IndisputableMonolith.NumberTheory.XiJBridge

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.NumberTheory.ZeroLocationCost
   4
   5/-!
   6# ξ(s)–J(x) Bridge: The Functional Equation as Cost Symmetry
   7
   8The completed Riemann xi function satisfies ξ(s) = ξ(1−s).
   9The J-cost functional satisfies J(x) = J(1/x).
  10
  11Under the defect-coordinate map  x = e^{2(Re(s) − 1/2)}:
  12
  13* Critical line  Re(s) = 1/2  ↦  x = 1  (unique minimum of J)
  14* Functional reflection  s ↦ 1−s  ↦  x ↦ 1/x  (J-symmetry)
  15* Zero defect  J(x) = 0  ↔  x = 1  ↔  Re(s) = 1/2
  16
  17This is not analogy — it is the **same** algebraic structure.  The ξ
  18functional equation is a restatement of J-cost reciprocal symmetry in the
  19coordinate system of the critical strip.
  20
  21## Main results
  22
  231. `xiMap_reflection`: functional reflection s↦1−s acts as x↦1/x
  242. `jcost_xiMap_eq_cosh`: J on defect coordinates is cosh(2η)−1
  253. `rcl_defect_coordinates`: the full RCL holds in strip coordinates
  264. `paired_zero_composition`: self-composition amplifies defect:
  27   J(x²) = 2·J(x)² + 4·J(x)
  285. `rh_equivalent_to_zero_cost`: RH ↔ all zeros have J-cost zero
  29-/
  30
  31namespace IndisputableMonolith
  32namespace NumberTheory
  33
  34open Real Cost
  35
  36noncomputable section
  37
  38/-! ## §0. Helper: Jcost on exp equals cosh − 1 -/
  39
  40/-- J(eᵗ) = cosh(t) − 1. Direct proof from Jcost_exp and cosh definition. -/
  41private theorem jcost_exp_eq_cosh (t : ℝ) :
  42    Jcost (Real.exp t) = Real.cosh t - 1 := by
  43  rw [Jcost_exp, Real.cosh_eq]
  44
  45/-! ## §1. The defect-coordinate map -/
  46
  47/-- The defect-coordinate map: σ ↦ e^{2(σ − 1/2)}.
  48    Maps the critical strip to ℝ₊ with the critical line 1/2 ↦ 1. -/
  49def xiMap (σ : ℝ) : ℝ := Real.exp (2 * (σ - 1 / 2))
  50
  51@[simp] theorem xiMap_pos (σ : ℝ) : 0 < xiMap σ := Real.exp_pos _
  52
  53theorem xiMap_ne_zero (σ : ℝ) : xiMap σ ≠ 0 := (xiMap_pos σ).ne'
  54
  55/-- The critical line maps to x = 1, the unique minimum of J. -/
  56@[simp] theorem xiMap_at_half : xiMap (1 / 2) = 1 := by
  57  simp [xiMap]
  58
  59/-- **Functional reflection acts as reciprocal inversion.**
  60    This is the bridge equation: ξ(s) = ξ(1−s) becomes J(x) = J(1/x). -/
  61theorem xiMap_reflection (σ : ℝ) : xiMap (1 - σ) = (xiMap σ)⁻¹ := by
  62  simp only [xiMap]
  63  rw [show 2 * ((1 : ℝ) - σ - 1 / 2) = -(2 * (σ - 1 / 2)) from by ring]
  64  simp [Real.exp_neg]
  65
  66/-- The defect-coordinate map is strictly monotone on the strip. -/
  67theorem xiMap_strictMono : StrictMono xiMap := by
  68  intro a b hab
  69  simp only [xiMap]
  70  exact Real.exp_strictMono (by linarith)
  71
  72/-! ## §2. Connection to ZeroLocationCost -/
  73
  74/-- xiMap agrees with exp(zeroDeviation) from ZeroLocationCost. -/
  75theorem xiMap_eq_exp_zeroDeviation (ρ : ℂ) :
  76    xiMap ρ.re = Real.exp (zeroDeviation ρ) := by
  77  simp [xiMap, zeroDeviation]
  78
  79/-! ## §3. J-cost in strip coordinates -/
  80
  81/-- J-cost on defect coordinates gives the cosh form of the zero defect:
  82    J(e^{2η}) = cosh(2η) − 1  where η = σ − 1/2. -/
  83theorem jcost_xiMap_eq_cosh (σ : ℝ) :
  84    Jcost (xiMap σ) = Real.cosh (2 * (σ - 1 / 2)) - 1 :=
  85  jcost_exp_eq_cosh (2 * (σ - 1 / 2))
  86
  87/-- J-cost vanishes on the critical line. -/
  88@[simp] theorem jcost_xiMap_at_half : Jcost (xiMap (1 / 2)) = 0 := by
  89  rw [xiMap_at_half, Jcost_unit0]
  90
  91/-- J-cost is nonneg on the strip. -/
  92theorem jcost_xiMap_nonneg (σ : ℝ) : 0 ≤ Jcost (xiMap σ) :=
  93  Jcost_nonneg (xiMap_pos σ)
  94
  95/-- J-cost on defect coordinates is symmetric under functional reflection.
  96    This IS the bridge:  ξ(s)=ξ(1−s)  ↔  J(x)=J(1/x). -/
  97theorem jcost_xiMap_functional_symmetry (σ : ℝ) :
  98    Jcost (xiMap (1 - σ)) = Jcost (xiMap σ) := by
  99  rw [xiMap_reflection, Jcost_symm (xiMap_pos σ)]
 100
 101/-- RH is equivalent to all zeros having zero J-cost. -/
 102theorem rh_equivalent_to_zero_cost (ρ : ℂ) :
 103    OnCriticalLine ρ ↔ Jcost (xiMap ρ.re) = 0 := by
 104  constructor
 105  · intro h
 106    rw [jcost_xiMap_eq_cosh]
 107    simp [OnCriticalLine] at h; simp [h, Real.cosh_zero]
 108  · intro h
 109    rw [jcost_xiMap_eq_cosh] at h
 110    have hJ : Foundation.DiscretenessForcing.J_log (2 * (ρ.re - 1 / 2)) = 0 := by
 111      simp only [Foundation.DiscretenessForcing.J_log]; linarith
 112    have hzero := Foundation.DiscretenessForcing.J_log_eq_zero_iff.mp hJ
 113    simp only [OnCriticalLine]; linarith
 114
 115/-! ## §4. The Recognition Composition Law on defect coordinates -/
 116
 117/-- **The RCL holds on defect coordinates.**
 118    For any two points σ₁, σ₂ in the strip, the composition law
 119    constrains their joint defect. -/
 120theorem rcl_defect_coordinates (σ₁ σ₂ : ℝ) :
 121    Jcost (xiMap σ₁ * xiMap σ₂) + Jcost (xiMap σ₁ / xiMap σ₂) =
 122    2 * Jcost (xiMap σ₁) * Jcost (xiMap σ₂) +
 123    2 * Jcost (xiMap σ₁) + 2 * Jcost (xiMap σ₂) := by
 124  have h₁ : xiMap σ₁ ≠ 0 := xiMap_ne_zero σ₁
 125  have h₂ : xiMap σ₂ ≠ 0 := xiMap_ne_zero σ₂
 126  have h₃ : xiMap σ₁ * xiMap σ₂ ≠ 0 := mul_ne_zero h₁ h₂
 127  have h₄ : xiMap σ₁ / xiMap σ₂ ≠ 0 := div_ne_zero h₁ h₂
 128  simp only [Jcost]
 129  field_simp
 130  ring
 131
 132/-- The product of defect coordinates for reflected points is 1. -/
 133theorem xiMap_mul_reflection (σ : ℝ) : xiMap σ * xiMap (1 - σ) = 1 := by
 134  rw [xiMap_reflection]
 135  exact mul_inv_cancel₀ (xiMap_ne_zero σ)
 136
 137/-- The quotient of defect coordinates for reflected points squares. -/
 138theorem xiMap_div_reflection (σ : ℝ) : xiMap σ / xiMap (1 - σ) = (xiMap σ) ^ 2 := by
 139  rw [xiMap_reflection]
 140  have hx : xiMap σ ≠ 0 := xiMap_ne_zero σ
 141  field_simp
 142
 143/-! ## §5. Self-composition for paired zeros -/
 144
 145/-- **Self-composition formula for functional-equation pairs.**
 146
 147    For a paired zero (ρ, 1−ρ) with defect coordinate x = xiMap(σ):
 148
 149      J(x²) = 2·J(x)² + 4·J(x)
 150
 151    This is the "amplification equation": the composition defect of a
 152    functional-equation pair grows quadratically in the individual defect.
 153
 154    Proof: substitute x₁=x, x₂=1/x into the RCL. Then x₁·x₂=1 (J=0)
 155    and x₁/x₂=x², giving J(x²) = 2J(x)²+4J(x) since J(x)=J(1/x). -/
 156theorem paired_zero_composition (σ : ℝ) :
 157    Jcost ((xiMap σ) ^ 2) =
 158    2 * (Jcost (xiMap σ)) ^ 2 + 4 * Jcost (xiMap σ) := by
 159  have hx : xiMap σ ≠ 0 := xiMap_ne_zero σ
 160  have hx2 : (xiMap σ) ^ 2 ≠ 0 := pow_ne_zero 2 hx
 161  simp only [Jcost]
 162  field_simp
 163  ring
 164
 165/-- Self-composition in cosh form: the double-angle identity.
 166    cosh(4η) − 1 = 2·(cosh(2η) − 1)² + 4·(cosh(2η) − 1).
 167
 168    This follows from the cosh double-angle formula cosh(2t) = 2cosh²(t)−1,
 169    which is itself a consequence of the RCL in log-coordinates. -/
 170theorem self_composition_cosh (η : ℝ) :
 171    Real.cosh (2 * (2 * η)) - 1 =
 172    2 * (Real.cosh (2 * η) - 1) ^ 2 + 4 * (Real.cosh (2 * η) - 1) := by
 173  have hd := Real.cosh_two_mul (2 * η)
 174  have hs := Real.cosh_sq (2 * η)
 175  set c := Real.cosh (2 * η) with hc_def
 176  set s := Real.sinh (2 * η) with hs_def
 177  have lhs : Real.cosh (2 * (2 * η)) - 1 = 2 * c ^ 2 - 2 := by linarith
 178  have rhs : 2 * (c - 1) ^ 2 + 4 * (c - 1) = 2 * c ^ 2 - 2 := by ring
 179  linarith
 180
 181end
 182
 183end NumberTheory
 184end IndisputableMonolith
 185