beat_diff_prime

formal source

  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