IndisputableMonolith.NumberTheory.ProxyPhysicalizationBridge

   1import IndisputableMonolith.NumberTheory.DirectedEulerLedger
   2import IndisputableMonolith.NumberTheory.ZetaLedgerBridge
   3
   4/-!
   5# Proxy Physicalization Bridge
   6
   7This module isolates the exact remaining gap after constructing the concrete
   8directed Euler-ledger system and connecting it to the already-proved
   9admissibility / realizability infrastructure.
  10
  11## Proven earlier
  12
  13For every `DefectSensor`, we now have:
  14
  15* a concrete directed Euler ledger over finite prime supports,
  16* an admissible Euler trace,
  17* a T1-bounded realizability proxy (`PhysicallyRealizableLedger`).
  18
  19## Remaining bridge
  20
  21What is still missing for `RSPhysicalThesis` is the transport from the bounded
  22proxy state supplied by `PhysicallyRealizableLedger` to the actual
  23`PhysicallyExists` predicate, which is defined using `eulerLedgerScalarState`.
  24
  25This file packages that as an explicit assumption and proves that, once this
  26bridge is supplied for the zeta-zero sensors, `RSPhysicalThesis` and hence
  27`RiemannHypothesis` follow.
  28-/
  29
  30namespace IndisputableMonolith
  31namespace NumberTheory
  32
  33open IndisputableMonolith.Unification.UnifiedRH
  34
  35/-- The remaining sensor-level bridge: the bounded T1-defect of the
  36realizability proxy coming from the concrete Euler-ledger ontology package
  37must imply `PhysicallyExists` for the same sensor. -/
  38def ProxyPhysicalizationBridge (sensor : DefectSensor) : Prop :=
  39  let pkg := concreteEulerLedgerOntologyInterface sensor
  40  letI : PhysicallyRealizableLedger sensor := pkg.realizableProxy
  41  (∃ K : ℝ, ∀ N : ℕ,
  42    IndisputableMonolith.Foundation.LawOfExistence.defect
  43      (PhysicallyRealizableLedger.scalarState (sensor := sensor) N) ≤ K) →
  44    PhysicallyExists sensor
  45
  46/-- If the proxy physicalization bridge holds for a sensor, then the concrete
  47Euler-ledger ontology package yields `PhysicallyExists` for that sensor. -/
  48theorem physicallyExists_of_ProxyPhysicalizationBridge
  49    (sensor : DefectSensor) (bridge : ProxyPhysicalizationBridge sensor) :
  50    PhysicallyExists sensor := by
  51  let pkg := concreteEulerLedgerOntologyInterface sensor
  52  letI : PhysicallyRealizableLedger sensor := pkg.realizableProxy
  53  exact bridge (EulerLedgerOntologyInterface.scalarDefectBounded pkg)
  54
  55/-- Zero-induced proxy physicalization: every strip zero of `ζ` produces a
  56sensor for which the proxy-to-physical-existence bridge holds. -/
  57def ZeroInducedProxyPhysicalizationBridge : Prop :=
  58  ∀ (ρ : ℂ), riemannZeta ρ = 0 →
  59    ∀ (hlo : 1/2 < ρ.re) (hhi : ρ.re < 1),
  60      ProxyPhysicalizationBridge (zetaDefectSensor ρ.re ⟨hlo, hhi⟩ 1)
  61
  62/-- The zero-induced proxy physicalization bridge implies the RS Physical
  63Thesis. -/
  64theorem rsPhysicalThesis_of_ZeroInducedProxyPhysicalizationBridge
  65    (bridge : ZeroInducedProxyPhysicalizationBridge) :
  66    RSPhysicalThesis := by
  67  intro ρ hzero hlo hhi
  68  exact physicallyExists_of_ProxyPhysicalizationBridge
  69    (zetaDefectSensor ρ.re ⟨hlo, hhi⟩ 1) (bridge ρ hzero hlo hhi)
  70
  71/-- Hence the full Riemann Hypothesis follows once the proxy physicalization
  72bridge is proved for the zeta-zero sensors. -/
  73theorem rh_from_ZeroInducedProxyPhysicalizationBridge
  74    (bridge : ZeroInducedProxyPhysicalizationBridge) :
  75    RiemannHypothesis :=
  76  rh_from_rs_thesis (rsPhysicalThesis_of_ZeroInducedProxyPhysicalizationBridge bridge)
  77
  78/-! ## Equivalence Theorems
  79
  80The bounded-proxy hypothesis in `ProxyPhysicalizationBridge` is
  81unconditionally proved (`EulerLedgerOntologyInterface.scalarDefectBounded`).
  82Therefore the implication collapses to its conclusion, and the bridge
  83is equivalent to `PhysicallyExists`, to `charge = 0`, and (for the
  84zeta-zero specialization) to `RSPhysicalThesis` and to `RiemannHypothesis`.
  85
  86These equivalences are the definitive reduction: the bridge proposition is
  87neither weaker nor stronger than RH — it IS RH, expressed through the
  88T1-bounded realizability architecture. -/
  89
  90/-- The proxy physicalization bridge is equivalent to physical existence.
  91The bounded-proxy hypothesis is unconditionally true, so the implication
  92reduces to its conclusion. -/
  93theorem proxyPhysicalizationBridge_iff_physicallyExists (sensor : DefectSensor) :
  94    ProxyPhysicalizationBridge sensor ↔ PhysicallyExists sensor := by
  95  constructor
  96  · exact physicallyExists_of_ProxyPhysicalizationBridge sensor
  97  · intro hphys
  98    let pkg := concreteEulerLedgerOntologyInterface sensor
  99    letI : PhysicallyRealizableLedger sensor := pkg.realizableProxy
 100    exact fun _ => hphys
 101
 102/-- The proxy physicalization bridge is equivalent to charge zero.
 103Combines the bridge-to-existence equivalence with the ontological
 104dichotomy `charge = 0 ↔ PhysicallyExists`. -/
 105theorem proxyPhysicalizationBridge_iff_charge_zero (sensor : DefectSensor) :
 106    ProxyPhysicalizationBridge sensor ↔ sensor.charge = 0 :=
 107  (proxyPhysicalizationBridge_iff_physicallyExists sensor).trans
 108    (charge_zero_iff_physicallyExists sensor).symm
 109
 110/-- The bridge holds unconditionally for charge-zero sensors. -/
 111theorem proxyPhysicalizationBridge_of_charge_zero (sensor : DefectSensor)
 112    (h : sensor.charge = 0) : ProxyPhysicalizationBridge sensor :=
 113  (proxyPhysicalizationBridge_iff_charge_zero sensor).mpr h
 114
 115/-- The bridge fails for any sensor with nonzero charge: the cost scalar
 116collapses and T1 defect diverges, so `PhysicallyExists` is false. -/
 117theorem not_proxyPhysicalizationBridge_of_charge_ne_zero (sensor : DefectSensor)
 118    (hm : sensor.charge ≠ 0) : ¬ ProxyPhysicalizationBridge sensor := by
 119  intro hbridge
 120  exact nonzero_charge_not_physical sensor hm
 121    (physicallyExists_of_ProxyPhysicalizationBridge sensor hbridge)
 122
 123/-- `ZeroInducedProxyPhysicalizationBridge` is logically equivalent to
 124`RSPhysicalThesis`: the bridge at zeta-zero sensors reduces to the RS
 125claim that zeta zeros are physical recognition events. -/
 126theorem zeroInducedBridge_iff_rsPhysicalThesis :
 127    ZeroInducedProxyPhysicalizationBridge ↔ RSPhysicalThesis := by
 128  constructor
 129  · intro hzipb ρ hzero hlo hhi
 130    exact (proxyPhysicalizationBridge_iff_physicallyExists _).mp (hzipb ρ hzero hlo hhi)
 131  · intro hrs ρ hzero hlo hhi
 132    exact (proxyPhysicalizationBridge_iff_physicallyExists _).mpr (hrs ρ hzero hlo hhi)
 133
 134/-- `ZeroInducedProxyPhysicalizationBridge` is equivalent to the absence of
 135strip zeros of ζ: for charge-1 sensors, the bridge evaluates to `False`,
 136so quantifying over strip zeros asserts their non-existence. -/
 137theorem zeroInducedBridge_iff_no_strip_zeros :
 138    ZeroInducedProxyPhysicalizationBridge ↔
 139      (∀ ρ : ℂ, riemannZeta ρ = 0 → 1/2 < ρ.re → ρ.re < 1 → False) := by
 140  constructor
 141  · intro hzipb ρ hzero hlo hhi
 142    exact not_proxyPhysicalizationBridge_of_charge_ne_zero
 143      (zetaDefectSensor ρ.re ⟨hlo, hhi⟩ 1)
 144      (zetaDefectSensor_charge_ne_zero ρ.re ⟨hlo, hhi⟩)
 145      (hzipb ρ hzero hlo hhi)
 146  · intro hno ρ hzero hlo hhi
 147    exact (hno ρ hzero hlo hhi).elim
 148
 149/-- Mathlib's `RiemannHypothesis` implies `ZeroInducedProxyPhysicalizationBridge`.
 150Any strip zero would violate `Re(ρ) = 1/2`, so the bridge holds vacuously. -/
 151theorem zeroInducedBridge_of_rh (hrh : RiemannHypothesis) :
 152    ZeroInducedProxyPhysicalizationBridge := by
 153  rw [zeroInducedBridge_iff_no_strip_zeros]
 154  intro ρ hzero hlo hhi
 155  have hntrivial : ¬∃ n : ℕ, ρ = -2 * (↑n + 1) := by
 156    rintro ⟨n, hn⟩
 157    have h1 : ρ.re = (-2 * ((n : ℂ) + 1)).re := congrArg Complex.re hn
 158    have h2 : (-2 * ((n : ℂ) + 1)).re = -2 * ((n : ℝ) + 1) := by
 159      rw [Complex.mul_re, Complex.add_re, Complex.natCast_re, Complex.one_re,
 160          Complex.add_im, Complex.natCast_im, Complex.one_im]
 161      simp [Complex.neg_re, Complex.neg_im]
 162    linarith [Nat.cast_nonneg (α := ℝ) n]
 163  have hne1 : ρ ≠ 1 := by
 164    intro h; rw [h, Complex.one_re] at hhi; linarith
 165  linarith [hrh ρ hzero hntrivial hne1]
 166
 167/-- **`ZeroInducedProxyPhysicalizationBridge ↔ RiemannHypothesis`.**
 168
 169The bridge proposition is exactly equivalent to Mathlib's `RiemannHypothesis`.
 170Forward: `rh_from_ZeroInducedProxyPhysicalizationBridge` (via RS thesis + functional equation).
 171Backward: `zeroInducedBridge_of_rh` (RH eliminates all strip zeros, making the bridge vacuous).
 172
 173This closes the reduction: the directed-ledger infrastructure correctly isolates
 174the gap, and the gap is precisely RH — no more, no less. -/
 175theorem zeroInducedBridge_iff_rh :
 176    ZeroInducedProxyPhysicalizationBridge ↔ RiemannHypothesis :=
 177  ⟨rh_from_ZeroInducedProxyPhysicalizationBridge, zeroInducedBridge_of_rh⟩
 178
 179/-! ## Axiom audit -/
 180
 181#print axioms proxyPhysicalizationBridge_iff_physicallyExists
 182#print axioms proxyPhysicalizationBridge_iff_charge_zero
 183#print axioms zeroInducedBridge_iff_rsPhysicalThesis
 184#print axioms zeroInducedBridge_iff_no_strip_zeros
 185#print axioms zeroInducedBridge_iff_rh
 186
 187end NumberTheory
 188end IndisputableMonolith
 189