IndisputableMonolith.NumberTheory.PrimeLedgerStructure

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Prime Numbers as Irreducible Ledger Transactions
   7
   8In RS, reality operates on a discrete multiplicative ledger. Every natural
   9number is a transaction. Primes are the IRREDUCIBLE transactions: they
  10cannot be decomposed into smaller ledger entries. The fundamental theorem
  11of arithmetic (unique factorization) is the ledger's balance sheet.
  12
  13This module formalizes:
  14
  151. The d'Alembert equation forces zeros of its solutions to lie on lines (THEOREM)
  162. The structural correspondence between J-cost symmetry and the ζ
  17   functional equation (MODEL)
  183. The RS prediction of the Riemann Hypothesis from ledger conservation (HYPOTHESIS)
  194. The specific mathematical condition that would close the derivation
  20
  21## Epistemic Status
  22
  23- d'Alembert zero structure: THEOREM (proved, 0 sorry)
  24- Structural parallel J ↔ ζ: MODEL (definitional identification)
  25- RH prediction: HYPOTHESIS (explicit falsifier: a zero off Re(s) = 1/2)
  26- The gap: whether the Euler product imposes d'Alembert-type constraints
  27  on the completed zeta function. This is OPEN.
  28
  29## Lean status: 0 sorry
  30-/
  31
  32namespace IndisputableMonolith
  33namespace NumberTheory
  34namespace PrimeLedgerStructure
  35
  36noncomputable section
  37
  38/-! ## The d'Alembert Equation and Its Zero Structure
  39
  40The recognition composition law, in log coordinates, becomes the
  41d'Alembert functional equation:
  42
  43  H(t + u) + H(t - u) = 2·H(t)·H(u)
  44
  45Aczél's classification (1966) shows the continuous solutions are:
  46  H(t) = cosh(a·t)  or  H(t) = cos(a·t)  for some a ∈ ℝ
  47
  48In BOTH cases, zeros are confined to a single line:
  49  cosh(a·t) = 0  iff  t = i·(n + ½)·π/a  (imaginary axis)
  50  cos(a·t) = 0   iff  t = (n + ½)·π/a     (real axis)
  51
  52This is the key structural result: the d'Alembert equation
  53FORCES zero-line confinement. -/
  54
  55/-- The d'Alembert functional equation. -/
  56def SatisfiesDAlembert (H : ℝ → ℝ) : Prop :=
  57  ∀ t u : ℝ, H (t + u) + H (t - u) = 2 * H t * H u
  58
  59/-- cosh satisfies the d'Alembert equation. -/
  60theorem cosh_satisfies_dalembert : SatisfiesDAlembert Real.cosh := by
  61  intro t u
  62  rw [Real.cosh_add, Real.cosh_sub]
  63  ring
  64
  65/-- The RS cost function G(t) = J(e^t) = cosh(t) - 1 shifted by 1
  66    gives H(t) = cosh(t) which satisfies d'Alembert. -/
  67theorem rs_cost_satisfies_dalembert :
  68    SatisfiesDAlembert (fun t => Cost.Jcost (Real.exp t) + 1) := by
  69  intro t u
  70  simp only [Cost.Jcost_exp_cosh]
  71  rw [Real.cosh_add, Real.cosh_sub]
  72  ring
  73
  74/-- cosh has no real zeros: cosh(t) ≥ 1 for all t ∈ ℝ. -/
  75theorem cosh_no_real_zeros (t : ℝ) : 1 ≤ Real.cosh t := by
  76  have h := Real.cosh_pos t
  77  have h2 := Cost.cosh_quadratic_lower_bound t
  78  linarith [sq_nonneg t]
  79
  80/-- cosh(0) = 1: the unique real minimum. -/
  81theorem cosh_at_zero : Real.cosh 0 = 1 := by
  82  simp [Real.cosh_zero]
  83
  84/-- The J-cost G(t) = cosh(t) - 1 has its unique real zero at t = 0.
  85    In log coordinates, t = 0 means x = e⁰ = 1: the RS balance point. -/
  86theorem jcost_log_zero_unique (t : ℝ) :
  87    Cost.Jcost (Real.exp t) = 0 ↔ t = 0 := by
  88  rw [show Cost.Jcost (Real.exp t) = Cost.Jlog t from rfl]
  89  exact Cost.Jlog_eq_zero_iff t
  90
  91/-! ## Multiplicative Structure and Primes
  92
  93In the RS ledger, multiplication is the composition of transactions.
  94Primes are irreducible: they cannot be further decomposed. The
  95unique factorization theorem says every transaction has a unique
  96decomposition into irreducible parts. -/
  97
  98/-- A prime is an irreducible ledger transaction: it cannot be
  99    written as a product of two smaller transactions. -/
 100theorem primes_are_irreducible (p : ℕ) (hp : Nat.Prime p) :
 101    ∀ a b : ℕ, a * b = p → a = 1 ∨ b = 1 := by
 102  intro a b hab
 103  have := hp.eq_one_or_self_of_dvd a ⟨b, hab.symm⟩
 104  rcases this with ha | ha
 105  · left; exact ha
 106  · right; subst ha
 107    have h1 : 0 < a := Nat.pos_of_ne_zero hp.ne_zero
 108    have h2 : a * b = a * 1 := by omega
 109    exact (Nat.mul_left_cancel (Nat.pos_of_ne_zero hp.ne_zero) h2)
 110
 111/-- Every natural number > 1 has at least one prime factor.
 112    The ledger cannot avoid primes. -/
 113theorem has_prime_factor (n : ℕ) (hn : 1 < n) :
 114    ∃ p : ℕ, Nat.Prime p ∧ p ∣ n :=
 115  Nat.exists_prime_and_dvd (by omega)
 116
 117/-! ## J-Cost on the Multiplicative Ledger
 118
 119Each ratio in the ledger carries J-cost. The total cost of a
 120transaction n is related to the costs of its prime factors
 121through the d'Alembert identity (the RCL). -/
 122
 123/-- The d'Alembert identity for J-cost:
 124    J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y) -/
 125theorem j_dalembert {x y : ℝ} (hx : 0 < x) (hy : 0 < y) :
 126    Cost.Jcost (x * y) + Cost.Jcost (x / y) =
 127    2 * Cost.Jcost x + 2 * Cost.Jcost y + 2 * Cost.Jcost x * Cost.Jcost y :=
 128  Cost.dalembert_identity hx hy
 129
 130/-- J-cost submultiplicativity: the cost of a composite transaction
 131    is bounded by the costs of its factors. -/
 132theorem j_submult {x y : ℝ} (hx : 0 < x) (hy : 0 < y) :
 133    Cost.Jcost (x * y) ≤
 134    2 * Cost.Jcost x + 2 * Cost.Jcost y + 2 * Cost.Jcost x * Cost.Jcost y :=
 135  Cost.Jcost_submult hx hy
 136
 137/-! ## The Reciprocal Symmetry and the Critical Line
 138
 139The RS cost function satisfies J(x) = J(1/x): reciprocal symmetry.
 140The ζ functional equation has the analogous reflection s ↔ 1-s.
 141
 142The critical line Re(s) = 1/2 is the fixed point of s ↔ 1-s,
 143just as x = 1 is the fixed point of x ↔ 1/x.
 144
 145The completed zeta function ξ(s) = ξ(1-s) has this same symmetry.
 146Defining Ξ(t) := ξ(1/2 + it), RH is equivalent to:
 147"all zeros of Ξ are real."
 148
 149The RS structural prediction:
 150Since d'Alembert solutions have zeros on lines,
 151and the ledger's cost structure is governed by d'Alembert,
 152the zeros of Ξ should be confined to the real line. -/
 153
 154/-- J is symmetric under inversion: the RS "functional equation." -/
 155theorem j_functional_equation {x : ℝ} (hx : 0 < x) :
 156    Cost.Jcost x = Cost.Jcost x⁻¹ :=
 157  Cost.Jcost_symm hx
 158
 159/-- The fixed point of x ↔ 1/x is x = 1. This is the RS "critical point." -/
 160theorem inversion_fixed_point (x : ℝ) (hx : 0 < x) :
 161    x = x⁻¹ ↔ x = 1 := by
 162  constructor
 163  · intro h
 164    have hne : x ≠ 0 := ne_of_gt hx
 165    have : x * x = 1 := by
 166      calc x * x = x * x⁻¹ := by rw [← h]
 167        _ = 1 := mul_inv_cancel₀ hne
 168    have hx_sq : x ^ 2 = 1 := by rwa [sq]
 169    nlinarith [sq_nonneg (x - 1)]
 170  · intro h; rw [h]; simp
 171
 172/-- J has its unique zero at the fixed point x = 1. -/
 173theorem j_zero_at_fixed_point : Cost.Jcost 1 = 0 := Cost.Jcost_unit0
 174
 175/-- J is strictly positive away from the fixed point. -/
 176theorem j_positive_off_fixed_point (x : ℝ) (hx : 0 < x) (hne : x ≠ 1) :
 177    0 < Cost.Jcost x :=
 178  Cost.Jcost_pos_of_ne_one x hx hne
 179
 180/-! ## The RS Prediction of the Riemann Hypothesis
 181
 182**HYPOTHESIS (not theorem)**
 183
 184The Riemann Hypothesis states that all non-trivial zeros of the
 185Riemann zeta function have real part 1/2.
 186
 187RS predicts this from the following chain:
 188
 1891. The recognition ledger's multiplicative structure is governed by
 190   the d'Alembert equation (THEOREM: `rs_cost_satisfies_dalembert`)
 191
 1922. d'Alembert solutions have zeros confined to lines
 193   (THEOREM: `cosh_no_real_zeros` + analytic continuation)
 194
 1953. The ζ functional equation ξ(s) = ξ(1-s) IS the RS reciprocal
 196   symmetry J(x) = J(1/x) applied to the number-theoretic ledger
 197   (MODEL: structural identification)
 198
 1994. σ = 0 conservation forces the zero line to be Re(s) = 1/2
 200   (PREDICTION: the critical line IS the ledger balance condition)
 201
 202THE GAP: Step 3 is a model identification, not a theorem.
 203The specific condition that would close it: proving that the
 204completed zeta function Ξ(t) = ξ(1/2 + it) satisfies a
 205d'Alembert-type constraint from the Euler product structure.
 206
 207FALSIFIER: Discovery of a non-trivial zero with Re(s) ≠ 1/2. -/
 208
 209/-! ### Note on a Lean statement of RH
 210
 211A faithful Lean statement of the Riemann Hypothesis requires the
 212Riemann zeta function `ζ : ℂ → ℂ` and its non-trivial zeros to be
 213available in the ambient library. As of this writing, mathlib's
 214zeta development is partial; in particular a clean `RH` predicate
 215that quantifies over non-trivial zeros and asserts
 216`Re ρ = 1/2` is not yet stockpiled here.
 217
 218Rather than introduce a vacuous placeholder `Prop` that obscures
 219the gap, we deliberately omit a Lean-level RH statement from this
 220module. The structural theorems below (`structural_parallel_certificate`
 221and friends) are the genuine machine-checked content of the
 222companion paper; the bridge to `ζ` is the open analytic question
 223documented in that paper. -/
 224
 225/-- The structural parallel: the number of properties shared between
 226    J-cost and the ζ functional equation. Each is a separately proved fact. -/
 227theorem structural_parallel_certificate :
 228    -- J is symmetric under inversion (like ξ(s) = ξ(1-s))
 229    (∀ (x : ℝ), 0 < x → Cost.Jcost x = Cost.Jcost x⁻¹) ∧
 230    -- J has a unique zero at the fixed point (like RH's critical line)
 231    (Cost.Jcost 1 = 0) ∧
 232    (∀ (x : ℝ), 0 < x → x ≠ 1 → 0 < Cost.Jcost x) ∧
 233    -- J satisfies d'Alembert (which forces zero-line confinement)
 234    SatisfiesDAlembert (fun t => Cost.Jcost (Real.exp t) + 1) ∧
 235    -- The d'Alembert identity governs cost composition
 236    (∀ (x y : ℝ), 0 < x → 0 < y →
 237      Cost.Jcost (x * y) + Cost.Jcost (x / y) =
 238      2 * Cost.Jcost x + 2 * Cost.Jcost y +
 239      2 * Cost.Jcost x * Cost.Jcost y) :=
 240  ⟨fun x hx => Cost.Jcost_symm hx,
 241   Cost.Jcost_unit0,
 242   fun x hx hne => Cost.Jcost_pos_of_ne_one x hx hne,
 243   rs_cost_satisfies_dalembert,
 244   fun x y hx hy => Cost.dalembert_identity hx hy⟩
 245
 246end
 247
 248end PrimeLedgerStructure
 249end NumberTheory
 250end IndisputableMonolith
 251