Thermodynamics Carnot Efficiency Body Rs

The Carnot limit and a shared template

The Carnot efficiency is the upper bound on how much work a heat engine can extract from a temperature difference. For a reservoir at 310 kelvin (about body temperature) and a cooler sink at 293 kelvin (room temperature), the classical formula gives 1 minus 293/310, about 5.5 percent. That is the ceiling: no real engine can beat it, and most do far worse.

The module named thermodynamics carnot efficiency body rs in Recognition Science does not prove anything about that ceiling. Its ledger, a discrete record of recognition events, defines a generic cost function domainCost(m, e) = Jcost(m / e), where Jcost is the framework's forced cost function. The Lean file proves three general facts about this function: it is zero when the two inputs are equal, it is never negative for positive inputs, and the constant phi - 3/2 is positive. The docstring notes that if you plug in m as a metabolic rate and e as an energy supply, the cost at the golden ratio gives about 5.9 percent, close to the Carnot number and to measured human metabolic efficiency. But that paragraph is a research note, not a theorem.

What the module actually establishes is that this same cost template, shared verbatim across 2383 sibling modules, satisfies its three defining properties here. The structure CarnotBodyRS bundles those properties, and a certificate shows the bundle is inhabited. The template is stated once, universally, in DomainCostTemplate; this file instantiates it without giving m and e any thermodynamic meaning. The docstring is explicit: to make this a theorem about Carnot efficiency, one would need to define m and e in the subject's own terms.

In plain language, the module is a placeholder with a proof of its own generality. It shows the cost function behaves sensibly (zero at equality, nonnegative, with a positive threshold) for any positive inputs, but it does not connect those inputs to heat, work, or temperature. The Carnot comparison remains an idea for future work, not a derived result. Readers should treat the 5.9 percent near-match as a suggestive coincidence, not a framework theorem.

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

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)

structure CarnotBodyRS 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 module derives the Carnot efficiency from Recognition Science principles. The 5.9 percent near-match is a proved result. The cost function has any thermodynamic interpretation in this module.

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

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Derived articles

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