Chemistry Ligand Field From Phi Ladder Ligand Field Cert

The ligand field certificate

In coordination chemistry, the ligand field splitting 10Dq measures the energy gap between otherwise equal electron orbitals when a metal ion sits inside a cage of surrounding molecules. For a weak-field octahedral complex, textbooks place 10Dq/B in the range of 20 to 30, where B is the Racah interelectronic repulsion parameter; a typical value is 10Dq near 1000 to 2000 cm^-1. The Recognition Science framework's ledger, a discrete record of recognition events, proposes a specific value: 10Dq/B = φ^5 ≈ 11.09, which with B ≈ 800 cm^-1 gives 10Dq ≈ 8800 cm^-1, a mid-range prediction.

The declaration LigandFieldCert does not establish that prediction. It is a structure, a formal container, that bundles three general theorems about the function Jcost, the framework's forced cost of recognition. The first theorem states that when the two inputs are equal, the cost is zero: domainCost r r = 0 for any nonzero r. The second states that the cost is never negative for positive inputs: 0 ≤ domainCost m e whenever m and e are both positive. The third states that the constant φ − 3/2 is positive, roughly 0.118. The library proves all three, and the certificate simply packages them into one object with a proof that such a certificate exists.

The chemistry never enters the proof. The function domainCost is defined as Jcost (m / e) with no reference to a metal, a ligand, or an orbital. The numbers m and e are just real variables. The research note attached to the module records the intention to identify m with a metal property and e with a ligand property, but no such identification is made in the Lean code. The certificate is therefore a theorem about the cost function, not about crystal field theory.

What the certificate does establish is a consistency check. Any future definition of m and e in chemical terms that plugs into this cost function will automatically inherit three facts: equal inputs cost nothing, costs are nonnegative, and the golden-ratio threshold is positive. That is a small but real constraint on how the framework might attach its cost function to ligand field theory. The 10Dq value itself, and the mapping of m and e to physical quantities, remains a research note, not a result.

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 certificate does not prove the 10Dq value of 8800 cm^-1. The certificate does not identify m and e with any chemical quantities. The certificate does not derive the Racah parameter B 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/LigandFieldFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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