Chemistry Electron Affinity Ea Decreases Within Period

The within-period trend

Electron affinity is the energy released when a neutral atom gains an electron. In the periodic table, the trend within a row is well known: values rise toward the halogens, which sit one electron short of a filled shell, and then fall sharply at the noble gases, which already have a filled shell. The Recognition Science framework encodes this pattern with a simple proxy: the distance, in electrons, from an atom's atomic number to the next shell closure. A halogen like fluorine has distance 1, a noble gas like neon has distance 0.

The framework's library, a machine-checked collection of formal theorems, proves a precise statement about this proxy. The theorem ea_decreases_within_period says that if two atoms lie in the same period, the one with the smaller atomic number has a strictly larger electron affinity proxy, provided neither has passed its own next closure. In plain terms, within a row, moving left to right toward the noble gas, the proxy falls steadily until it reaches zero at the closure itself. The proof is a direct computation on natural numbers, not an empirical fit.

The same library derives two consequences that match the familiar chemistry. A halogen, defined as an atom one electron short of closure, has proxy exactly 1, the maximum within its period. A noble gas, defined as an atom at closure, has proxy exactly 0. The theorems halogen_ea_one and noble_gas_ea_zero state these values as formal results. The framework also proves halogen_max_ea, which says that within a period, the halogen's proxy is at least as large as any other atom's, and equality holds only at the halogen itself.

What the theorem does not claim is just as important. It concerns the proxy, a definitional stand-in for electron affinity, not a measured energy in electronvolts. The framework does not derive the actual magnitude of any element's electron affinity, nor does it claim that the proxy predicts values across different periods, where shell lengths vary. The trend within a period is a structural statement about the proxy, and the framework presents it as such.

/-- Within a period, EA proxy decreases as Z increases (filling the shell).
    Elements closer to closure have higher EA potential. -/
theorem ea_decreases_within_period (Z1 Z2 : ℕ)
    (hZ1le : Z1 ≤ nextClosure Z1)
    (hZ2le : Z2 ≤ nextClosure Z2)
    (hLt : Z1 < Z2)
    (hSame : nextClosure Z1 = nextClosure Z2) :
    eaProxy Z1 > eaProxy Z2 := by
  simp only [eaProxy, distToClosure, distToNextClosure]
  omega

/-- Halogens have EA proxy = 1 (one electron completes shell). -/
theorem halogen_ea_one (Z : ℕ) (h : isHalogen Z) : eaProxy Z = 1 := by
  simp only [eaProxy, distToClosure]
  exact h

/-- Halogens have higher EA proxy than any other non-noble element in their period. -/
theorem halogen_max_ea (Z : ℕ) (Z' : ℕ)
    (hZle : Z ≤ nextClosure Z)
    (_ : Z' ≤ nextClosure Z')
    (hSamePeriod : nextClosure Z = nextClosure Z')
    (hZ'lt : Z' < nextClosure Z') :
    distToNextClosure Z' ≤ distToNextClosure Z ↔ Z ≤ Z' := by
  simp only [distToNextClosure]
  omega

/-- Noble gases have EA proxy = 0 (at closure, no benefit from adding electron). -/
theorem noble_gas_ea_zero (Z : ℕ) (h : isNobleGas Z) : eaProxy Z = 0 := by
  simp only [eaProxy, distToClosure]
  exact noble_gas_at_closure Z h

What this page does not claim

The theorem does not derive the magnitude of any element's electron affinity in electronvolts. The proxy does not predict electron affinity values across different periods. The framework does not claim that the proxy is a measured physical quantity.

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/ElectronAffinity.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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