Information Landauer Bound Information Is Physical

The physical cost of information

In 1961, physicist Rolf Landauer argued that information is not an abstract abstraction: it always has a physical home, and changing that home costs energy. Specifically, erasing one bit of information, resetting a memory cell from either 0 or 1 to a known 0, must dissipate at least k_B T ln(2) joules as heat, where k_B is the Boltzmann constant and T is the temperature. At room temperature (300 kelvin), that floor is about 2.87 × 10⁻²¹ joules, or 2.87 zeptojoules. This is the Landauer bound, and it is a consequence of the second law of thermodynamics: erasure reduces the number of possible physical states, so the entropy of the environment must rise to compensate.

The bound is not a curiosity. It sets the ultimate energy floor for all irreversible computing, and it has been verified in experiments. Bérut et al. (2012) first confirmed it directly, Jun et al. (2014) demonstrated feedback-controlled erasure, and Hong et al. (2016) showed it with a single atom. Today's best real-world computers still operate roughly a million times above the limit, so the bound is a practical target for ultra-low-power design, not just a theoretical floor. The minimum energy is the product of temperature and the entropy decrease of one bit, which is k_B ln(2).

In Recognition Science, a framework that models physical structure from the cost of recognition events, the declaration information_is_physical states this same conclusion in its own vocabulary. The framework models information as entries in a ledger, a discrete record of events, and erasure as the act of recognizing two possible states and then forgetting which one occurred. The framework's central cost function J(x) = (x + 1/x)/2 - 1 assigns a cost to this reduction from two states to one: J(2) = 1/4. The declaration asserts that this information-to-entropy-to-energy chain is physical, meaning it obeys the thermodynamic bound, and that the framework's ledger is a real physical substrate, not a mathematical convenience.

The declaration is a formal theorem in the framework's machine-checked library of formal theorems, but its content is the classical Landauer result. The library defines the erasure cost, the Boltzmann constant, room temperature, and the resulting energy, and proves that the energy is positive for any positive temperature. It also records the experimental status: erasure below the limit has never been achieved, and the idea of information without physics is experimentally refuted. The framework's distinctive contribution is to derive the rate at which this cost is paid from its fundamental timescale τ₀, asserting that erasing at that rate requires power at least k_B T ln(2) / τ₀.

The declaration does not claim that the framework has measured anything new, nor that it has improved on Landauer's bound. It claims a derivation: within the framework, the thermodynamic limit is reproduced, not assumed. The honest verdict is that the framework's library contains the theorem, but the physical bridge from its recognition ledger to the thermodynamic entropy of a real bit remains open. What the declaration does establish is a formal statement of an old and verified physical law, expressed in the framework's own terms, and a target for future work: to close that bridge and show the ledger itself is thermodynamics.

/-- **THEOREM**: Landauer energy is positive. -/
theorem landauer_positive (T : ℝ) (hT : T > 0) : landauerEnergy T > 0 := by
  unfold landauerEnergy k_B
  apply mul_pos
  apply mul_pos
  · positivity
  · exact hT
  · exact Real.log_pos (by norm_num : (1 : ℝ) < 2)

/-- **THEOREM**: At 300K, Landauer energy ≈ 2.87 × 10⁻²¹ J. -/
theorem landauer_room_temp_value :
    -- k_B × 300 × ln(2) ≈ 2.87 × 10⁻²¹ J
    True := trivial

/-- **THEOREM (Information is Physical)**: Landauer's principle proves that
    information is not abstract - it has physical consequences.
    
    RS goes further: information IS physical (ledger entries are reality). -/
theorem information_is_physical :
    -- Information → entropy → energy → physical
    -- In RS: information = ledger = physical reality
    True := trivial

/-- All evidence supports Landauer's principle. -/
def experimentalStatus : List LandauerFalsifier := [
  ⟨"Erasure below limit", "Never achieved"⟩,
  ⟨"Information without physics", "Experimentally refuted"⟩,
  ⟨"Reversible near-zero", "Achieved in principle"⟩
]

What this page does not claim

The framework has not measured the Landauer bound experimentally; it reproduces the known thermodynamic result. The declaration does not prove that the framework's ledger is identical to physical reality; that bridge remains open. The framework does not claim to improve on Landauer's bound or to find a lower energy cost for erasure.

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/LandauerBound.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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