Chemistry Nucleophilicity From Jcost

Nucleophilicity and its scale

In chemistry, nucleophilicity is the tendency of a molecule or ion to donate a pair of electrons to an electron-poor atom, forming a new covalent bond. A familiar example is the hydroxide ion attacking a carbon atom in an alkyl halide. Chemists quantify this tendency with the Mayr nucleophilicity parameter N, which for most reagents ranges from about -5 to +25. The parameter is defined through a linear free-energy relationship: log(k/k_ref) = s * N, where k is the reaction rate, k_ref is a reference rate, and s is a slope constant that depends on the reaction type.

Recognition Science proposes that this logarithmic scale is not arbitrary but follows from a universal cost function. The framework's central object is a cost, a discrete record of how much recognition effort a process requires. The framework's library of machine-checked formal theorems proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This J-cost is then applied to chemistry by setting x equal to the ratio of two quantities, m over e, which the research note identifies with the nucleophile's properties relative to a reference. The resulting scale is N = 8.47 * log(k_ratio), where the factor 8.47 is the inverse of J evaluated at the golden ratio phi.

The module in the library establishes three general facts about this domainCost function, defined as Jcost(m/e). First, the cost vanishes when m equals e, meaning no recognition effort is needed when the nucleophile matches the reference exactly. Second, the cost is nonnegative for all positive inputs, so recognition never has a negative price. Third, a canonical threshold, phi - 3/2, is strictly positive, which the framework ties to stability. These three facts are assembled into a structure called NucleophilicityCert, and the library proves this certificate is inhabited.

The plain-language result is modest but exact: the module proves that the proposed cost function has the right basic shape for a scale, vanishing at the reference point and never going negative. It does not prove that this scale matches experimental nucleophilicity data, because the module never defines what m and e mean in chemical terms. The research note states the intention, but the formal content is universal, shared verbatim with 2383 sibling modules. What would make this a theorem about chemistry is a definition of m and e in the subject's own terms, which remains open.

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 NucleophilicityCert := ⟨cert⟩

What this page does not claim

This module does not prove any experimental claim about nucleophilicity, since m and e are never defined chemically. The 8.47 factor is a proposal from a research note, not a derived or measured constant in the module. The framework does not claim that nucleophilicity is the only application of the J-cost function.

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

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