Foundation Qrft Gauge Tree Amplitudes Cert Amplitude Nonneg

The nonnegative amplitude

In quantum field theory, a scattering amplitude is a complex number whose square gives the probability that a particular particle collision produces a particular outcome. A physically sensible amplitude cannot be negative when it is interpreted as a rate or a probability weight. The declaration amplitude_nonneg in the Recognition Science library is a machine-checked theorem stating that, for three canonical gauge-theory processes, the framework's amplitude is always greater than or equal to zero.

The three processes are Compton scattering (a photon and an electron exchanging energy), pair annihilation (an electron and a positron turning into two photons), and W-boson pair annihilation into Z-bosons. In the framework, each amplitude is defined by a single function called the cost function, which measures the price of a recognition event: a discrete record of an interaction. The cost function is J(r) = (r + 1/r)/2 - 1, where r is the ratio of the relevant couplings. The theorem amplitude_nonneg states that for any positive ratio r, this cost is never negative. It is proved in the machine-checked library of formal theorems, with zero axioms beyond the standard logical kernel.

The nonnegativity is not an isolated fact. It is one of four properties that together form the structural certificate for gauge tree amplitudes: the number of processes is exactly three, the amplitude vanishes at threshold (r = 1), the amplitude is symmetric under exchanging r with 1/r, and the amplitude is nonnegative. The certificate is a structural opening: it records the shape of the amplitudes without fitting any free parameters. The full derivation from continuum quantum field theory, the Wightman/OS limit, is still in progress.

What the theorem does not claim is just as important. It does not claim that the framework's amplitudes match the Standard Model's numerical predictions; that comparison is a separate, empirical check, not a proved theorem. It does not claim that the three processes are the only gauge tree amplitudes, only that these three are the canonical ones. And it does not claim that the amplitude is positive everywhere; it only claims nonnegativity, with strict positivity reserved for ratios away from 1. The theorem is a shape constraint, not a full physical derivation.

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

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

structure GaugeTreeAmplitudesCert where
  process_count : Fintype.card GaugeTreeProcess = 3
  threshold_zero : processAmplitude 1 = 0
  reciprocal_symm : ∀ {r : ℝ}, 0 < r → processAmplitude r = processAmplitude r⁻¹
  amplitude_nonneg : ∀ {r : ℝ}, 0 < r → 0 ≤ processAmplitude r

What this page does not claim

The theorem does not claim that the framework's amplitudes match Standard Model numerical predictions. The theorem does not claim that the three processes are the only gauge tree amplitudes. The theorem does not claim strict positivity for all ratios, only nonnegativity.

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: