Chemistry Nucleophilicity From Jcost Nucleophilicity Cert

The certificate and its limits

In chemistry, nucleophilicity measures how readily a species donates an electron pair to form a new bond. A common scale, the Mayr scale, assigns nucleophilicity parameter N, with values ranging from about -5 for weak nucleophiles to +25 for strong ones. The Recognition Science framework proposes to derive such scales from a single cost function, but the formal declaration NucleophilicityCert does not itself connect to any chemical measurement.

The certificate is a structure in the framework's machine-checked library of formal theorems. It bundles three proved facts about a function called domainCost, defined as Jcost(m/e), where m and e are real numbers. The three facts are: the cost is zero when m equals e, the cost is nonnegative when both inputs are positive, and a certain threshold phi - 3/2 is positive. These are general properties of the cost function, not properties of any particular chemical system.

The library proves these facts once, universally, in a shared template module. The certificate simply packages them for reuse. The docstring attached to the module records a research intention: that the Mayr scale might be expressed as N = 8.47 * log(k/k_ref), where 8.47 is the inverse of J(phi). But that is a research note, not a result. The formal file proves nothing about nucleophilicity because it never defines m and e in chemical terms.

What the certificate does establish is that the cost function has the minimal properties one would want from a measure of mismatch: it is zero when the two quantities are equal, it never goes negative for positive inputs, and the golden-ratio threshold is positive. These are necessary conditions for a useful scale, but they are far from sufficient. A reader should not take the certificate as evidence that the Mayr scale has been derived from first principles.

In plain language, the certificate is a small, correct, and almost empty box. It proves three inequalities and an equality. The chemical content, if any, would come from a separate definition that identifies m and e with specific molecular quantities. Until that definition exists, the certificate remains a template with no chemical teeth.

structure NucleophilicityCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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 claim that nucleophilicity is derived from J-cost. The certificate does not claim that the Mayr scale N = 8.47 * log(k/k_ref) is a theorem. The certificate does not claim that the cost function is a valid nucleophilicity measure.

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)

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.

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