Chemistry Atomic Radii Lower Z More Remaining

What the theorem shows

In the periodic table, atomic radius follows a familiar pattern. Moving left to right across a period, atoms get smaller as the nucleus pulls electrons closer. Moving down a group, atoms get larger because each new shell sits farther out. The Recognition Science framework encodes this trend in a machine-checked library of formal theorems, and one of its results, lower_z_more_remaining, states a precise version of the left-to-right shrinkage.

The theorem compares two elements in the same shell, meaning they share the same next closure point, the atomic number where a shell completes. If one element has a lower atomic number than another, and both are still inside that same shell, then the lower element has a larger distance to the closure. In plain terms, the element closer to the start of a row has more room left before the row ends. The proof is a direct calculation on the definition of distance, and it holds for all such pairs, not just for particular elements.

This result is a theorem about the framework's own model of radii, not a measurement of real atoms. The framework defines a radius proxy, a stand-in for radius, as a shell term multiplied by a screening factor. The shell term grows with the period number, and the screening factor shrinks as valence electrons fill the shell. The theorem works entirely inside that definition. It confirms that the model reproduces the expected trend: within a period, higher atomic number means less distance to the next shell closure.

The framework's library also proves supporting facts about specific elements. It shows that lithium has a larger normalized radius than fluorine, that sodium is larger than chlorine, and that each alkali metal has exactly one valence electron. These are consequences of the same definitions, checked by computation. They illustrate the model's internal consistency, not new chemistry.

What the theorem does not claim is that these proxies equal measured atomic radii. It does not compare its numbers against experimental data, and it does not derive the periodic trend from quantum mechanics. The framework's contribution is a formal structure that mirrors the trend, not a new physical law. A reader should take the theorem as a statement about the model's definitions, with the classical chemistry remaining the empirical ground truth.

/-- Within a period, lower Z has more "remaining capacity" to next closure.
    This is the basic principle of across-period radius contraction. -/
theorem lower_z_more_remaining (Z1 Z2 : ℕ) (hLt : Z1 < Z2)
    (hZ1_le : Z1 ≤ nextClosure Z1)
    (hZ2_le : Z2 ≤ nextClosure Z2)
    (hSameNext : nextClosure Z1 = nextClosure Z2) :
    distToNextClosure Z1 > distToNextClosure Z2 := by
  simp only [distToNextClosure]
  omega

/-- Atomic radius proxy: combines shell and screening effects.
    radius = shellRadius * screeningFactor -/
def radiusProxy (Z : ℕ) : ℝ :=
  shellRadiusProxy Z * screeningFactor Z

/-- Li (Z=3) has larger radius than F (Z=9) in Period 2.
    Li: (1 - (3-2)/(10-2)) = 7/8
    F:  (1 - (9-2)/(10-2)) = 1/8 -/
theorem li_larger_than_f : normalizedRadius 3 > normalizedRadius 9 := by
  simp only [normalizedRadius, prevClosure, nextClosure]
  norm_num

/-- Na (Z=11) has larger radius than Cl (Z=17) in Period 3.
    Na: (1 - (11-10)/(18-10)) = 7/8
    Cl: (1 - (17-10)/(18-10)) = 1/8 -/
theorem na_larger_than_cl : normalizedRadius 11 > normalizedRadius 17 := by
  simp only [normalizedRadius, prevClosure, nextClosure]
  norm_num

/-- Alkali metals start new shells, so have valence = 1. -/
theorem li_valence_one : valenceElectrons 3 = 1 := by native_decide

What this page does not claim

The theorem does not state that the radius proxy equals any measured atomic radius. It does not derive the periodic trend from quantum mechanics or electron configuration. It does not claim that the framework's model is the only way to encode the trend.

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

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