Foundation Qrft Gauge Tree Amplitudes Cert Gauge Tree Process Count

Three structural processes

In particle physics, a tree-level amplitude is the leading approximation to the probability that two particles scatter or transform into others, calculated without the quantum loops that add higher-order corrections. The standard model of particle physics organizes these processes through gauge theories, where force-carrying particles like photons and W bosons mediate interactions. A familiar example is Compton scattering, in which a photon strikes an electron and both emerge changed in direction and energy.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, gauge_tree_process_count, that establishes a simple structural fact: the framework recognizes exactly three canonical gauge tree processes. These are Compton scattering (γe⁻ → γe⁻), pair annihilation (e⁺e⁻ → γγ), and W⁺W⁻ → ZZ unitarisation. The declaration proves that the count of these processes is three, a result checked by computation in the library's formal system. This count matches the framework's internal claim that the configuration dimension D minus two equals three, though the physical significance of that match is a matter of ongoing derivation.

Each of the three processes has an associated amplitude defined as a cost function J(r), where r is the ratio of the relevant coupling constants. The framework proves three properties of this amplitude: it is zero when the ratio equals one (the threshold), it is unchanged when the ratio is replaced by its reciprocal, and it is non-negative for positive ratios. These properties hold for all three processes uniformly, meaning the framework treats them as instances of a single structural pattern rather than as independent calculations.

The declaration does not claim that these amplitudes match any measured or computed standard model values. The framework's docstring states that the amplitudes "match SM leading-order results in the canonical sector," but this is a structural opening, not a proven equivalence. The full derivation requires a Wightman/OS continuum limit that is marked as in progress. The count theorem itself is a statement about the framework's own classification of processes, not about the physical world's particle content.

In Recognition Science, this count is a certificate: a compact, machine-checked record that the framework's structural predictions for gauge tree amplitudes are internally consistent. It is a stepping stone toward a larger goal of deriving standard model results from the framework's first principles, but the step it takes is only to fix the number of canonical processes and their shared amplitude properties. The leap from that structure to actual scattering cross-sections remains open.

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

/-- 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_nonneg {r : ℝ} (hr : 0 < r) :
    0 ≤ processAmplitude r := Jcost_nonneg hr

What this page does not claim

The amplitudes do not match any measured standard model values in this declaration. The count of three does not derive from physical principles; it is a classification within the framework. No claim is made about the physical existence of exactly three gauge tree processes in nature.

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: