Chemistry Complexation Constant From Jcost

The stability constant

A complexation constant, also called a stability constant, is a number that describes how strongly a metal ion binds to surrounding molecules or ions, called ligands. For a reaction like Ca²⁺ + EDTA⁴⁻ ⇌ [Ca(EDTA)]²⁻, the equilibrium constant K_f is large, meaning the complex forms readily. For calcium with EDTA, log K_f is about 18, a familiar textbook value. The constant depends on the metal, the ligand, the solvent, and temperature; it is a measured quantity, not something derived from first principles.

In Recognition Science, the framework attempts to derive such constants from its central object, the cost function, a measure of the forced expense of a recognition event. The framework defines a domain cost as J(m/e), where J(x) = (x + 1/x)/2 - 1, and m and e are real numbers. The machine-checked library of formal theorems proves three general facts about this cost: it equals zero when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are properties of the cost function itself, not of any chemical system.

The research note sketches a path to the EDTA stability constant: log K_f = J(phi)^(-1) × n_donors, where n_donors = 6, giving about 21, which is within an order of magnitude of the measured 18. This is a hypothesis, not a theorem. The formal proofs establish nothing specific to chemistry, because the definition of m and e is left open. The framework's own docstring states this plainly: it proves general facts about J(m/e), not about calcium or EDTA.

What the framework does establish is a template. It shows that if one can define m and e in chemical terms, then the cost function's properties (zero at equality, nonnegativity, positive threshold) carry over. The structure ComplexKCert packages these three facts as a certificate, and the theorem cert_inhabited shows such a certificate exists. This is a formal scaffold awaiting a chemical definition, not a derivation of a stability constant.

The honest verdict: the framework models the stability constant as a function of a cost ratio, and the numerical agreement with EDTA is suggestive but unproven. The gap is the definition of m and e. Until that definition exists, the framework is a template with a research note attached, not a result about chemistry.

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 cert_inhabited : Nonempty ComplexKCert := ⟨cert⟩

What this page does not claim

The framework does not prove that log K_f for EDTA is 21. The framework does not derive the stability constant from first principles. The numerical agreement with EDTA is not a theorem, only a research note.

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/ComplexationConstantFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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