Chemistry Photocatalysis From Jcost Photocat Qycert

A certificate with a narrow scope

Photocatalysis is the acceleration of a chemical reaction by light, and quantum yield is the fraction of absorbed photons that produce a desired chemical change. For titanium dioxide (TiO2) under ultraviolet light, a common benchmark value is about 11.8 percent. The Recognition Science declaration PhotocatQYCert is a machine-checked certificate, a formal object that bundles three proved facts about a cost function, not about chemistry itself.

The certificate proves three general facts. First, the cost function Jcost, defined as J(x) = (x + 1/x)/2 - 1, vanishes when its two inputs are equal and nonzero. Second, it is nonnegative for all positive inputs. Third, the golden-ratio constant phi minus 1.5 is positive. These are properties of the cost function alone, true for any positive real numbers, with no reference to photons, catalysts, or chemical reactions.

The framework's library defines a domain cost as Jcost applied to the ratio m/e, where m and e are intended to stand for some measured and expected quantities. The certificate proves facts about this ratio, but the meaning of m and e in chemistry is never defined in the formal code. The research note records the intended direction: that TiO2 quantum yield might equal J(phi), approximately 0.118, matching the 11.8 percent benchmark. That identification is a research aspiration, not a proved theorem.

What the certificate does not claim is the central point. It does not establish that TiO2 quantum yield equals 11.8 percent, nor that the cost function applies to photocatalysis at all. The formal proof would become a theorem about photocatalysis only if someone defined m and e in chemical terms, such as measured reaction rate and expected photon count, and then proved the connection. Until that definition exists, the certificate remains a general mathematical object with a chemical name attached.

theorem cert_inhabited : Nonempty PhotocatQYCert := ⟨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]

What this page does not claim

The certificate does not prove that TiO2 quantum yield equals 11.8 percent. The certificate does not establish that the cost function applies to photocatalysis at all. The research note's numerical identification is a hypothesis, not a theorem.

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/PhotocatalysisFromJCost.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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