Chemistry Catalysis Activation Energy2 Volcano Plot Cert

What the certificate proves

A volcano plot in catalysis shows how reaction rate rises then falls as the catalyst's binding strength changes; the peak sits at an intermediate value, a pattern known as the Sabatier principle. The Recognition Science framework models that peak as the minimum of a cost, a number measuring how far a system is from a perfect match. Its machine-checked library of formal theorems contains a declaration called VolcanoPlotCert, which bundles three proved facts about that cost when it is written as J(m/e), where m and e are two positive real numbers.

First, the certificate proves that the cost is zero when the two inputs are equal: J(r/r) = 0 for any nonzero r. Second, it proves the cost is never negative for positive inputs. Third, it proves that a certain threshold, phi - 3/2, is positive. These three facts are general properties of the cost function; they hold for any positive numbers m and e, not just for catalysis.

The certificate does not claim that m and e stand for any specific physical quantities. In the framework's source, the cost is defined simply as J(m/e) with no reference to adsorption energy, reaction rate, or any catalyst. The declaration's own documentation says it proves nothing specific to this subject. What would make it a theorem about catalysis is a definition of m and e in that subject's own terms, for example binding energy and an optimal reference value. That step remains open.

So the certificate is a small, clean piece of mathematics: it records that the cost function has a zero at equality, is nonnegative, and has a positive threshold. It is a building block, not a finished theory of catalysis. The Sabatier principle and the volcano shape are the motivation, but the formal result stops short of connecting them to real chemistry.

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 certificate does not prove that any real catalyst follows the volcano plot. The certificate does not define m or e in terms of adsorption energy or any other chemical quantity. The certificate does not derive the Sabatier principle from first principles.

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/CatalysisActivationEnergy2.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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