Foundation Qrft Gauge Tree Amplitudes Cert Process Count Equals 3

The three process certificate

In particle physics, a tree amplitude is the simplest approximation to a scattering event's probability, ignoring quantum loops. The Standard Model's gauge bosons (photons, W and Z particles) mediate these interactions. A structural certificate in the Recognition Science framework's machine-checked library of formal theorems establishes that exactly three canonical gauge tree processes exist: Compton scattering (γe⁻ → γe⁻), pair annihilation (e⁺e⁻ → γγ), and W⁺W⁻ → ZZ unitarisation.

The theorem process_count_equals_3 proves that the inductive type enumerating these three processes has cardinality exactly three. This is a recognition framework result: a discrete record of events where each event has a forced cost. The count of three is not arbitrary; the certificate links it to the framework's spatial dimension, configDim D - 2, where D is the dimension of space.

Each of the three amplitudes is defined as the J-cost function, a specific mathematical form that satisfies five plain conditions including reciprocal symmetry and zero cost at unity. The certificate proves three properties: the amplitude is zero at threshold (r = 1), it is symmetric under r → 1/r, and it is non-negative for positive ratios. These properties hold for all three processes uniformly.

The structural claim is that these RS-native amplitudes match Standard Model leading-order results in the canonical sector. This is a zero-parameter match: no free constants are fitted. The certificate is a structural opening, not a complete derivation; the full continuum limit remains in progress.

What this theorem does not claim: it does not prove that these are the only possible scattering processes in the Standard Model, only that within this framework's canonical set, exactly three are recognized. It does not derive the numerical values of cross sections, only the structural form of the amplitudes. It does not establish the physical recognition-to-linking bridge that would connect this count to observable three-dimensional space; that bridge remains open.

/-- The gauge tree amplitude triad has 3 processes = configDim D - 2. -/
theorem process_count_equals_3 :
    Fintype.card GaugeTreeProcess = 3 :=
  gauge_tree_process_count

/-- 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

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 prove these are the only possible scattering processes in the Standard Model, only that within this framework's canonical set, exactly three are recognized. The certificate does not derive numerical cross sections, only the structural form of the amplitudes. The physical recognition-to-linking bridge connecting this count to observable three-dimensional space remains open.

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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