Foundation Qrft Gauge Tree Amplitudes Cert Amplitude Zero At Threshold

The threshold condition

In quantum field theory, a scattering amplitude is a number that tells how likely a process is. At threshold, the minimum energy needed for a reaction to occur, many amplitudes vanish or simplify. The Recognition Science declaration amplitude_zero_at_threshold establishes the same behavior for a specific class of processes: when the ratio of the relevant couplings is exactly 1, the amplitude is 0.

Concretely, the framework models three canonical gauge tree processes: Compton scattering (γe⁻ → γe⁻), pair annihilation (e⁺e⁻ → γγ), and W⁺W⁻ → ZZ unitarisation. For each, the amplitude is defined as a function of a single real ratio r, and the theorem states that at r = 1 the amplitude equals zero. This is not an accident of a particular model; it follows from the defining property of the cost function J, which is zero at unity. The same function also guarantees the amplitude is nonnegative and symmetric under r ↔ 1/r, meaning the amplitude for a process and its reciprocal are equal.

What the theorem does not claim is broader. It does not assert that these amplitudes match the full Standard Model at all energies, nor that the framework has derived the complete scattering matrix. The docstring explicitly notes that the full derivation requires a Wightman/OS continuum limit, which is in progress. The declaration is a structural certificate: it records that the zero-at-threshold property holds for the three canonical processes in the framework's discrete setting, with zero axioms and zero sorrys in the machine-checked library.

The practical consequence is a checkable constraint. Any future extension of the framework that claims to describe these processes must reproduce this threshold zero, just as any correct quantum field theory must. The declaration pins down one number at one point, and that point is the boundary where the process becomes possible at all.

theorem amplitude_zero_at_threshold : processAmplitude 1 = 0 := Jcost_unit0

theorem amplitude_nonneg {r : ℝ} (hr : 0 < r) :
    0 ≤ processAmplitude r := Jcost_nonneg hr

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

What this page does not claim

The declaration does not prove that the framework's amplitudes match Standard Model results at all energies. It does not derive the full scattering matrix or the complete gauge theory. It does not claim that the Higgs particle is unnecessary in the full Standard Model.

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.

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