Thermodynamics Recognition Heat3 Engine From Jcost

A proposed engine

Thermodynamics studies how heat becomes work, and the classical limit on that conversion is the Carnot efficiency, which depends only on the temperatures of the hot and cold reservoirs. The recognition heat3 engine from jcost is a proposal within Recognition Science to attach that classical limit to the framework's cost function. The idea, recorded in a research note, is that the maximum work per recognition cycle equals the cost function evaluated at the golden ratio, multiplied by Boltzmann's constant and the hot reservoir temperature, giving an efficiency of about 7.3 percent.

The machine-checked library of formal theorems proves three general facts about the cost function J(x) = (x + 1/x)/2 - 1. It vanishes when its two inputs are equal, it is nonnegative for all positive inputs, and the golden ratio minus 1.5 is positive. These are true for any positive real numbers, because the cost function is defined as J(m/e) without any reference to a specific engine, a reservoir, or a cycle.

What the module does not establish is that this cost function describes an actual heat engine. The definition of the two quantities m and e in thermodynamic terms is absent. The certificate structure merely packages the three general facts into a single object, which is inhabited because the facts are proved. The engine itself remains a target, not a theorem.

In Recognition Science, the cost function is forced by five plain conditions, and its uniqueness is a proved theorem. The engine proposal tries to connect that abstract result to Carnot's efficiency, but the connection is a definitional choice, not a derivation. The plain language verdict is that the module proves three harmless facts about a cost function, and the engine is an idea awaiting a real definition of its inputs.

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]

What this page does not claim

The module proves the engine exists or that its efficiency is physically realized. The 7.3 percent figure is a measured or derived efficiency, rather than a research-note proposal. The cost function's uniqueness theorem is proved in this module, only its three general properties.

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/RecognitionHeat3_Engine_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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