Chemistry Molecular Orbital Gap From Jcost

The orbital gap

The HOMO-LUMO gap is the energy difference between the highest occupied molecular orbital and the lowest unoccupied molecular orbital. It is approximated as ionization energy minus electron affinity, and it governs a molecule's color, conductivity, and reactivity. A small gap lets visible light excite an electron; a large gap leaves the molecule transparent and inert.

In Recognition Science, the framework proposes that this gap is not arbitrary. Its ledger, a discrete record of recognition events, assigns a forced cost to any ratio. The cost function is J(x) = (x + 1/x)/2 - 1, which vanishes at x = 1 and grows as x moves away from unity. The framework's research note suggests that for a well-designed chromophore, the optimal gap equals J(φ) times the ionization energy, roughly 0.118 times IP, placing the absorption in the visible range.

What the machine-checked library of formal theorems actually proves is narrower. The module defines domainCost(m, e) as Jcost(m / e), then proves three general facts: the cost is zero when m equals e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These hold for any positive real numbers, not specifically for molecular orbitals.

The library proves nothing about chemistry because the variables m and e are never defined in chemical terms. The module is a template, shared verbatim with 2383 sibling modules, that would become a theorem about orbital gaps only if m and e were given a chemical meaning. The research note records the intended direction, not a result.

What a reader can take away: the framework offers a concrete, testable proposal for why chromophores absorb where they do, and it provides a clean mathematical cost function. But the gap between that proposal and a proved chemical theorem remains open.

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

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 does not prove that any real molecule has a HOMO-LUMO gap equal to 0.118 times its ionization energy. The framework does not derive the HOMO-LUMO gap from first principles; it only proposes a formula.

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)

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Derived articles

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