Chemistry Photocatalysis Efficiency2 From Jcost Photocat Eff2 Cert

What the certificate proves

Photocatalysis is the process where light drives a chemical reaction, such as splitting water into hydrogen and oxygen. A good solar-to-hydrogen catalyst might convert 1 to 10 percent of incoming sunlight into fuel. The declaration PhotocatEff2Cert is a machine-checked certificate inside the Recognition Science framework, a formal library of theorems. The certificate proves three facts about a cost function, not about any real catalyst.

The cost function, written domainCost, is defined as J(m/e), where J is a fixed mathematical function and m and e are two positive numbers. The three proved facts are: when m equals e, the cost is zero; for positive m and e, the cost is never negative; and a certain threshold, phi minus 3/2, is positive. Here phi is the golden ratio, about 1.618, so the threshold is about 0.118. These are general properties of the cost function, proved for any positive inputs.

The certificate bundles these three facts into a single structure named PhotocatEff2Cert, and proves that this structure exists. The proofs are machine-checked, meaning a computer verified every step. The certificate is a formal object: it records that these three statements are true in the framework's logic.

What the certificate does not claim is any connection to actual photocatalysis. The definition of domainCost uses m and e as abstract numbers, not as mass and energy of a real system. The research note attached to the module mentions solar-to-hydrogen efficiency, but that note is not a theorem. To make this a theorem about photocatalysis, one would need to define m and e in terms of real physical quantities, such as the mass of hydrogen produced and the energy of incident light. That step is not present.

In plain terms: the certificate proves a small piece of mathematics about a cost function, and it leaves open the entire question of whether that mathematics describes any real photocatalytic process.

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]

structure PhotocatEff2Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The certificate does not prove any efficiency value for any real photocatalyst. The certificate does not define m and e in terms of physical quantities. The certificate does not derive the golden ratio from photocatalysis.

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/PhotocatalysisEfficiency2FromJCost.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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