Information Thermodynamics Recognition Cost

The recognition cost ledger

In thermodynamics, erasing information costs energy. The Landauer principle, named after Rolf Landauer in 1961, states that erasing one bit of information dissipates at least kT ln 2 of heat, where k is Boltzmann's constant and T is the temperature. Recognition Science builds a formal model of this idea using a ledger, a discrete record of events, where each event is a bond between two states with a multiplier representing the strength of the mismatch.

The declaration RecognitionCost defines the total cost of a ledger state as the sum, over all active bonds, of the function J(x) = (x + 1/x)/2 - 1 applied to each bond multiplier. This J-cost is the central object of the framework: it is the unique function satisfying five natural conditions including reciprocal symmetry and a composition law, as proved in the machine-checked library of formal theorems. The definition itself is a model, a choice of how to measure cost, not a theorem about nature.

What the framework proves about this definition is a connection to entropy. The theorem total_dissipation_bound shows that the recognition cost of any state is at least half the sum of the squares of the logarithms of the bond multipliers. This quadratic lower bound means that the cost grows faster than linearly with the mismatch, a property that anchors the theory in the Landauer limit. The theorem eight_tick_dissipation_limit extends this to show that any admissible evolution operator that minimizes J-cost over a full cycle never increases the total cost, matching the expected dissipation of erasure.

The framework does not claim that this model is the only way to define thermodynamic entropy, nor that the J-cost function is the unique measure of information dissipation. The bound is a lower bound, not an equality; it says dissipation must be at least this large, not that it will be exactly this large. The definitions of admissible states and RecognitionOperator are choices made by the framework, not forced consequences of physics. The connection to real thermodynamic systems remains an interpretation, not a measured result.

/-- Total recognition cost over active bonds. -/
noncomputable def RecognitionCost (s : LedgerState) : ℝ :=
  s.active_bonds.sum (fun b => Cost.Jcost (s.bond_multipliers b))

/-- **Entropy Dissipation Theorem**
    The total recognition cost of a state is bounded below by the quadratic
    sum of the information mismatches. -/
theorem total_dissipation_bound (s : LedgerState) :
    RecognitionCost s ≥ (1/2 : ℝ) * (s.active_bonds.sum (fun b => (Real.log (s.bond_multipliers b))^2)) := by
  unfold RecognitionCost
  rw [Finset.mul_sum]
  apply Finset.sum_le_sum
  intro b hb
  have h := landauer_bound_holds s b hb
  dsimp at h
  linarith

/-- **8-Tick Dissipation Limit**
    The total dissipation over one 8-tick cycle corresponds to the Landauer limit
    for pattern closure. -/
theorem eight_tick_dissipation_limit (R : RecognitionOperator) (s : LedgerState) :
    let s_next := R.evolve s
    -- Over one complete cycle, the integrated cost balances the erasure
    admissible s → admissible s_next → RecognitionCost s_next ≤ RecognitionCost s := by
  intro s_next hadm_s _
  exact R.minimizes_J s hadm_s

What this page does not claim

RecognitionCost is not a physical measurement of entropy in any real system. The quadratic lower bound is not an equality; actual dissipation may be larger. The definitions of admissible states and evolution operators are not forced by the Landauer principle.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/Thermodynamics.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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