Chemistry Atomic Radii Shell Radius Increases With Period

The shell radius theorem

In chemistry, atomic radius is the distance from the nucleus to the outermost electron shell. The periodic table shows a clear pattern: as you move down a group, each new element has a larger atomic radius because it gains an additional electron shell. This is a fundamental observation of atomic structure, taught in every introductory chemistry course.

The Recognition Science framework contains a formal, machine-checked library of theorems. Within that library, the declaration shell_radius_increases_with_period proves a specific mathematical statement: if one shell number is less than another, then the framework's model of the shell radius for the smaller shell is less than the model's radius for the larger shell. In the framework's model, the shell radius is defined as the golden ratio (approximately 1.618) raised to the power of the shell number. The theorem states that since the golden ratio is greater than one, a larger exponent yields a larger value. This is a purely mathematical result about the model's definition, proved using the standard properties of real exponentiation.

The theorem is a formal consequence of the model's definitions, not a measurement of actual atoms. It establishes an internal consistency of the framework: the model's shells are ordered by size in the same way they are ordered by number. The proof does not compare its predictions to experimental data, and it does not claim that the golden ratio scaling accurately reproduces the measured atomic radii of real elements. The theorem operates entirely within the abstract definitions of the library.

In Recognition Science, this theorem is part of a larger effort to derive physical constants and structures from a single principle of recognition cost. The framework models atomic radii using a shell number and a screening factor that accounts for electron shielding. The theorem about shell radius increasing with period is one small, verified piece of that larger model. It shows that the model is logically coherent with respect to the ordering of shells, but it leaves the empirical question of whether the model matches reality completely open.

/-- Shell radius increases with period number.
    Period 2 elements are smaller than Period 3 counterparts. -/
theorem shell_radius_increases_with_period (n m : ℕ)
    (hLt : n < m) :
    Constants.phi ^ (n : ℝ) < Constants.phi ^ (m : ℝ) := by
  have hphi_gt_1 : (1 : ℝ) < Constants.phi := by
    have h := Constants.phi_gt_onePointFive
    linarith
  apply Real.rpow_lt_rpow_of_exponent_lt hphi_gt_1
  exact Nat.cast_lt.mpr hLt

/-- Raw shell radius proxy: φ^(shell_number).
    Higher shell = larger base radius. -/
def shellRadiusProxy (Z : ℕ) : ℝ :=
  Constants.phi ^ (shellNumber Z : ℝ)

What this page does not claim

The theorem does not claim that the framework's radius model matches any measured atomic radius. The theorem does not establish that real atoms follow the golden ratio scaling. The theorem does not prove that the framework's model of chemistry is physically correct.

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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Derived articles

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