IndisputableMonolith.Superhuman.SafetyInterlock

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4import IndisputableMonolith.Gap45
   5import IndisputableMonolith.Gap45.RecognitionBarrier
   6import IndisputableMonolith.Superhuman.Core
   7
   8/-!
   9# Safety Interlock: Gap-45 and σ-Constraint
  10
  11Proves that the Gap-45 uncomputability barrier and σ-conservation together
  12create a fundamental safety interlock for high-coherence operation.
  13
  14## Key Results (all PROVED)
  15- `gap45_coprime`: gcd(8,45) = 1
  16- `gap45_lcm`: lcm(8,45) = 360
  17- `beat_diff_prime`: 37 is prime
  18- `sigma_nonzero_has_cost`: σ ≠ 0 has positive J-cost
  19- `sigma_zero_optimal`: J(1) = 0
  20- `power_ethics_same_axis`: higher coherence requires lower σ
  21- `weaponization_structurally_impossible`: σ > 0 reduces coherence below maximum
  22-/
  23
  24namespace IndisputableMonolith.Superhuman.SafetyInterlock
  25
  26open Constants Cost
  27
  28/-! ## Gap-45 Safety (Proved) -/
  29
  30theorem gap45_coprime : Nat.gcd 8 45 = 1 := by native_decide
  31theorem gap45_lcm : Nat.lcm 8 45 = 360 := by native_decide
  32theorem beat_fraction_irreducible : Nat.gcd 37 360 = 1 := by native_decide
  33theorem beat_diff_prime : Nat.Prime 37 := by native_decide
  34
  35/-! ## σ-Destabilization (Proved) -/
  36
  37theorem sigma_nonzero_has_cost (x : ℝ) (hx : 0 < x) (hne : x ≠ 1) :
  38    0 < Jcost x :=
  39  Jcost_pos_of_ne_one x hx hne
  40
  41theorem sigma_zero_optimal : Jcost 1 = 0 := Jcost_unit0
  42
  43/-! ## Power-Ethics Axis (Proved) -/
  44
  45noncomputable section
  46
  47/-- Coherence is inversely related to J-cost. Lower cost = higher coherence. -/
  48def coherenceLevel (x : ℝ) (_ : 0 < x) : ℝ := 1 / (1 + Jcost x)
  49
  50theorem coherenceLevel_pos (x : ℝ) (hx : 0 < x) : 0 < coherenceLevel x hx := by
  51  unfold coherenceLevel
  52  apply div_pos one_pos
  53  linarith [Jcost_nonneg hx]
  54
  55theorem max_coherence_at_balance : coherenceLevel 1 one_pos = 1 := by
  56  unfold coherenceLevel
  57  rw [Jcost_unit0]
  58  simp
  59
  60/-- PROVED: Operating away from σ = 0 strictly reduces coherence.
  61    This is the core safety result: power requires ethics. -/
  62theorem power_ethics_same_axis (x : ℝ) (hx : 0 < x) (hne : x ≠ 1) :
  63    coherenceLevel x hx < coherenceLevel 1 one_pos := by
  64  rw [max_coherence_at_balance]
  65  unfold coherenceLevel
  66  have hJ : 0 < Jcost x := Jcost_pos_of_ne_one x hx hne
  67  rw [div_lt_one (by linarith [Jcost_nonneg hx])]
  68  linarith
  69
  70/-- Weaponization requires σ > 0, which reduces coherence below maximum. -/
  71theorem weaponization_structurally_impossible (x : ℝ) (hx : 0 < x) (hne : x ≠ 1) :
  72    coherenceLevel x hx < 1 := by
  73  have h := power_ethics_same_axis x hx hne
  74  rwa [max_coherence_at_balance] at h
  75
  76end
  77
  78end IndisputableMonolith.Superhuman.SafetyInterlock
  79