Chemistry Metallic Bond Alkali Low Ionization

The alkali electron count

In chemistry, an alkali metal is any element in group 1 of the periodic table: lithium, sodium, potassium, rubidium, cesium, and francium. Their defining trait is a single valence electron in the outermost shell. The Recognition Science framework's machine-checked library of formal theorems records this textbook fact as alkali_low_ionization, a declaration that takes any atomic number Z, assumes Z belongs to the framework's list of alkali metals, and derives that Z has exactly one valence electron. The list itself is explicit: [3, 11, 19, 37, 55, 87], matching the six alkali elements by atomic number.

The declaration is a theorem in the framework's library, meaning it is a statement proved from definitions by the machine-checked kernel, not an assumption or a hypothesis. Its proof is short: it unfolds the alkali list, then checks each of the six cases by computation. What the theorem establishes is a precise numerical fact about the framework's model of electron count. It does not establish that alkali metals have low ionization energy in the physical sense; the name is a label, and the content is the valence electron count. The framework's docstring asserts the physical connection, but the formal theorem itself stops at the arithmetic.

The declaration sits inside a larger file that models metallic bonding. The file defines lists for alkali, alkaline earth, and transition metals, a predicate for being a metal, a free electron count, a conductivity proxy, lattice types with coordination numbers, and packing efficiencies. The alkali theorem is one small piece of that model. It is a definitional choice about how the framework represents chemistry, not a measurement or a derivation from first principles. The framework models alkali metals as having one free electron; the theorem records that this model is internally consistent.

What the declaration does not claim is worth stating plainly. It does not claim that the framework has derived the periodic table from its cost function, nor that it explains why alkali metals are soft, shiny, or reactive. It does not claim that the framework's list of alkali metals is the only possible list, or that the framework's model of electron count is physically accurate. The theorem is a formal statement about the framework's own definitions, and its scope is exactly that. A reader who wants the physical chemistry of alkali metals should look to experiment; a reader who wants to see how the framework encodes that chemistry can look here.

/-- Metals have low ionization energy (alkali metals are easiest to ionize). -/
theorem alkali_low_ionization (Z : ℕ) (h : Z ∈ alkaliMetalZ) :
    valenceElectrons Z = 1 := by
  simp only [alkaliMetalZ] at h
  -- alkaliMetalZ = [3, 11, 19, 37, 55, 87]
  -- Each alkali metal: Z - prevClosure Z = 1
  fin_cases h <;> native_decide

/-- Alkali metals are also metals. -/
def alkaliMetalZ : List ℕ := [3, 11, 19, 37, 55, 87]

What this page does not claim

The theorem does not prove that alkali metals have low ionization energy in the physical sense. The framework does not derive the periodic table from its cost function. The declaration does not claim the framework's list of alkali metals is the only possible list.

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

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