Foundation Qrft Gauge Tree Amplitudes Cert Gauge Tree Amplitudes Cert

Gauge tree amplitudes

In particle physics, a tree amplitude is the simplest possible calculation of a collision outcome: it ignores quantum loops and virtual particle showers, retaining only the direct, classical-looking exchange of force carriers. The Standard Model's tree amplitudes are textbook results, computed from its Lagrangian by Feynman rules. The declaration GaugeTreeAmplitudesCert, part of the Recognition Science framework's machine-checked library of formal theorems, asks whether those amplitudes can be reproduced from a single structural principle rather than from the full Standard Model Lagrangian.

The framework's answer is a ledger, a discrete record of recognition events, whose cost function J(r) = (r + 1/r)/2 - 1 measures the price of a recognition at ratio r. The declaration defines three canonical gauge tree processes: Compton scattering (γe⁻ → γe⁻), pair annihilation (e⁺e⁻ → γγ), and W⁺W⁻ → ZZ unitarisation. For each, the amplitude is set equal to the J-cost evaluated on the relevant coupling ratio. The formal theorems prove that this amplitude is zero at threshold (r = 1), symmetric under r → 1/r, and positive away from threshold. The three-process count is also proved to equal 3, matching the framework's configuration dimension minus 2.

In Recognition Science, the structural claim is that these RS-native amplitudes match Standard Model leading-order results in the canonical sector, reproducing the SM unitarity bound for W⁺W⁻ → ZZ without a Higgs particle at the zero-parameter level. The declaration is a structural certificate: it packages the three theorems into a single object, with 0 sorry and 0 axiom in the Lean kernel audit. It is explicitly a next layer beyond the fermion-kinetic cert, and it is explicitly not a full derivation. The complete derivation requires the Wightman/OS continuum limit, which the framework marks as in progress.

What the declaration does not claim is as important as what it proves. It does not claim that the full Standard Model Lagrangian has been derived from the ledger; it claims only that three canonical tree amplitudes take the J-cost form in the canonical sector. It does not claim that the Higgs mechanism is unnecessary for the full SM; it claims only that the unitarity bound for W⁺W⁻ → ZZ is reproduced at the structural, zero-parameter level. And it does not claim that any of these amplitudes have been measured or compared against experiment; the declaration is purely formal, establishing properties of a defined function.

/-- The three canonical gauge tree processes. -/
inductive GaugeTreeProcess where
  | comptonScattering   -- γe⁻ → γe⁻
  | pairAnnihilation    -- e⁺e⁻ → γγ
  | wwzzUnitarisation   -- W⁺W⁻ → ZZ
  deriving DecidableEq, Repr, BEq, Fintype

/-- 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 amplitude_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    processAmplitude r = processAmplitude r⁻¹ := Jcost_symm hr

theorem amplitude_pos_off_threshold {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < processAmplitude r := Jcost_pos_of_ne_one r hr hne

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

What this page does not claim

The full Standard Model Lagrangian has been derived from the ledger. The Higgs mechanism is unnecessary for the full Standard Model. Any of the three amplitudes has been measured or compared against experimental data.

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)

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: