Physics Landauer Principle From Jcost

Landauer's principle from J-cost

Landauer's principle, proposed by Rolf Landauer in 1961, states that erasing one bit of information necessarily dissipates at least k_B T ln(2) of heat, where k_B is Boltzmann's constant and T is the temperature. This is not about the physical energy of the bit itself, but about the thermodynamic cost of resetting a memory to a known state. The principle connects information theory to thermodynamics, implying that information is physical and has a minimum energy price.

In the Recognition Science framework, the cost of a recognition event is forced by a proved theorem to equal J(x) = (x + 1/x)/2 - 1. The framework models the cost of erasing one bit as J(φ) × k_B T, where φ is the golden ratio. This gives approximately 0.118 k_B T, compared to Landauer's ln(2) ≈ 0.693 k_B T. Both are subliminal, meaning they are smaller than the thermal energy scale k_B T itself.

The module LandauerPrincipleFromJCost in the machine-checked library of formal theorems defines a domain cost as J(m/e), and proves three general facts: the cost vanishes when m equals e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are universal properties of the J function, not specific to Landauer's principle, because the module defines the cost without linking m and e to physical quantities like energy or temperature.

The module itself does not establish the Landauer connection. Its docstring notes that the idea was meant to go further, but the formal content proves only general facts about J. What would make it a theorem about Landauer's principle is a definition of m and e in physical terms, such as energy and temperature. That step remains open.

What the module does establish is a template: any system whose cost is J(m/e) inherits these three properties. The framework's contribution is the specific value J(φ) ≈ 0.118, which is a derived constant, not a free parameter. The comparison to Landauer's ln(2) is an empirical observation, not a proved equality.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove that erasing a bit costs J(φ) × k_B T; it only defines a cost function with general properties. The framework does not derive the value ln(2) from first principles; the comparison to Landauer's constant is empirical. No claim is made that the framework's cost applies to physical memory devices without further assumptions.

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

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