Physics Maxwell Demon From Jcost

The demon's ledger

Maxwell's demon is a thought experiment from 1867. James Clerk Maxwell imagined a tiny being guarding a door between two chambers of gas. By opening the door for fast molecules and closing it for slow ones, the demon could make one side hot and the other cold, apparently without doing work. This seemed to violate the second law of thermodynamics, which says entropy, a measure of disorder, tends to increase. Later physicists resolved the paradox by noticing that the demon must store information about each molecule's speed, and erasing that stored information costs energy. The standard limit, named after Rolf Landauer, is k_B T ln(2), about 0.693 k_B T, where k_B is Boltzmann's constant and T is temperature.

In Recognition Science, the framework models the demon's memory as a ledger, a discrete record of events. The framework's central cost function, written J(x) = (x + 1/x)/2 - 1, measures the price of recognition, the act of matching a record to a state. The framework proves that this cost is zero when the record matches the state exactly, and that it is never negative for positive inputs. The module MaxwellDemon_FromJCost applies this cost to the demon's bookkeeping. Its domainCost is defined as J(m/e), where m is the memory state and e is the event state. The module proves three facts: the cost vanishes when m equals e, the cost is nonnegative, and the quantity phi - 3/2 is positive, where phi is the golden ratio.

The third fact is the one with physical weight. The golden ratio phi is approximately 1.618, so phi - 3/2 is about 0.118. In the framework's units, this number multiplies k_B T to give the cost of erasing one bit of demon memory. That gives roughly 0.118 k_B T, which is smaller than Landauer's 0.693 k_B T. The framework's account suggests the demon can operate below the Landauer limit because its recognition cost function is more efficient than a naive binary switch. This is a research note, not a proved theorem about physics. The framework proves general facts about the cost function, but it does not define m and e in terms of actual gas molecules or demon operations, so the physical claim about sub-Landauer erasure remains a hypothesis.

What the framework does establish, in plain language, is a clean mathematical skeleton. The cost of recognition is zero when the record matches the event. The cost is never negative. And the golden ratio appears as a natural threshold in the cost function. These are proved facts about the framework's own definitions. The step from that skeleton to a physical demon in a gas chamber is not yet made. The framework's own documentation states that the physical content is a research note recording where the idea was meant to go, not a result. The framework's contribution here is a precise vocabulary for talking about the cost of information, not a new measurement or a new law of thermodynamics.

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

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]

structure MaxwellDemonCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The framework does not prove that a physical Maxwell demon can operate below the Landauer limit. The framework does not define m and e in terms of gas molecules or demon operations. The framework does not claim to have measured any thermodynamic quantity.

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

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