Chemistry Atomic Radii K Larger Shell Than Li

Shell numbers and atomic radii

In the standard periodic table, elements are arranged in rows called periods, and each row corresponds to a new electron shell. Lithium, atomic number 3, sits in period 2. Potassium, atomic number 19, sits in period 4. The Recognition Science framework's declaration k_larger_shell_than_li is a formal theorem stating that the shell number of potassium (19) is greater than the shell number of lithium (3). This is a statement about the framework's internal model of atomic structure, not a new experimental measurement.

The framework defines a shell number for each atomic number Z as the period number plus one. For lithium, that is 2; for potassium, it is 4. The theorem is checked by a computer verifying the arithmetic: 4 is greater than 2. The verification is recorded in the framework's machine-checked library of formal theorems, meaning the statement is verified by a program that checks every logical step.

What the theorem does not claim is more interesting than what it does. It does not say that potassium atoms are physically larger than lithium atoms. Atomic radius depends on more than shell number; it also depends on how strongly the nucleus pulls electrons inward. The framework's own model includes a screening factor for this, and it has separate theorems about normalized radius. The theorem about shell numbers alone establishes only the ordering of shells, not the ordering of measured radii.

The framework also shows that the shell radius proxy, defined as the golden ratio raised to the shell number, increases with shell number. This is a monotonicity result: a higher shell number means a larger base radius in the model. But again, this is about the proxy, not about a measured atomic radius in picometers.

In plain terms, the declaration is a small, verified piece of bookkeeping: it records that the framework's model places potassium in a higher shell than lithium. It is a building block for larger claims about periodic trends, not a claim about the physical size of real atoms.

/-- K (Z=19) has larger shell number than Li (Z=3) due to more shells. -/
theorem k_larger_shell_than_li : shellNumber 19 > shellNumber 3 := by
  native_decide

/-- Shell number (1-indexed) for atomic radii scaling. -/
def shellNumber (Z : ℕ) : ℕ := periodOf Z + 1

/-- 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

What this page does not claim

This theorem does not claim potassium atoms are physically larger than lithium atoms. It does not claim any measured atomic radius value in picometers. It does not claim the framework's shell number is the same as the conventional principal quantum number.

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)

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.

Derived articles

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