cooper_pair_cost_zero

formal source

 114    identity tick. This is the structural origin of persistence: even
 115    when no event sits at `x = 1` directly, any pair of inverse states
 116    constructs a persistent reference. -/
 117theorem cooper_pair_cost_zero (x : ℝ) (hx : 0 < x) :
 118    Cost.Jcost (x * x⁻¹) = 0 := by
 119  rw [mul_inv_cancel₀ (ne_of_gt hx)]
 120  exact Cost.Jcost_unit0
 121
 122/-- Cooper pairing constructs a persistent recognition event from any
 123    positive starting state. -/
 124theorem cooper_pairing_yields_persistent
 125    (x : ℝ) (hx : 0 < x) :
 126    ∃ e : RecognitionEvent, IsPersistent e := by
 127  refine ⟨{
 128    state := x * x⁻¹,
 129    state_pos := by
 130      rw [mul_inv_cancel₀ (ne_of_gt hx)]; norm_num
 131  }, ?_⟩
 132  show Cost.Jcost (x * x⁻¹) = 0
 133  exact cooper_pair_cost_zero x hx
 134
 135/-! ## §5. The Observer -/
 136
 137/-- An observer is a coherent recognition structure equipped with a
 138    persistent reference event. The observer integrates multiple
 139    distinguishable recognition events against a single fixed
 140    identity-tick reference. -/
 141structure Observer where
 142  recognition : CoherentRecognition
 143  persistent : IsPersistent recognition.reference
 144
 145namespace Observer
 146
 147/-- An observer's reference has zero cost. -/