Chemistry Molecular Orbitals4 From Jcost

Molecular orbital gaps and the phi ladder

In chemistry, the HOMO-LUMO gap is the energy difference between the highest occupied and lowest unoccupied molecular orbitals. For small diatomic molecules, measured gaps are roughly: H2 15.5 eV, N2 10.8 eV, O2 5.1 eV. The ratio of the H2 gap to the O2 gap is about 3.04, which is close to phi^2.7, where phi is the golden ratio (1.618...). This observation suggests that molecular orbital energy spacings might follow a phi-based ladder, a pattern seen elsewhere in Recognition Science.

Recognition Science starts from the idea that reality keeps a ledger, a discrete record of recognition events, and that the cost of recognition is forced by a few plain conditions. The cost function J(x) = (x + 1/x)/2 - 1 is proved in a machine-checked library of formal theorems to be the unique function satisfying those conditions. The module in question, MolecularOrbitals4FromJCost, applies this cost function to molecular orbital energies by defining a domain cost as J(m/e), where m and e are two energy values.

The module proves three general facts about this cost function. First, when the two energies are equal, the cost is zero: J(r/r) = 0. Second, for positive energies, the cost is never negative: J(m/e) ≥ 0. Third, the quantity phi - 3/2 is positive, which provides a threshold. These are all proved in the library, but they are generic properties of the cost function, not specific to chemistry. The module does not define m and e in terms of molecular orbitals, so it does not yet prove that the HOMO-LUMO gap follows the phi ladder; that remains a research note.

The practical upshot is that the framework gives a candidate explanation for why molecular orbital gaps might follow a phi-based pattern: the same cost function that forces the golden ratio in other contexts also applies to energy ratios. The next step would be to define m and e specifically as orbital energies, which would turn the generic cost properties into a theorem about chemistry. Until then, the empirical pattern remains a motivation, not a derivation.

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

This module does not prove that molecular orbital gaps follow the phi ladder; that remains a research note. The module does not define m and e in terms of molecular orbitals, so its theorems are generic to the cost function, not specific to chemistry. The measured HOMO-LUMO gaps are empirical values, not derived from the framework.

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)

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