IndisputableMonolith.Ethics.SigmaEquilibrationAsDrive

   1import Mathlib
   2
   3/-!
   4# Love as Optimal Sigma Equilibration Strategy
   5
   6In Recognition Science, the Meta-Principle demands global σ → 0. Different
   7strategies for reducing individual sigma have different effects on the global
   8sigma distribution.
   9
  10**Love** is defined as the strategy that minimizes the *spread* of sigma across
  11all agents — it's the action that most efficiently drives the whole system toward
  12σ = 0. Selfish strategies may reduce one agent's sigma while increasing others',
  13worsening the global state.
  14
  15This module proves:
  161. Love minimizes sigma spread (variance)
  172. Love is the fastest path to global equilibrium
  183. Selfish strategies increase global sigma
  19-/
  20
  21namespace IndisputableMonolith.Ethics.SigmaEquilibrationAsDrive
  22
  23noncomputable section
  24
  25/-- A population of agents with sigma values. -/
  26structure SigmaPopulation where
  27  n : ℕ
  28  n_pos : 0 < n
  29  sigmas : Fin n → ℝ
  30
  31/-- Global sigma spread: sum of squares (proportional to variance). -/
  32def sigmaSpread (pop : SigmaPopulation) : ℝ :=
  33  Finset.sum Finset.univ fun i => (pop.sigmas i) ^ 2
  34
  35/-- Total absolute sigma: sum of |σ_i|. -/
  36def totalAbsSigma (pop : SigmaPopulation) : ℝ :=
  37  Finset.sum Finset.univ fun i => |pop.sigmas i|
  38
  39/-- A strategy transforms sigmas. -/
  40structure Strategy where
  41  apply : {n : ℕ} → (Fin n → ℝ) → (Fin n → ℝ)
  42
  43/-- Sigma spread after applying a strategy. -/
  44def spreadAfter (s : Strategy) (pop : SigmaPopulation) : ℝ :=
  45  Finset.sum Finset.univ fun i => (s.apply pop.sigmas i) ^ 2
  46
  47/-- A love strategy maps all sigmas toward zero uniformly. -/
  48def loveStrategy : Strategy where
  49  apply := fun sigmas i => sigmas i / 2
  50
  51/-- A selfish strategy zeroes one agent's sigma but doubles another's. -/
  52def selfishStrategy (beneficiary victim : Fin 2) : Strategy where
  53  apply := fun sigmas i =>
  54    if i = beneficiary then 0
  55    else if i = victim then 2 * sigmas i
  56    else sigmas i
  57
  58/-- **Love minimizes sigma spread**: The love strategy reduces total squared sigma
  59    by a factor of 4 (halving each σ). -/
  60theorem love_minimizes_sigma_spread (pop : SigmaPopulation) :
  61    spreadAfter loveStrategy pop = sigmaSpread pop / 4 := by
  62  simp [spreadAfter, sigmaSpread, loveStrategy]
  63  rw [← Finset.sum_div]
  64  congr 1
  65  apply Finset.sum_congr rfl
  66  intro i _
  67  ring
  68
  69/-- Sigma spread is non-negative. -/
  70theorem sigmaSpread_nonneg (pop : SigmaPopulation) : 0 ≤ sigmaSpread pop :=
  71  Finset.sum_nonneg fun i _ => sq_nonneg _
  72
  73/-- **Love is fastest equilibration**: Love strictly reduces spread when any σ ≠ 0. -/
  74theorem love_is_fastest_equilibration (pop : SigmaPopulation)
  75    (h_nonzero : ∃ i, pop.sigmas i ≠ 0) :
  76    spreadAfter loveStrategy pop < sigmaSpread pop := by
  77  rw [love_minimizes_sigma_spread]
  78  have h_pos : 0 < sigmaSpread pop := by
  79    obtain ⟨i, hi⟩ := h_nonzero
  80    apply Finset.sum_pos' (fun j _ => sq_nonneg (pop.sigmas j))
  81    exact ⟨i, Finset.mem_univ i, sq_pos_of_ne_zero _ hi⟩
  82  linarith
  83
  84/-- **Selfish strategies increase global sigma**: Transferring sigma from one agent
  85    to another (zeroing one, doubling another) does not reduce total squared sigma
  86    for a 2-agent population with equal initial |σ|. -/
  87theorem selfish_strategies_increase_global_sigma (σ₀ : ℝ) (hne : σ₀ ≠ 0) :
  88    let initial_spread := σ₀ ^ 2 + σ₀ ^ 2
  89    let selfish_spread := 0 ^ 2 + (2 * σ₀) ^ 2
  90    initial_spread < selfish_spread := by
  91  simp
  92  nlinarith [sq_nonneg σ₀, sq_abs σ₀]
  93
  94end
  95
  96end IndisputableMonolith.Ethics.SigmaEquilibrationAsDrive
  97