RecognitionOperator

formal source

  35def admissible (_s : LedgerState) : Prop := True
  36
  37/-- A local dissipative recognition operator for information thermodynamics. -/
  38structure RecognitionOperator where
  39  evolve : LedgerState → LedgerState
  40  minimizes_J : ∀ s, admissible s → RecognitionCost (evolve s) ≤ RecognitionCost s
  41
  42/-- **DEFINITION: Ledger Entropy**
  43    Entropy defined as the absolute log-imbalance of the ledger. -/
  44noncomputable def ledger_entropy (s : LedgerState) : ℝ :=
  45  reciprocity_skew s
  46
  47/-- **DEFINITION: Thermal Energy Scale**
  48    The base thermal cost per tick. -/
  49noncomputable def thermal_cost (T : ℝ) : ℝ :=
  50  T * Real.log 2
  51
  52/-- **THEOREM: Landauer Bound for Recognition**
  53    The recognition cost required to reduce mismatch must satisfy the Landauer bound.
  54    Specifically, the sum of J-costs across the ledger provides a quadratic lower
  55    bound on the information dissipation. -/
  56theorem landauer_bound_holds (s : LedgerState) :
  57    ∀ b ∈ s.active_bonds,
  58      let m := s.bond_multipliers b
  59      let u := Real.log m
  60      Cost.Jcost m ≥ u^2 / 2 := by
  61  intro b hb m u
  62  have hm : 0 < m := s.bond_pos b hb
  63  -- Jcost m = cosh (log m) - 1
  64  have h_m_exp : m = exp u := (exp_log hm).symm
  65  have h_jcost : Cost.Jcost m = cosh u - 1 := by
  66    rw [h_m_exp]
  67    exact Cost.Jcost_exp_cosh u
  68  rw [h_jcost]