Chemistry Organocatalysis3 From Jcost

Organocatalysis and the cost threshold

Organocatalysis is a branch of chemistry in which small organic molecules, rather than metals or enzymes, accelerate chemical reactions. A common rule of thumb for predicting whether a molecule will act as a reactive electrophilic catalyst involves the energy gap between its lowest unoccupied molecular orbital (LUMO) and its highest occupied molecular orbital (HOMO). The smaller this gap, the more readily the molecule accepts electrons and activates a substrate. In a research note attached to a formal module, the idea was to set a threshold: a catalyst is predicted to be active when its LUMO energy falls below a value derived from the golden ratio times its HOMO energy, corresponding to a LUMO-HOMO gap below about 4 electron volts.

The module itself, named Organocatalysis3_FromJCost, does not prove that chemical rule. It defines a cost function cost, a measure of recognition effort, as J(m/e), where J is a fixed function and m and e are real numbers. The machine-checked library of formal theorems proves three general facts about this cost: it equals zero when m equals e, it is never negative for positive inputs, and a certain threshold constant phi minus 3/2 is positive. These facts hold for any positive numbers m and e, because the cost function is defined without any reference to orbitals, electrons, or catalysts. The chemical content exists only in a research note, not in the formal statements.

In Recognition Science, the framework models recognition events as discrete records in a ledger, and the cost of recognition is forced by a proved theorem to take the form J(x) = (x + 1/x)/2 - 1. The module applies this general cost to a ratio m/e, but it leaves m and e undefined in chemical terms. To turn the module into a theorem about organocatalysis, one would need to define m as the LUMO energy and e as the HOMO energy, or some similar identification, and then prove that the cost threshold corresponds to the empirical 4 eV gap. That step is absent. The module is a template, shared verbatim with 2383 sibling modules, each one awaiting a subject-specific definition.

What the module does establish, in plain language, is that the cost function has three basic properties: it vanishes when its two inputs are equal, it stays nonnegative for positive inputs, and the golden-ratio-derived threshold is a positive number. These are general facts about the cost function, not about chemistry. The chemical prediction remains a research note, an open target for a future definition that connects m and e to molecular orbital energies.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module proves the organocatalyst LUMO threshold rule. The module identifies m and e with molecular orbital energies. The empirical 4 eV gap is a measured result in this module.

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)

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Derived articles

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