Information Computation Limits Structure Computation Has Nonzero Energy Cost

The energy floor

In thermodynamics, Landauer's principle says that erasing one bit of information must dissipate at least k_B T ln(2) of energy, where k_B is the Boltzmann constant and T is the temperature of the system. This is not a statement about a particular computer; it is a physical floor on any information-processing device, because erasing a bit means reducing the number of possible states, which decreases entropy and therefore requires energy to be expelled as heat. The principle, proposed by Rolf Landauer in 1961, is one of the foundational links between information theory and thermodynamics.

Within the Recognition Science framework, this principle is formalized as a theorem in a machine-checked library of formal theorems. The declaration computation_has_nonzero_energy_cost states that for any real temperature T greater than zero, the product k_B * T * Real.log 2 is greater than zero. In plain language: at any temperature above absolute zero, the energy cost of erasing one bit is strictly positive. The theorem is proved from the positivity of the Boltzmann constant, the positivity of temperature, and the fact that the natural logarithm of 2 is positive. The library also proves that this energy cost scales linearly with temperature: a hotter system pays a higher cost per erased bit.

The theorem does not claim that all computation requires this energy, only erasure. Logical operations that are reversible, such as swapping two bits, can in principle be done without dissipating energy. The cost appears when information is discarded. The theorem also does not claim that the Landauer floor is the only limit on computation; the same library contains separate results about a minimum time tick and about the irrationality of the golden ratio, which together impose additional constraints on what can be computed and how fast.

The practical consequence is that computation has an unavoidable thermodynamic price. A machine that runs at room temperature pays a fixed tax per bit erased, and the tax rises with temperature. This is why cooling a computer is not just a convenience but a way to lower the fundamental cost of its operations. The framework's contribution is to make this physical principle part of a single formal system alongside its other results about the structure of computation.

/-- **THEOREM IC-002.10**: The Landauer bound is strictly greater than zero.
    No computation can be done for free (in thermodynamic equilibrium). -/
theorem computation_has_nonzero_energy_cost :
    ∀ T : ℝ, T > 0 → k_B * T * Real.log 2 > 0 :=
  landauer_energy_pos

/-- **THEOREM IC-002.9**: The Landauer energy grows linearly with temperature. -/
theorem landauer_scales_with_temp (T₁ T₂ : ℝ) (hT₁ : T₁ > 0) (hT₂ : T₂ > 0) (h : T₂ > T₁) :
    k_B * T₂ * Real.log 2 > k_B * T₁ * Real.log 2 := by
  unfold k_B
  have hlog : Real.log 2 > 0 := Real.log_pos (by norm_num)
  have hkB : (1.380649e-23 : ℝ) > 0 := by norm_num
  nlinarith

What this page does not claim

The theorem does not claim that all computation requires energy, only erasure. The theorem does not claim that the Landauer floor is the only limit on computation. The theorem does not establish the value of the Boltzmann constant; it uses the defined constant.

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

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