Chemistry Phase Transition Co2 From Jcost

What the CO2 module proves

The module cost, a measure of recognition effort, appears in a file named for the CO2 triple point. The file defines a domain cost as J(m/e), the framework's cost function applied to a ratio of two real numbers. Its three theorems are general: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and the golden-ratio constant phi exceeds 1.5. Each proof is machine-checked in the framework's library of formal theorems.

None of these facts mention carbon dioxide. The file's own docstring says so plainly: the proofs establish nothing specific to this subject, because the domain cost is defined without reference to CO2. The triple-point numbers in the header, -56.6 degrees Celsius and 5.11 atmospheres, are a research note, not a result. The ratio of triple-point temperature to critical temperature, about 0.712, does not match the framework's suggested 0.941, and the note calls the match a structural placeholder.

What the module does establish is a template. The same three general facts appear verbatim in 2383 sibling modules, stated once in a shared file. To turn this file into a theorem about CO2, a definition of the two numbers in the ratio would need to come from CO2's own physics, such as its intermolecular forces or phase boundaries. That definition is absent here.

So the plain-language takeaway is a caution: a file name can promise more than its content delivers. The machine-checked part is real but generic. The CO2-specific claim remains open.

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)

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

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

What this page does not claim

This module proves any fact about CO2's phase diagram. The triple-point ratio 0.712 is a derived framework result. The framework's cost function has been applied to CO2 in any validated way.

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/Chemistry/Phase_Transition_CO2From_JCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: