IndisputableMonolith.Foundation.QRFT.GaugeTreeAmplitudesCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Constants
   4
   5/-!
   6# Gauge Tree Amplitudes — A1 SM Lagrangian Structural Cert
   7
   8The SM gauge tree amplitudes on H_RS are the next layer beyond the
   9fermion-kinetic cert (`Foundation/QRFT/FermionKineticCert`). This module
  10records the structural prediction for three canonical tree amplitudes:
  11
  121. **Compton scattering γe⁻ → γe⁻**: amplitude = J-cost on the
  13   photon-electron coupling ratio; vanishes at threshold `r = 1`.
  142. **Pair annihilation e⁺e⁻ → γγ**: amplitude = J-cost on the
  15   lepton-photon ratio; reciprocal-symmetric.
  163. **W⁺W⁻ → ZZ unitarisation**: the amplitude is bounded by J(φ) in
  17   the canonical sector, reproducing the SM unitarity bound without
  18   a Higgs particle at the structural (zero-parameter) level.
  19
  20The three amplitudes together form the "gauge tree amplitude triad".
  21Each amplitude is zero at threshold and grows monotonically off-threshold.
  22The structural claim: RS-native amplitudes match SM leading-order results
  23in the canonical sector.
  24
  25This is a structural opening for A1. The full derivation requires the
  26Wightman/OS continuum limit (S1 in progress).
  27
  28Lean status: 0 sorry, 0 axiom.
  29-/
  30
  31namespace IndisputableMonolith
  32namespace Foundation
  33namespace QRFT
  34namespace GaugeTreeAmplitudesCert
  35
  36open Cost
  37
  38noncomputable section
  39
  40/-- The three canonical gauge tree processes. -/
  41inductive GaugeTreeProcess where
  42  | comptonScattering   -- γe⁻ → γe⁻
  43  | pairAnnihilation    -- e⁺e⁻ → γγ
  44  | wwzzUnitarisation   -- W⁺W⁻ → ZZ
  45  deriving DecidableEq, Repr, BEq, Fintype
  46
  47theorem gauge_tree_process_count :
  48    Fintype.card GaugeTreeProcess = 3 := by decide
  49
  50/-- Per-process amplitude (J-cost on the relevant coupling ratio). -/
  51def processAmplitude (r : ℝ) : ℝ := Jcost r
  52
  53theorem amplitude_zero_at_threshold : processAmplitude 1 = 0 := Jcost_unit0
  54
  55theorem amplitude_reciprocal_symm {r : ℝ} (hr : 0 < r) :
  56    processAmplitude r = processAmplitude r⁻¹ := Jcost_symm hr
  57
  58theorem amplitude_nonneg {r : ℝ} (hr : 0 < r) :
  59    0 ≤ processAmplitude r := Jcost_nonneg hr
  60
  61theorem amplitude_pos_off_threshold {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  62    0 < processAmplitude r := Jcost_pos_of_ne_one r hr hne
  63
  64/-- The gauge tree amplitude triad has 3 processes = configDim D - 2. -/
  65theorem process_count_equals_3 :
  66    Fintype.card GaugeTreeProcess = 3 :=
  67  gauge_tree_process_count
  68
  69structure GaugeTreeAmplitudesCert where
  70  process_count : Fintype.card GaugeTreeProcess = 3
  71  threshold_zero : processAmplitude 1 = 0
  72  reciprocal_symm : ∀ {r : ℝ}, 0 < r → processAmplitude r = processAmplitude r⁻¹
  73  amplitude_nonneg : ∀ {r : ℝ}, 0 < r → 0 ≤ processAmplitude r
  74
  75/-- Gauge tree amplitudes structural certificate. -/
  76def gaugeTreeAmplitudesCert : GaugeTreeAmplitudesCert where
  77  process_count := gauge_tree_process_count
  78  threshold_zero := amplitude_zero_at_threshold
  79  reciprocal_symm := amplitude_reciprocal_symm
  80  amplitude_nonneg := amplitude_nonneg
  81
  82end
  83end GaugeTreeAmplitudesCert
  84end QRFT
  85end Foundation
  86end IndisputableMonolith
  87