Information Landauer Bound Jcost Equals Thermodynamic

The thermodynamic link

Landauer's principle, proposed by Rolf Landauer in 1961, states that erasing one bit of information must dissipate at least k_BT ln(2) of energy as heat, where k_B is the Boltzmann constant and T is the temperature. This is not a statement about a particular computer; it is a fundamental floor set by the second law of thermodynamics. The information lost in erasing a bit increases entropy by k_B ln(2), and that entropy increase must be paid for in heat. Experiments have confirmed the bound: Bérut et al. (2012) gave the first verification, and later work pushed to within about ten times the limit.

Recognition Science (RS) is a framework that starts from a different premise: reality keeps a discrete record of recognition events, and the cost of each recognition is forced by a proved theorem. That cost function, J(x) = (x + 1/x)/2 - 1, is unique under five plain conditions. In the framework's machine-checked library of formal theorems, a declaration named jcost_equals_thermodynamic states that the J-cost framework reproduces thermodynamics. The declaration is a theorem in the library, but its proof is the trivial proof of the proposition True; the library does not yet contain a derivation that connects J to the Landauer bound in a formal chain.

What the framework does provide is a set of definitions and informal arguments that point toward the connection. The library defines erasureJCost as J(2), which equals (2 + 1/2)/2 - 1 = 1/4. It defines landauerEnergy(T) as k_B * T * ln(2), and it defines minimumErasurePower(T) as that energy divided by tau0_seconds, where tau0 is the framework's fundamental recognition timescale. The informal notes in the library say that erasing is recognizing then forgetting, and that this has a J-cost. The library also records theorems with trivial proofs stating that erasing at rate 1/tau0 requires at least that power, and that erasing ledger entries has minimum cost. These are statements of intent, not yet formal derivations.

The honest summary is plain: the framework has a unique cost function and a timescale, and it has defined the Landauer energy and power in terms of them. The declaration jcost_equals_thermodynamic is a placeholder in the formal library, a statement that the connection is intended, not a proof that it holds. The framework's claims about Landauer are at the level of a research program with definitions and informal arguments, not a completed theorem. What a reader can take away is that the framework has identified the quantities it would need to connect its cost function to thermodynamics, and has stated the connection as a target.

/-- The Landauer energy at room temperature.
    E = k_B × T × ln(2) ≈ 2.87 × 10⁻²¹ J ≈ 0.018 eV -/
noncomputable def landauerEnergy (T : ℝ) : ℝ := k_B * T * Real.log 2

/-- **THEOREM**: The J-cost of erasure equals the thermodynamic cost.
    J_erase ∝ ln(2) (the information content of 1 bit) -/
theorem jcost_equals_thermodynamic :
    -- The J-cost framework reproduces thermodynamics
    True := trivial

/-- **THEOREM**: The J-cost of erasure equals the thermodynamic cost.
    J_erase ∝ ln(2) (the information content of 1 bit) -/
theorem jcost_equals_thermodynamic :
    -- The J-cost framework reproduces thermodynamics
    True := trivial

/-- Erasing a bit has a J-cost.
    J_erase = cost of recognizing the current state + cost of resetting -/
noncomputable def erasureJCost : ℝ := (2 + 1/2)/2 - 1  -- Jcost(2) = 2 states → 1 state

What this page does not claim

The framework has formally derived the Landauer bound from its axioms. The jcost_equals_thermodynamic declaration contains a substantive proof of the connection. The framework's definitions of Landauer energy and power have been experimentally validated.

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)

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