Information Thermodynamics Landauer Bound Holds

The Landauer bound

The Landauer bound is a principle from information theory and thermodynamics. It states that erasing one bit of information must dissipate at least a minimum amount of heat. The standard value is kT ln 2, where k is Boltzmann's constant and T is the temperature. Rolf Landauer proposed this limit in 1961 at IBM, arguing that logical irreversibility, the fact that erased information cannot be recovered, has a physical cost. The bound is not about the speed of a computer or the efficiency of a particular design; it is a fundamental floor on energy loss that any physical system must pay when it discards information.

In Recognition Science, the framework models physical structure through a ledger, a discrete record of events and their costs. The relevant declaration, landauer_bound_holds, proves a specific version of this principle. It shows that for any state of a local information ledger, the recognition cost, defined as the sum of a particular cost function over active bonds, is bounded below by a quadratic function of the information mismatches. In plainer terms, the theorem establishes that the cost of reducing mismatch in the ledger cannot go below a floor set by the square of the logarithmic imbalance. This is a formal, machine-checked proof, meaning the logic has been verified by a computer, not just argued informally.

The framework's library also derives a related total dissipation bound. It states that the total recognition cost of a state is at least half the sum of the squares of the logarithms of the bond multipliers. This is a stronger, aggregate version of the same idea: the more imbalanced the ledger, the higher the minimum cost. The declarations define the ledger state, the recognition cost, and the entropy proxy, and then prove the bounds as theorems. The eight-tick dissipation limit, another theorem, shows that over one complete cycle of a recognition operator, the cost does not increase. This is consistent with the idea of a dissipative process that tends toward lower cost states.

What the declaration does not claim is just as important as what it proves. It does not claim that the Landauer bound is derived from first principles in the framework; rather, it formalizes the bound within the framework's own definitions. The theorem does not prove that physical systems actually obey this bound, nor does it connect the framework's entropy proxy to thermodynamic entropy in a way that has been empirically verified. The declarations are mathematical results about a defined structure, not experimental measurements. The framework treats conventional physics as itself derived, but this particular theorem is a formalization, not an independent derivation of thermodynamics.

The practical consequence is that the framework can now reason about information erasure in its own terms. The theorem gives a precise, checkable statement of a known physical limit, expressed in the language of the ledger. This allows the framework to build further results on a solid foundation, knowing that the cost of erasure is not arbitrary but has a provable lower bound. For a reader, the takeaway is that the Landauer bound is not just a physical principle but also a formal theorem within this framework, with all the rigor that machine-checked proof provides.

/-- **THEOREM: Landauer Bound for Recognition**
    The recognition cost required to reduce mismatch must satisfy the Landauer bound.
    Specifically, the sum of J-costs across the ledger provides a quadratic lower
    bound on the information dissipation. -/
theorem landauer_bound_holds (s : LedgerState) :
    ∀ b ∈ s.active_bonds,
      let m := s.bond_multipliers b
      let u := Real.log m
      Cost.Jcost m ≥ u^2 / 2 := by
  intro b hb m u
  have hm : 0 < m := s.bond_pos b hb
  -- Jcost m = cosh (log m) - 1
  have h_m_exp : m = exp u := (exp_log hm).symm
  have h_jcost : Cost.Jcost m = cosh u - 1 := by
    rw [h_m_exp]
    exact Cost.Jcost_exp_cosh u
  rw [h_jcost]
  have h_lb := Cost.cosh_quadratic_lower_bound u
  linarith

/-- **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

The theorem does not prove that physical systems obey the Landauer bound. The framework's entropy proxy is not shown to be equivalent to thermodynamic entropy. The bound is a formalization within the framework, not an independent derivation of thermodynamics.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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