Chemistry Lewis Acid From Jcost Lewis Acid Cert

What the certificate proves

A Lewis acid is a molecule or ion that accepts a pair of electrons. The classical strength order for boron trihalides is BF3 > BCl3 > BBr3, and chemists explain it through electronegativity and orbital overlap. In Recognition Science, the framework's library of machine-checked formal theorems contains a file named LewisAcidFromJCost. Its central structure, LewisAcidCert, is a certificate: a packaged collection of three proved statements about a cost function applied to a ratio of two real numbers.

The cost function here is the framework's cost, a measure of the forced price of a recognition event, defined as J(x) = (x + 1/x)/2 - 1. The file defines domainCost(m, e) = Jcost(m / e). The three proved facts are: the cost is zero when the two inputs are equal, the cost is nonnegative when both inputs are positive, and the golden-ratio-derived constant phi - 3/2 is positive. These are general properties of the cost function, not facts about any particular chemical system.

The file's own docstring is explicit: it proves nothing specific to Lewis acids, because the definition of domainCost does not reference electronegativity, boron, or any chemical quantity. The structure LewisAcidCert is inhabited, meaning such a certificate exists, but the certificate's content is purely about the cost function's formal properties. To make this a theorem about Lewis acidity, one would need to define m and e in chemical terms, for instance as electronegativities of the halogen and boron.

What the framework's library does not claim is equally important. It does not assert that Lewis acid strength follows the inverse of Jcost, nor that fluorine's high electronegativity leads to the weakest acid. The research note in the file records that as an intended direction, not a result. The soft-hard acid-base distinction, where the reverse pattern can appear, is also not addressed by the certificate. The file is a template shared verbatim with 2383 sibling modules, each awaiting a subject-specific definition to become a theorem about its own domain.

For the reader, the practical takeaway is precise: LewisAcidCert is a reusable formal scaffold, not a chemical discovery. It proves that a certain cost function behaves well on positive inputs, and it packages that proof for future use. The chemical interpretation remains open, waiting for a definition that connects the abstract ratio to measurable properties like electronegativity.

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]

structure LewisAcidCert 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

What this page does not claim

Lewis acid strength follows the inverse of Jcost. Fluorine's high electronegativity leads to the weakest Lewis acid. The soft-hard acid-base distinction is captured by the certificate.

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)

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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