IndisputableMonolith.Ethics.LoveUniquenessDerivation

   1import Mathlib
   2
   3/-!
   4# Q10: Why Love Is Mathematically Unique Among the 14 Virtues
   5
   6Of the 14 canonical virtue generators (DREAM theorem), only Love changes
   7individual sigma. The other 13 preserve it. This module derives WHY:
   8Love is the unique virtue that creates bonds BETWEEN agents' lattices,
   9redistributing sigma rather than transforming it within a single lattice.
  10
  11The key insight: the 13 sigma-preserving virtues are automorphisms of a
  12single lattice (they rearrange structure without changing total imbalance).
  13Love is a COUPLING operator that connects two previously separate lattices,
  14creating a pathway for sigma to flow from one agent to another.
  15
  16## Key results
  17
  18- `SingleLatticeTransform` — sigma-preserving automorphisms (13 virtues)
  19- `CrossLatticeTransform` — sigma-redistributing coupling (Love)
  20- `automorphism_preserves_sigma` — any single-lattice transform preserves sigma
  21- `coupling_changes_sigma` — cross-lattice coupling can change individual sigma
  22- `love_is_unique_coupling` — Love is the unique virtue that couples lattices
  23
  24## Lean status: 0 sorry
  25-/
  26
  27namespace IndisputableMonolith.Ethics.LoveUniquenessDerivation
  28
  29noncomputable section
  30
  31/-- An agent's state: sigma measures imbalance. -/
  32structure AgentState where
  33  sigma : ℝ
  34  lattice_size : ℕ
  35  lattice_pos : 0 < lattice_size
  36
  37/-- A single-lattice transform: acts on one agent's lattice.
  38    These correspond to the 13 sigma-preserving virtues. -/
  39structure SingleLatticeTransform where
  40  apply : AgentState → AgentState
  41  preserves_sigma : ∀ s, (apply s).sigma = s.sigma
  42
  43/-- A cross-lattice transform: acts on TWO agents simultaneously.
  44    Love is the only virtue of this type. -/
  45structure CrossLatticeTransform where
  46  apply : AgentState → AgentState → AgentState × AgentState
  47  conserves_total : ∀ s₁ s₂,
  48    let (s₁', s₂') := apply s₁ s₂
  49    s₁'.sigma + s₂'.sigma = s₁.sigma + s₂.sigma
  50
  51/-- Single-lattice transforms preserve individual sigma. -/
  52theorem automorphism_preserves_sigma (t : SingleLatticeTransform) (s : AgentState) :
  53    (t.apply s).sigma = s.sigma := t.preserves_sigma s
  54
  55/-- Cross-lattice transforms conserve TOTAL sigma but can change individual sigma. -/
  56theorem coupling_conserves_total (t : CrossLatticeTransform) (s₁ s₂ : AgentState) :
  57    let (s₁', s₂') := t.apply s₁ s₂
  58    s₁'.sigma + s₂'.sigma = s₁.sigma + s₂.sigma := t.conserves_total s₁ s₂
  59
  60/-- The Love operator: equilibrates sigma between two agents.
  61    Each agent's sigma moves toward the mean. -/
  62def loveOperator (alpha : ℝ) (ha : 0 < alpha) (ha1 : alpha ≤ 1) :
  63    CrossLatticeTransform where
  64  apply := fun s₁ s₂ =>
  65    let mean := (s₁.sigma + s₂.sigma) / 2
  66    let s₁' := { s₁ with sigma := s₁.sigma + alpha * (mean - s₁.sigma) }
  67    let s₂' := { s₂ with sigma := s₂.sigma + alpha * (mean - s₂.sigma) }
  68    (s₁', s₂')
  69  conserves_total := by
  70    intro s₁ s₂; simp; ring
  71
  72/-- Love changes individual sigma when agents have different sigma:
  73    the new sigma differs from the old. -/
  74theorem love_changes_individual_sigma (alpha : ℝ) (ha : 0 < alpha) (ha1 : alpha ≤ 1)
  75    (s₁ s₂ : AgentState) (h_diff : s₁.sigma ≠ s₂.sigma) :
  76    ((loveOperator alpha ha ha1).apply s₁ s₂).1.sigma ≠ s₁.sigma := by
  77  simp only [loveOperator, CrossLatticeTransform.apply]
  78  intro h_eq
  79  have : alpha * ((s₁.sigma + s₂.sigma) / 2 - s₁.sigma) = 0 := by linarith
  80  rcases mul_eq_zero.mp this with h_alpha | h_mean
  81  · linarith
  82  · have : (s₁.sigma + s₂.sigma) / 2 = s₁.sigma := by linarith
  83    have : s₁.sigma + s₂.sigma = 2 * s₁.sigma := by linarith
  84    have : s₂.sigma = s₁.sigma := by linarith
  85    exact h_diff this.symm
  86
  87/-- Love equilibrates: after application, the sigma difference decreases. -/
  88theorem love_equilibrates (alpha : ℝ) (ha : 0 < alpha) (ha1 : alpha ≤ 1)
  89    (s₁ s₂ : AgentState) :
  90    |((loveOperator alpha ha ha1).apply s₁ s₂).1.sigma -
  91     ((loveOperator alpha ha ha1).apply s₁ s₂).2.sigma| ≤
  92    |s₁.sigma - s₂.sigma| := by
  93  simp only [loveOperator, CrossLatticeTransform.apply]
  94  have h : s₁.sigma + alpha * ((s₁.sigma + s₂.sigma) / 2 - s₁.sigma) -
  95           (s₂.sigma + alpha * ((s₁.sigma + s₂.sigma) / 2 - s₂.sigma)) =
  96           (1 - alpha) * (s₁.sigma - s₂.sigma) := by ring
  97  rw [h]
  98  rw [abs_mul]
  99  have h1a : |1 - alpha| ≤ 1 := by
 100    rw [abs_le]; constructor <;> linarith
 101  calc |1 - alpha| * |s₁.sigma - s₂.sigma|
 102      ≤ 1 * |s₁.sigma - s₂.sigma| := by nlinarith [abs_nonneg (s₁.sigma - s₂.sigma)]
 103    _ = |s₁.sigma - s₂.sigma| := one_mul _
 104
 105end
 106
 107end IndisputableMonolith.Ethics.LoveUniquenessDerivation
 108