Chemistry Solvent Extraction From Jcost Solvent Extraction Cert

The solvent extraction certificate

Liquid-liquid extraction separates a dissolved substance between two immiscible liquids, usually an aqueous phase and an organic phase. The distribution ratio D is the concentration of the analyte in the organic phase divided by its concentration in the aqueous phase. A chemist tunes the system so that D is large enough to pull most of the analyte out, but not so large that the extraction becomes hard to reverse. In the Recognition Science framework, the cost of recognition is a specific function J(x) = (x + 1/x)/2 - 1, and the framework models the distribution ratio as the argument of that cost.

The declaration SolventExtractionCert is a bundle of three machine-checked facts about the cost function applied to a ratio m/e, where m and e are real numbers. First, when m equals e, the cost is exactly zero. Second, for positive m and e, the cost is never negative. Third, the golden-ratio constant phi minus 3/2 is positive, which is a small numerical threshold. These three facts are proved in the machine-checked library of formal theorems, and the certificate simply packages them into one object. The certificate is inhabited, meaning such a bundle exists.

What the certificate does not do is say anything specific about chemistry. The cost is defined as Jcost (m / e) without any reference to an analyte, a solvent, or a real extraction process. The docstring records a research note that at D = phi, the cost is about 0.118, suggesting 11.8% remaining in the aqueous phase and 88.2% extracted, but that note is not a theorem. To turn the certificate into a chemistry result, one would need a definition of m and e in terms of real concentrations and a proof that the cost model applies. That step is open.

In plain terms, the certificate proves three general properties of a cost function, and it packages them under a chemistry-sounding name. The name is a promise, not a result. The honest summary is that the certificate is a small, correct piece of mathematics waiting for a chemical interpretation.

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)

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

What this page does not claim

The certificate does not prove that D = phi is optimal for any real solvent system. The certificate does not establish that the cost function J models any actual extraction process. The certificate does not claim that the 88.2% extraction figure is a measured or derived result.

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/SolventExtractionFromJCost.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

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