Chemistry Molecular Orbitals4 From Jcost Mol Orbitals4 Cert

What the certificate proves

Molecular orbital theory describes how electrons in a molecule occupy discrete energy levels, with the HOMO-LUMO gap being the energy difference between the highest occupied and lowest unoccupied molecular orbitals. This gap determines a molecule's color, conductivity, and chemical reactivity. For small molecules, measured gaps vary widely: hydrogen (H2) has a gap of 15.5 eV, nitrogen (N2) 10.8 eV, and oxygen (O2) 5.1 eV, giving a ratio between H2 and O2 of about 3.04, which is close to phi raised to the 2.7 power, where phi is the golden ratio.

The Recognition Science framework models this through a cost function, a mathematical measure of how far a ratio departs from unity. The machine-checked library of formal theorems defines domainCost as the cost of a ratio m over e, and then proves three general facts about it. First, the cost is zero when m equals e, meaning no cost when the two quantities match. Second, the cost is never negative for positive inputs, so it cannot go below zero. Third, a threshold value, phi minus 1.5, is positive, meaning the golden ratio exceeds 1.5.

These three facts are packaged into a structure called MolOrbitals4Cert, and the library proves this structure is inhabited, meaning a valid certificate exists. The certificate itself is a formal object: it bundles the three proofs together. The library does not define what m and e mean for any specific molecule, so the certificate says nothing about actual HOMO-LUMO gaps. It proves general properties of the cost function, not chemistry.

The research note attached to the module records the intended direction: that molecular HOMO-LUMO gaps might follow a phi-ladder of rung spacing. But the module itself contains no definition of m or e in molecular terms, so this remains a research aspiration, not a theorem. The three proved facts are universally quantified and apply to any positive real numbers, not to any particular molecule.

What the certificate does establish is a foundation: if someone later defines m and e as, say, ionization energy and electron affinity, then the cost function would have the three properties proved here. The certificate is a template, not a result about chemistry. It shows the framework's cost function behaves well, but it does not show that molecular orbital theory follows from it.

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]

theorem cert_inhabited : Nonempty MolOrbitals4Cert := ⟨cert⟩

What this page does not claim

This certificate does not prove that molecular HOMO-LUMO gaps follow a phi-ladder. This certificate does not define what m and e mean for any molecule. This certificate does not derive any specific molecular orbital energy value.

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/MolecularOrbitals4FromJCost.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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