zeroCompositionLaw_forces_eta_zero

formal source

  69
  70/-- A zero-composition law forces the corresponding point onto the critical
  71line once the observable attains its minimum at that point's deviation. -/
  72theorem zeroCompositionLaw_forces_eta_zero
  73    (zc : ZeroCompositionLaw) (ρ : ℂ) :
  74    zc.H (zeroDeviation ρ) = 1 ↔ OnCriticalLine ρ := by
  75  constructor
  76  · intro h
  77    have hz : zeroDeviation ρ = 0 :=
  78      (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mp h
  79    exact (zeroDeviation_eq_zero_iff_on_critical_line ρ).mp hz
  80  · intro h
  81    have hz : zeroDeviation ρ = 0 :=
  82      (zeroDeviation_eq_zero_iff_on_critical_line ρ).mpr h
  83    exact (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mpr hz
  84
  85/-- A concrete Vector C witness at a specific complex point. -/
  86structure ZeroCompositionWitness (ρ : ℂ) where
  87  law : ZeroCompositionLaw
  88  value_at_deviation : law.H (zeroDeviation ρ) = 1
  89
  90/-- Any such witness forces the corresponding point onto the critical line. -/
  91theorem zeroCompositionWitness_forces_on_critical_line
  92    {ρ : ℂ} (w : ZeroCompositionWitness ρ) :
  93    OnCriticalLine ρ :=
  94  (zeroCompositionLaw_forces_eta_zero w.law ρ).mp w.value_at_deviation
  95
  96/-- Therefore the zero-location defect must vanish there as well. -/
  97theorem zeroCompositionWitness_forces_zero_defect
  98    {ρ : ℂ} (w : ZeroCompositionWitness ρ) :
  99    zeroDefect ρ = 0 := by
 100  exact (zeroDefect_zero_iff_on_critical_line ρ).mpr
 101    (zeroCompositionWitness_forces_on_critical_line w)
 102