Chemistry Lewis Acid From Jcost

Lewis acidity and the cost of recognition

A Lewis acid is a chemical species that accepts a pair of electrons. The idea, named after Gilbert N. Lewis in 1923, classifies reactions by electron pair transfer rather than by proton transfer, which is the older Brønsted-Lowry view. Chemists rank Lewis acids by strength, often using the electronegativity of the attached atoms. For boron trihalides, the common order is BF3 > BCl3 > BBr3, meaning the fluoride is the strongest acid in this series.

The Recognition Science framework models a ledger, a discrete record of recognition events, and assigns a forced cost, a number measuring the price of recognizing one thing as another. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This J-cost is nonnegative, and it vanishes exactly when the two inputs are equal.

In Recognition Science, the module LewisAcidFromJCost defines a domain cost as J(m/e), the J-cost of the ratio between two quantities m and e. The machine-checked library of formal theorems proves three general facts about this definition: the cost is zero when m equals e, the cost is nonnegative for positive inputs, and the number phi - 3/2 is positive. These are properties of the cost function itself, not of any chemical system.

The module does not establish the Lewis acidity order BF3 > BCl3 > BBr3. The library proves nothing specific to boron trihalides, because the quantities m and e are never defined in chemical terms. The research note in the source mentions a plan to relate acid strength to the inverse of J(electronegativity_X / electronegativity_B), but that plan is a recorded intention, not a proved result.

What the module does establish is a template. The same three facts, zero at equality, nonnegativity, and a positive threshold, are shared verbatim across 2383 sibling modules. The universal statement lives in one place, DomainCostTemplate. To turn this into a theorem about Lewis acids, one would need a definition of m and e in the chemist's own terms, for example electronegativity values, and a proof that the resulting ratio behaves as the acidity order demands. That step remains open.

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 canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty LewisAcidCert := ⟨cert⟩

What this page does not claim

This answer does not claim that Recognition Science predicts any measured Lewis acidity order. This answer does not claim that the J-cost function itself encodes electronegativity or any chemical property.

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

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