IndisputableMonolith.NumberTheory.CompositionDivergence

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.NumberTheory.ZeroLocationCost
   4import IndisputableMonolith.NumberTheory.XiJBridge
   5import IndisputableMonolith.NumberTheory.ZeroCompositionLaw
   6
   7/-!
   8# Composition Divergence ⟹ Riemann Hypothesis
   9
  10**Classification: ALTERNATE** — separate conditional RH certificate.
  11
  12The CCH bridge ("each iterate is reflected in the carrier budget") is the
  13composition-law analogue of the EBBA bridge. Both are RH-equivalent.
  14The composition cascade is theoretically stronger (infinitely many cost
  15values from a single zero) but does not reduce EBBA or HonestPhaseCostBridge.
  16
  17## The Argument
  18
  19This module connects the zero composition law (ZeroCompositionLaw.lean) to
  20the Riemann Hypothesis via a finite carrier budget.
  21
  22### The Chain of Forcing
  23
  241. **RCL uniquely determines J** (T5, CostUniqueness):
  25   J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)  ⟹  J(x) = ½(x+1/x)−1
  26
  272. **ξ(s)=ξ(1−s) is J-symmetry** (XiJBridge):
  28   Under x = e^{2(σ−1/2)}, the functional equation becomes J(x)=J(1/x)
  29
  303. **RCL self-composition amplifies defect** (ZeroCompositionLaw):
  31   For any off-critical zero with defect d₀ > 0:
  32   dₙ₊₁ = 2dₙ(dₙ+2),  dₙ ≥ 4ⁿ·d₀ → ∞
  33
  344. **Divergent defect violates carrier budget** (this module):
  35   The carrier C(s) = det₂(I−A)² has finite budget (AnnularCost framework).
  36   The iterated defect grows as cosh(2ⁿ⁺¹η)−1, which exceeds any
  37   finite budget.
  38
  39### The Composition Closure Hypothesis
  40
  41The remaining bridge between steps 3 and 4 is:
  42
  43**CCH**: Each iterated defect dₙ is reflected in the annular excess of
  44the carrier at the corresponding scale. In particular, there exists a
  45finite bound B that all iterated defects must respect.
  46
  47Under CCH, the carrier budget is violated for any off-critical zero,
  48and all zeros must lie on the critical line.
  49
  50## Main Results
  51
  521. `CompositionClosureHypothesis`: the bridge from virtual to actual defect
  532. `composition_violates_budget`: divergent defect exceeds any finite bound
  543. `rh_from_composition_closure`: RH conditional on CCH
  55-/
  56
  57namespace IndisputableMonolith
  58namespace NumberTheory
  59
  60open Real Cost
  61
  62noncomputable section
  63
  64/-! ## §1. The Composition Closure Hypothesis -/
  65
  66/-- The **Composition Closure Hypothesis** (CCH).
  67
  68    For each nontrivial zero ρ off the critical line, the n-th iterate
  69    of the RCL self-composition produces a defect that must be
  70    absorbed by a finite carrier budget.
  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
 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 :=
 129  defectIterate_mono ht n
 130
 131/-- The cascade grows at least as fast as 4ⁿ · d₀. -/
 132theorem cascade_exponential_growth (t : ℝ) (n : ℕ) :
 133    (4 : ℝ) ^ n * defectIterate t 0 ≤ defectIterate t n :=
 134  defectIterate_exponential_lower t n
 135
 136/-- Doubly-exponential growth: the defect at level n involves
 137    cosh(2ⁿ · t), which for t ≠ 0 grows as exp(2ⁿ · |t|)/2. -/
 138theorem cascade_doubly_exponential_lower {t : ℝ} (_ht : 0 < t) (n : ℕ) :
 139    Real.exp ((2 : ℝ) ^ n * t) / 2 - 1 ≤ defectIterate t n := by
 140  simp only [defectIterate]
 141  have h : Real.exp ((2 : ℝ) ^ n * t) / 2 ≤ Real.cosh ((2 : ℝ) ^ n * t) := by
 142    rw [Real.cosh_eq]
 143    have hexp : 0 ≤ Real.exp (-((2 : ℝ) ^ n * t)) := Real.exp_nonneg _
 144    linarith
 145  linarith
 146
 147/-! ## §5. What remains
 148
 149The gap between the composition route and unconditional RH is precisely
 150the Composition Closure Hypothesis (CCH).
 151
 152CCH asserts that each iterate of the RCL self-composition corresponds
 153to an actual constraint on the carrier budget. This is the RS-native
 154version of the `EulerBoundaryBridgeAssumption`.
 155
 156Potential approaches to proving CCH:
 1571. **Explicit formula**: the Guinand-Weil formula connects zeros to primes;
 158   the iterated defect should map to prime correlations at scale 2ⁿ
 1592. **Hadamard product**: the convergence ∑ 1/|ρ|² < ∞ constrains the
 160   collective defect budget; the cascade from one zero may violate it
 1613. **Spectral**: the Hilbert-Pólya approach places zeros as eigenvalues;
 162   the RCL cascade maps to an operator norm constraint -/
 163
 164end
 165
 166end NumberTheory
 167end IndisputableMonolith
 168