Chemistry Ligand Field From Phi Ladder

Ligand field splitting

Ligand field theory describes how the energies of a metal ion's d-orbitals split when ligands approach. In an octahedral complex, the five d-orbitals split into two sets, and the energy gap between them is called 10Dq, the crystal field splitting parameter. A typical weak-field octahedral complex has 10Dq/B roughly 20 to 30, where B is the Racah interelectronic repulsion parameter, and 10Dq itself is often 1000 to 2000 cm^-1.

The framework Recognition Science starts from a forced cost function J(x) = (x + 1/x)/2 - 1, proved in its machine-checked library of formal theorems. The idea here is to model the ligand field splitting as a cost of recognition between the metal and ligand. The module defines domainCost as J(m/e), where m and e are positive real numbers representing the metal and ligand energies. The module proves three general facts: domainCost is zero when m equals e, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These are general properties of J, not specific to chemistry.

The research note attached to the module suggests a specific prediction: 10Dq/B = phi^5, about 11.09. With B around 800 cm^-1, this gives 10Dq about 8800 cm^-1, which the note calls mid-range. This numerical comparison is a research note, not a proved result, because the module never defines m and e in chemical terms.

What the module actually establishes is a template: it shows that the cost function J has the right formal properties to serve as a ligand field energy, but it does not connect those properties to any measured splitting. The chemical content, the identification of m and e, remains an open target.

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)

noncomputable def cert : LigandFieldCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module does not prove any chemical statement about ligand field splitting. The research note's numerical match is not a theorem; it is a hypothesis without a named falsifier. The framework does not derive the fine-structure constant alpha.

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/LigandFieldFromPhiLadder.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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