Foundation Qrft Gauge Tree Amplitudes Cert Amplitude Reciprocal Symm

The symmetry of exchange

In quantum field theory, a scattering amplitude is a complex number whose squared magnitude gives the probability that a particular collision outcome occurs. For processes like electron-positron annihilation into two photons, the amplitude possesses a fundamental property called crossing symmetry: the amplitude computed for one particle arrangement equals the amplitude computed when the roles of incoming and outgoing particles are exchanged. This symmetry is not accidental; it follows from the underlying Lorentz invariance and unitarity of the theory.

The Recognition Science framework models these amplitudes using its central cost function. The framework defines a process amplitude as the cost J(r) evaluated at the ratio r of the coupling constants for the two particles involved. The declaration amplitude_reciprocal_symm establishes that this amplitude is unchanged when r is replaced by its reciprocal 1/r. In plain language: the amplitude for a process with a given coupling ratio equals the amplitude for the same process with the ratio inverted. This is a theorem proved in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The symmetry holds for all positive values of the coupling ratio r, and it is a direct consequence of the reciprocal symmetry of the cost function itself. The theorem also implies that the amplitude is zero at the threshold r = 1, where the two couplings are equal, and that it is positive everywhere else. These properties together mean the amplitude grows monotonically as the coupling ratio moves away from unity in either direction.

What the declaration does not claim is that this reciprocal symmetry by itself determines the full structure of scattering amplitudes. The framework's structural certificate records the symmetry as one of several properties that the three canonical gauge tree amplitudes must satisfy, alongside the threshold zero and non-negativity conditions. The declaration does not assert that the amplitude formula J(r) is the unique possible form; it only establishes that the chosen form possesses the stated symmetry. The framework also does not claim that this symmetry alone reproduces the Standard Model's unitarity bounds or any other physical prediction; those claims require additional structure beyond what this single theorem establishes.

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

theorem amplitude_zero_at_threshold : processAmplitude 1 = 0 := Jcost_unit0

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

What this page does not claim

The declaration does not assert that the amplitude formula J(r) is the unique possible form for scattering amplitudes. The declaration does not by itself reproduce the Standard Model's unitarity bounds or other physical predictions. The reciprocal symmetry alone does not determine the full structure of the gauge tree amplitude triad.

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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