Foundation Maxwell Demon2 Deep From Jcost

The demon's minimum cost

Maxwell's demon is a thought experiment where a tiny intelligent being sorts fast and slow molecules to make heat flow backwards. The question it raises is whether an intelligent agent can extract work without paying an energy cost for the information it uses. In the Recognition Science framework, the demon's essential act is not sorting but recognition: telling one molecule from another. The framework models that act as a discrete record of events, a ledger, and assigns it a forced cost.

The module MaxwellDemon2DeepFromJCost takes the framework's central cost function, J(x) = (x + 1/x)/2 - 1, and applies it to the demon's problem. The cost is defined as J(m/e), where m is a count of recognized events and e is a count of something the demon must compare against. The module proves three general facts about this cost. First, when m equals e, the cost is zero. Second, for any positive counts, the cost is never negative. Third, the number phi - 3/2, about 0.118, is positive. These are the only theorems in the module, and they hold for any positive numbers m and e, not for any specific physical system.

The module is honest about its own limits. Its domainCost is defined as J(m/e) without saying what m and e mean for a real demon. The docstring states that the module proves nothing specific to Maxwell's demon, because the definition does not reference one. The research note records where the idea was meant to go: that a demon cannot reduce J-cost below J(phi) per bit processed, and that demon information gain equals J(phi) times the number of bits erased. But that claim is not a theorem in the module. It is a hypothesis, a prediction that would need a definition of m and e in the demon's own terms to become a result.

What the module does establish is a reusable template. The same three facts, cost zero at equality, nonnegativity, and a positive threshold, are stated once in a shared template and instantiated here. The structure MaxwellDemon2DeepCert packages these three facts as a certificate, and the module proves such a certificate exists. This means any future work that defines m and e for a real demon can inherit these three properties automatically. The threshold phi - 3/2 is the gap between the golden ratio and 1.5, and its positivity is what would let a later theorem say that some costs are strictly positive, not just nonnegative.

In plain language, the module is a scaffold. It proves the arithmetic of the cost function works as needed, and it leaves the physics for a later definition. The reader should see that the framework has a specific candidate for the demon's cost, and that the candidate has the right basic properties. Whether that candidate is the true cost of a real demon remains open, waiting for a definition of m and e that the module itself does not supply.

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]

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

What this page does not claim

The module does not prove that a real demon cannot reduce cost below J(phi) per bit. The module does not define what m and e mean for an actual physical demon. The module does not establish that the demon's information gain equals J(phi) times bits erased.

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/Foundation/MaxwellDemon2DeepFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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