Chemistry Phase Transition Co2 From Jcost Co2 Triple Pt Cert

The CO2 certificate

Carbon dioxide has a well-known triple point, the single temperature and pressure where solid, liquid, and gas can coexist: about -56.6°C and 5.11 atmospheres. A machine-checked library of formal theorems contains a declaration named CO2TriplePtCert, a certificate, a packaged bundle of three proved statements. In plain terms, the certificate proves that a certain cost function, written J(m/e), is zero when its two inputs are equal, that it never goes below zero for positive inputs, and that a particular constant built from the golden ratio is positive.

The cost function here is the Recognition Science cost, a measure of how expensive it is to recognize one quantity in terms of another; it equals (x + 1/x)/2 - 1. The certificate's three facts are general properties of this cost function, true for any positive real numbers m and e. They hold regardless of what m and e mean. The library proves them once, for all subjects at once, in a shared template module, and the CO2 module reuses that proof.

What the certificate does not do is connect those general facts to carbon dioxide. The module defines its cost as J(m/e) with no definition of m and e in chemistry's own terms. The docstring records a research note, not a result: a proposed ratio of CO2's triple point temperature to its critical temperature, about 0.712, compared to a framework-derived number near 0.941. The note itself says the match is not exact and calls the numbers a structural placeholder. No theorem in the module states anything about -56.6°C, 5.11 atm, or any phase boundary.

In Recognition Science, a certificate like this one is a building block waiting for a subject. The three proved facts are real and machine-checked, but they become a statement about CO2 only when someone defines m and e using CO2's own measured quantities. Until then, the certificate proves mathematics about a cost function, and the chemistry remains an aspiration recorded in a comment.

theorem cert_inhabited : Nonempty CO2TriplePtCert := ⟨cert⟩

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 certificate does not prove any fact about carbon dioxide's triple point, its phase diagram, or any measured chemical property. The research note's numerical comparison is not exact and is explicitly labeled a structural placeholder, not a derived result. The certificate does not derive the value -56.6°C or 5.11 atm from the cost function.

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.

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