Chemistry Crystal Field3 From Jcost

Crystal field energy from J-cost

In coordination chemistry, the crystal field stabilization energy (CFSE) measures how much a metal ion's d-orbital electrons are stabilized when ligands approach. For an octahedral complex with a d³ configuration, the standard empirical result is CFSE = -1.2 × Δ₀, where Δ₀ is the ligand field splitting parameter. This value comes from counting electrons in the lower-energy t₂g orbitals: three electrons each contribute -0.4 Δ₀.

Within Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1 is proposed as a universal measure of recognition cost. The module Crystal_Field3_FromJCost defines a domain cost as J(m/e), where m and e are real numbers representing some ratio. The Lean code proves three general facts: this cost vanishes when m = e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are theorems about J itself, not about chemistry.

What the module does not do is connect m and e to any physical quantity. The docstring records an intended application: CFSE = -J(φ) × (φ + 1) × Δ₀ = -0.309 × Δ₀. This number is off from the empirical -1.2 Δ₀ by a factor of 3.9. The module proves nothing that would justify this formula; it is a research note, not a derivation.

The honest takeaway is that the framework's cost function has general analytic properties, but the leap to crystal field theory is unformalized and numerically fails. The module is a template, shared verbatim with 2383 siblings, waiting for a definition of m and e in chemical terms. Until that definition exists, the page establishes only the three general facts about J, not any result about d³ octahedral complexes.

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 proves no theorem about crystal field theory or d³ octahedral complexes. The empirical CFSE value -1.2 Δ₀ is not derived from the framework. The factor 3.9 discrepancy is not explained by the module.

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/Crystal_Field3_FromJCost.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: