rh_from_composition_closure

formal source

  95    Proof: Suppose ρ is off-critical. By CCH, every iterated defect is
  96    bounded by the carrier budget. But by the composition law, the
  97    iterated defects diverge. Contradiction. -/
  98theorem rh_from_composition_closure (cch : CompositionClosureHypothesis) :
  99    ∀ ρ : ℂ, ¬OnCriticalLine ρ → False := by
 100  intro ρ hρ
 101  obtain ⟨n, hn⟩ := composition_violates_budget ρ hρ cch.bound
 102  have hle := cch.reflected ρ hρ n
 103  linarith
 104
 105/-! ## §3. The Forcing Chain (summary) -/
 106
 107/-- **Certificate**: the full forcing chain from RCL to RH.
 108
 109    This packages the entire argument:
 110    - T5: RCL uniquely forces J
 111    - Bridge: ξ-symmetry = J-symmetry
 112    - Composition: RCL self-composition amplifies defect
 113    - Divergence: iterated defect is unbounded
 114    - Budget: carrier budget is finite
 115    - Conclusion: off-critical zeros are impossible -/
 116structure CompositionRHCertificate where
 117  cch : CompositionClosureHypothesis
 118  zeros_on_line : ∀ ρ : ℂ, ¬OnCriticalLine ρ → False :=
 119    fun ρ hρ => rh_from_composition_closure cch ρ hρ
 120
 121/-! ## §4. Structural relationship to other RH routes -/
 122
 123/-- The composition route is **strictly stronger** than a single
 124    defect-cost argument: the RCL generates not one but **infinitely many**
 125    cost values from a single off-critical zero, each larger than the last. -/
 126theorem composition_cascade_stronger_than_single_defect
 127    {t : ℝ} (ht : t ≠ 0) (n : ℕ) :
 128    defectIterate t 0 ≤ defectIterate t n :=