Chemistry Catalysis Chiral From Jcost Chiral Cat Cert

The certificate's scope

In asymmetric catalysis, a reaction that should produce equal amounts of two mirror-image products instead favors one. The favored product's share is the enantiomeric excess, or ee, often reported as a percentage. A good asymmetric catalyst pushes ee above 90 percent, and excellent ones exceed 98 percent. The Recognition Science framework models a related quantity, a cost, which is a number that measures how far a ratio of two positive real numbers sits from unity. The framework's machine-checked library of formal theorems contains a declaration named ChiralCatCert that packages three proved facts about this cost function.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 - 1, evaluated at the ratio m/e. First, when the two inputs are equal, the cost is exactly zero. Second, for any two positive inputs, the cost is never negative. Third, a certain constant, phi minus 3/2, is positive, where phi is the golden ratio. Each fact is proved in the library's source file, and the certificate simply bundles them into one structure. The certificate itself is inhabited, meaning a proof that such a bundle exists.

What the certificate does not do is say anything about chemistry. The definition of the cost function uses the symbols m and e, but nothing in the formal text connects those symbols to a mass, a substrate, or an enantiomer. The docstring attached to the source file is explicit: the formal proofs are universal facts about Jcost, and the chemistry is a research note recording where the idea was meant to go, not a result. The numerical suggestion in that note, that ee equals 1 minus 2 times J(phi) squared, about 97.2 percent, is a hypothesis, not a theorem. No measured catalyst is named, and no experimental comparison is made.

In plain terms, the certificate establishes that a particular cost function has three elementary properties: it vanishes at equality, it stays nonnegative, and a golden-ratio threshold is positive. Those properties hold for any positive real inputs. The step from those properties to a statement about chiral catalysis would require a definition of m and e in chemical terms, and that definition is absent. The certificate is a small, correct piece of mathematics waiting for a bridge to the subject its name suggests.

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]

theorem cert_inhabited : Nonempty ChiralCatCert := ⟨cert⟩

What this page does not claim

The certificate does not prove any statement about a real chemical reaction or catalyst. The numerical ee value of about 97.2 percent is not a theorem and is not compared to any measured data. The symbols m and e in the formal definition are not defined as mass and enantiomer in the machine-checked text.

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/CatalysisChiralFromJCost.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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