Chemistry Electronegativity2 From Jcost Electroneg2 Cert

The certified threshold

Electronegativity is the tendency of an atom to attract a shared pair of electrons. A common rule of thumb says a bond is ionic when the difference in electronegativity between two atoms exceeds about 1.7 on the Pauling scale. The Recognition Science declaration Electroneg2Cert is a machine-checked certificate, a formal object in the framework's library, that records three general facts about a cost function. It does not, by itself, establish anything about chemistry.

The cost function is defined as J(m/e), where J is the framework's forced cost function and m and e are positive real numbers. The certificate proves three things. First, the cost vanishes when m equals e. Second, the cost is never negative when m and e are both positive. Third, the number phi minus 3/2 is positive, where phi is the golden ratio, approximately 1.618. These are the entire contents of the certificate, and each is a theorem about real numbers, not about atoms.

The research note attached to the module records an intended application: a threshold for ionic bonding at an electronegativity difference near phi, consistent with the empirical 1.7 rule. But the note is explicitly not a result. The definition of domainCost uses m and e as abstract real numbers with no reference to electronegativity, atomic properties, or bonding. The module is shared verbatim with 2383 sibling modules, and the docstring says plainly that what would make it a theorem about its subject is a definition of m and e in that subject's own terms.

In Recognition Science, the framework models the cost of recognition, and this certificate is a small piece of that modeling. It proves that a particular ratio-based cost has the basic properties one would want: zero at equality, nonnegative elsewhere, and a positive threshold constant. The framework's library contains the general theorem that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This certificate applies that general result to a ratio of two positive numbers.

The certificate is honest about its own limits. It proves three general facts and nothing specific to electronegativity. The connection to the 1.7 rule is a research note, a hypothesis about where the idea was meant to go, not a derived result. A reader should see the certificate as a formal prerequisite, a small checked step, and not as a chemical law.

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 prove any fact about actual chemical bonding. The 1.7 electronegativity threshold is not derived from the certificate. The module does not define electronegativity in terms of m and e.

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/Electronegativity2_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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