant coupling ratio). -/
def processAmplitude (r : ℝ) : ℝ := Jcost r

theorem amplitude_zero_at_threshold : processAmplitude 1 = 0 := Jcost_unit0

theorem gauge_tree_process_count :
    Fintype.card GaugeTreeProcess = 3 := by decide

What this page does not claim

This module does not prove that RS amplitudes match measured SM cross-sections. The full derivation from the Wightman/OS continuum limit is not yet formalized. The module does not derive the value of any coupling constant from first principles.

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/QRFT/GaugeTreeAmplitudesCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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