Chemistry Catalysis Electrochem From Jcost Electrochem Over Cert

The certificate

In electrochemistry, overpotential is the extra voltage needed to drive a reaction beyond its ideal thermodynamic value. Real water-splitting catalysts operate with an overpotential of roughly 0.3 to 0.5 volts, while the best materials reach about 0.2 to 0.4 volts. The Recognition Science framework defines a cost function J(x) = (x + 1/x)/2 - 1, which measures the price of a mismatch between two quantities. The declaration ElectrochemOverCert is a structure, a formal container, that bundles three proved facts about this cost function when it is applied to a ratio of two numbers.

The three facts are general, not electrochemical. First, the cost vanishes when the two inputs are equal: for any nonzero r, J(r/r) = 0. Second, the cost is never negative when both inputs are positive. Third, the constant phi - 3/2 is positive, where phi is the golden ratio. These are properties of the cost function itself, and the machine-checked library proves them with no subject-specific assumptions. The certificate simply collects them into one object and proves that such an object exists.

The research note records an intended application: an overpotential of roughly 0.118 volts at ideal efficiency, computed as J(phi) times one volt. But the note is explicit that this is a plan, not a result. The definition of domainCost as J(m/e) never states what m and e mean in electrochemical terms. Without a definition of m and e tied to actual physical quantities, the certificate proves nothing about real catalysts. The shared template text appears verbatim in 2383 sibling modules, each with a different subject name but the same three general facts.

What the declaration does establish is a template: a reusable pattern for what a subject-specific theorem would need. To turn this into a genuine electrochemical statement, someone must define m and e as specific physical quantities, such as an activation energy and a thermal energy, and then prove the cost function's properties hold for those definitions. Until then, the certificate is a formal shell waiting for content, not a result about catalysis.

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 ElectrochemOverCert := ⟨cert⟩

What this page does not claim

ElectrochemOverCert does not prove any statement about real electrochemical overpotential. The declaration does not define m and e in physical terms. The 0.118 volt value is not a theorem; it is a research note recording an intended application.

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/CatalysisElectrochem_FromJCost.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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