CompositionClosureHypothesis

formal source

  71
  72    The `bound` represents the carrier budget scale from the
  73    AnnularCost framework (carrierBudgetScale of a BudgetedCarrier). -/
  74structure CompositionClosureHypothesis where
  75  bound : ℝ
  76  reflected : ∀ (ρ : ℂ), ¬OnCriticalLine ρ →
  77    ∀ (n : ℕ), defectIterate (zeroDeviation ρ) n ≤ bound
  78
  79/-! ## §2. The contradiction -/
  80
  81/-- **The iterated defect exceeds any fixed bound.**
  82
  83    The composition law generates defect values that grow as
  84    cosh(2ⁿ·2η) − 1 ≥ 4ⁿ·(cosh(2η)−1), which exceeds any finite
  85    carrier budget for n large enough. -/
  86theorem composition_violates_budget (ρ : ℂ) (hρ : ¬OnCriticalLine ρ) (B : ℝ) :
  87    ∃ n : ℕ, B < defectIterate (zeroDeviation ρ) n :=
  88  zero_composition_diverges ρ hρ B
  89
  90/-- **Riemann Hypothesis from Composition Closure.**
  91
  92    If the Composition Closure Hypothesis holds, then every nontrivial
  93    zero of ζ(s) lies on the critical line Re(s) = 1/2.
  94
  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