Chemistry Molecular Orbital Gap From Jcost Homolumocert

What HOMOLUMOCert proves

In chemistry, the HOMO-LUMO gap is the energy difference between the highest occupied and lowest unoccupied molecular orbitals, a quantity that helps predict a molecule's color and reactivity. The Recognition Science framework contains a declaration called HOMOLUMOCert, and its name suggests it certifies something about this gap. The machine-checked library of formal theorems proves three general facts about a cost function, but the declaration itself contains no chemistry.

The cost function in question is J(x) = (x + 1/x)/2 - 1, which the framework derives from five plain conditions on recognition events. The library proves that J(r/r) = 0 for any nonzero r, that J(m/e) is nonnegative when m and e are positive, and that the number phi - 3/2 is positive, where phi is the golden ratio. These three facts form the certificate. They hold for any positive real numbers m and e, because the definition of the cost function does not refer to molecules, orbitals, or any chemical quantity.

The research note attached to the module records an intended application: the HOMO-LUMO gap might approximate ionization energy minus electron affinity, and the optimal gap might equal J(phi) times the ionization energy, about 0.118 times IP for a well-designed chromophore. That note is a research idea, not a proved result. The Lean code proves nothing about ionization energies, electron affinities, or chromophores, because the quantities m and e are never defined in chemical terms.

What the certificate does establish is a reusable template. The same three facts, vanishing at unity, nonnegativity, and a positive threshold, appear verbatim in 2383 sibling modules, each with a different subject name but the same universal content. The module becomes a theorem about its subject only when someone defines m and e in that subject's own terms. Until then, HOMOLUMOCert is a proof about numbers, not about molecules.

theorem cert_inhabited : Nonempty HOMOLUMOCert := ⟨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

HOMOLUMOCert proves nothing about actual molecules, ionization energies, electron affinities, or chromophores. The research note's approximation J(phi) times IP is not a proved result. The certificate does not establish that any real molecule has a HOMO-LUMO gap in the visible range.

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/MolecularOrbitalGapFromJCost.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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