Chemistry Organocatalysis3 From Jcost Organocatalysis3 Cert

A certificate with three facts

In the Recognition Science framework, a ledger (a discrete record of events) carries a forced cost of recognition. The declaration Organocatalysis3Cert is a machine-checked certificate: a bundled package of three proved facts about a cost function called domainCost, defined as J(m/e) for two positive real numbers m and e. The first fact is that the cost is zero when m equals e. The second is that the cost is never negative for positive inputs. The third is that the golden-ratio expression phi minus 3/2 is positive. Each fact is proved in the machine-checked library of formal theorems, and the certificate simply collects them into one object.

The certificate's own documentation is explicit about its limits. The three facts are general properties of the cost function; they do not reference organocatalysis, LUMO, HOMO, or any chemical quantity. The module defines domainCost as J(m/e) without connecting m and e to any chemical meaning. A research note above the code records an intended application: that reactive electrophilic catalysts have a LUMO-HOMO gap below about 4 eV, which is roughly J(phi) times 34 eV. But that note is a plan, not a result. The certificate proves nothing about that threshold.

What would make this declaration a theorem about organocatalysis is a definition of m and e in the subject's own terms, for example as molecular orbital energies. Without such a definition, the certificate is a shell: it proves three true statements about a generic cost function and stops there. The same body of text is shared verbatim across 2383 sibling modules, each with a different subject name in its title but the same three facts. The universally quantified version lives in a single template module, and this certificate is one instance of it.

The practical consequence is a clean separation. A reader can trust the three proved facts without trusting any chemical claim, because the certificate makes none. The research note is honest about being a note. The framework's library proves what it proves, and the chemical bridge, defining m and e as orbital energies, is not built here. That bridge is a target for future work, not a result of this declaration.

structure Organocatalysis3Cert 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

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 Organocatalysis3Cert 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 that any organocatalyst's LUMO-HOMO gap is below 4 eV. The certificate does not define m or e in chemical terms. The research note about the LUMO threshold is a plan, not a proved result.

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/Organocatalysis3_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.

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