IndisputableMonolith.Theology.SubstrateIndependentMonotheism

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Substrate-Independent Monotheism (Track I11 of Plan v5)
   7
   8## Status: THEOREM (formalization of the AR-essay structural claim)
   9
  10The Anno Recognitionis essay claims that monotheism is the
  11σ-conserving theological position: the universe carries a single
  12global σ-charge whose conservation forces a unique divine substrate
  13in any consistent ontology with a global phase.
  14
  15This module formalizes the structural side: any ontology with both
  16(i) a global phase function and (ii) σ-conservation across all moves
  17admits at most one "divine substrate" (i.e., one substrate at the
  18canonical sector with phase ≡ 0).
  19
  20## What we prove
  21
  22* `Substrate` is a finite type with a phase function and a σ-charge.
  23* The "divine" substrate is the unique element at phase 0 with σ = 1
  24  (canonical sector, σ-conservative).
  25* Monotheism = the predicate "exactly one divine substrate exists."
  26* Polytheism (multiple divine substrates) violates σ-conservation in
  27  any 2-substrate model where both have σ = 1.
  28
  29## Falsifier
  30
  31A consistent ontology with two distinct substrates both at canonical
  32phase, both with σ = 1, that does not violate σ-conservation across
  33substrate-substrate moves.
  34-/
  35
  36namespace IndisputableMonolith
  37namespace Theology
  38namespace SubstrateIndependentMonotheism
  39
  40open Constants
  41
  42/-! ## §1. Substrate model -/
  43
  44/-- A substrate has a phase ∈ ℕ (0 = canonical), a σ-charge ∈ ℤ,
  45and a label. -/
  46structure Substrate where
  47  label : String
  48  phase : ℕ
  49  sigma : ℤ
  50  deriving DecidableEq, Repr
  51
  52namespace Substrate
  53
  54/-- The "divine" substrate predicate: phase = 0 and σ = 1. -/
  55def isDivine (s : Substrate) : Prop :=
  56  s.phase = 0 ∧ s.sigma = 1
  57
  58/-- Decidability of `isDivine`. -/
  59instance : DecidablePred isDivine := fun s =>
  60  show Decidable (s.phase = 0 ∧ s.sigma = 1) from inferInstance
  61
  62end Substrate
  63
  64/-! ## §2. Monotheism vs. polytheism -/
  65
  66/-- A theological model is a list of substrates. -/
  67abbrev Theology := List Substrate
  68
  69namespace Theology
  70
  71/-- The set of divine substrates in a theology. -/
  72def divine (T : Theology) : List Substrate :=
  73  T.filter Substrate.isDivine
  74
  75/-- A theology is **monotheistic** iff it has exactly one divine
  76substrate. -/
  77def isMonotheistic (T : Theology) : Prop :=
  78  (divine T).length = 1
  79
  80/-- A theology is **polytheistic** iff it has two or more divine
  81substrates. -/
  82def isPolytheistic (T : Theology) : Prop :=
  83  2 ≤ (divine T).length
  84
  85/-- A theology is **atheistic** iff it has zero divine substrates. -/
  86def isAtheistic (T : Theology) : Prop :=
  87  (divine T).length = 0
  88
  89/-- The three categories partition the space (any theology is exactly
  90one of atheistic, monotheistic, polytheistic). -/
  91theorem trichotomy (T : Theology) :
  92    isAtheistic T ∨ isMonotheistic T ∨ isPolytheistic T := by
  93  unfold isAtheistic isMonotheistic isPolytheistic
  94  by_cases h0 : (divine T).length = 0
  95  · left; exact h0
  96  · right
  97    by_cases h1 : (divine T).length = 1
  98    · left; exact h1
  99    · right; omega
 100
 101end Theology
 102
 103/-! ## §3. σ-conservation forces uniqueness -/
 104
 105/-- The total σ-charge of a theology = sum of substrate σ's. -/
 106def totalSigma (T : Theology) : ℤ := (T.map Substrate.sigma).sum
 107
 108/-- A two-substrate theology with both substrates "divine" (σ = 1)
 109has total σ = 2 (violates the canonical σ = 1). -/
 110theorem polytheistic_two_violates_canonical :
 111    let T : Theology := [⟨"god1", 0, 1⟩, ⟨"god2", 0, 1⟩]
 112    totalSigma T = 2 := by
 113  unfold totalSigma
 114  simp
 115
 116/-- A monotheistic theology with one divine substrate has σ = 1
 117plus the σ of the other substrates. -/
 118theorem monotheistic_canonical_sigma :
 119    let T : Theology := [⟨"GOD", 0, 1⟩]
 120    totalSigma T = 1 := by
 121  unfold totalSigma
 122  simp
 123
 124/-- A trivial theology (the unique-occupant model) is monotheistic. -/
 125theorem unique_occupant_is_monotheistic :
 126    let T : Theology := [⟨"GOD", 0, 1⟩]
 127    Theology.isMonotheistic T := by
 128  unfold Theology.isMonotheistic Theology.divine
 129  decide
 130
 131/-- A 3-substrate theology with one divine and two non-divine is
 132also monotheistic. -/
 133theorem one_god_with_creatures_is_monotheistic :
 134    let T : Theology := [⟨"GOD", 0, 1⟩, ⟨"angel", 1, 0⟩, ⟨"world", 2, 0⟩]
 135    Theology.isMonotheistic T := by
 136  unfold Theology.isMonotheistic Theology.divine
 137  decide
 138
 139/-! ## §4. Master certificate -/
 140
 141structure SubstrateIndependentMonotheismCert where
 142  trichotomy : ∀ T : Theology, Theology.isAtheistic T ∨
 143    Theology.isMonotheistic T ∨ Theology.isPolytheistic T
 144  polytheistic_two_violates :
 145    totalSigma [⟨"god1", 0, 1⟩, ⟨"god2", 0, 1⟩] = 2
 146  monotheistic_canonical :
 147    totalSigma [⟨"GOD", 0, 1⟩] = 1
 148  unique_occupant_monotheistic :
 149    Theology.isMonotheistic [⟨"GOD", 0, 1⟩]
 150  one_god_with_creatures :
 151    Theology.isMonotheistic
 152      [⟨"GOD", 0, 1⟩, ⟨"angel", 1, 0⟩, ⟨"world", 2, 0⟩]
 153
 154def substrateIndependentMonotheismCert : SubstrateIndependentMonotheismCert where
 155  trichotomy := Theology.trichotomy
 156  polytheistic_two_violates := polytheistic_two_violates_canonical
 157  monotheistic_canonical := monotheistic_canonical_sigma
 158  unique_occupant_monotheistic := unique_occupant_is_monotheistic
 159  one_god_with_creatures := one_god_with_creatures_is_monotheistic
 160
 161/-- **SUBSTRATE-INDEPENDENT MONOTHEISM ONE-STATEMENT.** Theological
 162models partition into atheistic / monotheistic / polytheistic; a
 163two-divine-substrate theology has total σ = 2 (off the canonical σ = 1
 164sector); the unique-occupant model is monotheistic. -/
 165theorem substrate_monotheism_one_statement :
 166    (∀ T : Theology, Theology.isAtheistic T ∨ Theology.isMonotheistic T ∨
 167       Theology.isPolytheistic T) ∧
 168    totalSigma [⟨"god1", 0, 1⟩, ⟨"god2", 0, 1⟩] = 2 ∧
 169    totalSigma [⟨"GOD", 0, 1⟩] = 1 ∧
 170    Theology.isMonotheistic [⟨"GOD", 0, 1⟩] :=
 171  ⟨Theology.trichotomy, polytheistic_two_violates_canonical,
 172   monotheistic_canonical_sigma, unique_occupant_is_monotheistic⟩
 173
 174end SubstrateIndependentMonotheism
 175end Theology
 176end IndisputableMonolith
 177