Chemistry Ionization Energy Ionization Proxy

The ionization proxy

First ionization energy is the energy needed to remove one electron from a neutral atom. It rises and falls across the periodic table in a sawtooth pattern: low for alkali metals, high for noble gases. The Recognition Science framework's ionization proxy, a discrete record of the element's position within its period, captures that shape with a single number: the count of valence electrons. An alkali metal scores 1, a noble gas scores its full period length, and every element in between gets a rank that increases as the shell fills.

This proxy is a definition, not a measurement. The framework's machine-checked library of formal theorems proves three ordering facts about it: within any period the proxy increases with atomic number, an alkali metal always has the minimum value of 1, and a noble gas always has the maximum value equal to its period length. A fourth theorem shows the reset at each period boundary, where the element after a noble gas drops back to the minimum. These are exact statements about the proxy's definition, and they hold by construction.

The proxy deliberately stops short of predicting actual ionization energies. The framework defines a scaled version that multiplies the proxy by a φ-based factor and an energy anchor, but the numerical output depends on that anchor, which is not derived from the proxy alone. The docstring is explicit: the ordering is fit-free, the values are not. The falsification criteria target only the ordering, for example whether any noble gas fails to have the maximum proxy in its period when checked against NIST data.

What the proxy establishes is a structural claim about periodicity. It says that the sawtooth shape of ionization energy, the drop after each noble gas and the climb within each period, follows from counting valence electrons. It does not say how many electronvolts any element requires, and it does not derive the energy anchor. The proxy is a ladder rung, not a full prediction.

/-- Ionization proxy is monotone increasing within a period.
    If Z₁ < Z₂ are in the same period, then proxy(Z₁) < proxy(Z₂). -/
theorem ionization_monotone_within_period (Z1 Z2 : ℕ)
    (hZ1ge : Z1 ≥ prevClosure Z1)
    (hLt : Z1 < Z2) (hNotCross : prevClosure Z1 = prevClosure Z2) :
    ionizationProxy Z1 < ionizationProxy Z2 := by
  simp only [ionizationProxy, valenceElectrons]
  -- Z1 - prevClosure Z1 < Z2 - prevClosure Z2
  -- Since prevClosure Z1 = prevClosure Z2, this reduces to Z1 < Z2
  have hZ2ge : Z2 ≥ prevClosure Z2 := by omega
  omega

/-- Within any period, alkali metals have minimum ionization proxy. -/
theorem alkali_min_ionization (Z : ℕ) (hZ : valenceElectrons Z = 1) (hZ2 : Z > 2) :
    ionizationProxy Z = 1 := by
  simp only [ionizationProxy]
  exact hZ

/-- Noble gases have maximum ionization proxy (equal to period length). -/
theorem noble_max_ionization (Z : ℕ) (h : isNobleGas Z) :
    ionizationProxy Z = periodLength Z := by
  simp only [ionizationProxy]
  exact noble_gas_complete_shell Z h

/-- The sawtooth pattern: ionization resets at each period boundary.
    After a noble gas, the next element (alkali) has minimal ionization. -/
theorem sawtooth_reset (Znoble Zalkali : ℕ)
    (hNoble : isNobleGas Znoble)
    (hNext : Zalkali = Znoble + 1)
    (hValid : Zalkali ≤ 118) :
    ionizationProxy Zalkali < ionizationProxy Znoble := by
  -- Noble gas has maximum (= period length), alkali has 1
  -- Case by case on which noble gas
  unfold isNobleGas nobleGasZ at hNoble
  simp only [List.mem_cons, List.mem_nil_iff, or_false] at hNoble
  obtain rfl | rfl | rfl | rfl | rfl | rfl := hNoble <;>
    subst hNext <;>
    native_decide

What this page does not claim

The ionization proxy does not predict numerical ionization energies in electronvolts. The proxy does not derive the energy anchor E_coh used in predictedI1_eV. The ordering theorems do not assert that the proxy matches measured ionization energies, only that it defines a consistent ranking.

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/IonizationEnergy.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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