Chemistry Periodic Table Block Count Formula

Block electron counts

The periodic table organizes elements into four blocks, s, p, d, and f, named for the atomic orbitals being filled. Each block holds a fixed number of electrons per shell: the s block holds 2, the p block holds 6, the d block holds 10, and the f block holds 14. These are the familiar numbers that explain the table's shape, from the two-element first row to the fourteen-element lanthanide and actinide series.

In Recognition Science, a machine-checked library of formal theorems encodes these counts in a definition called blockElectronCount, a discrete record of how many electrons each block can contain. The library assigns the values directly: s maps to 2, p to 6, d to 10, and f to 14. This is a definitional choice, a model of the known chemistry, not a theorem that proves the numbers must be what they are. The library also defines a default set of block offsets, s=0, p=1, d=2, f=3, used in energy calculations, again without per-element tuning.

The framework's larger claim is that the noble gases, helium, neon, argon, krypton, xenon, and radon, are exactly the elements where a running sum of valence costs returns to zero over an eight-element window. The library proves that the cumulative shell closures match the noble gas atomic numbers, and that the period lengths derived from the gaps between them equal the standard sequence 2, 8, 8, 18, 18, 32. These are machine-checked identities within the library's own definitions.

What the block count definition does not do is explain why the blocks have these sizes. The numbers 2, 6, 10, and 14 are inputs to the framework, taken from observed chemistry, not outputs of its forcing chain. The library's theorems confirm that its definitions reproduce the known periodic table, but they do not derive the table from first principles. The framework's prediction about noble gases is a hypothesis with a named falsifier: any element outside the set {2, 10, 18, 36, 54, 86} that behaves as a noble gas would refute it.

/-- Block electron counts: s=2, p=6, d=10, f=14. -/
def blockElectronCount : Block → ℕ
  | Block.s => 2
  | Block.p => 6
  | Block.d => 10
  | Block.f => 14

/-- The cumulative closures match the noble gas sequence exactly. -/
theorem cumulative_closure_eq_noble (n : Fin 6) :
    cumulativeShellClosure n.val = nobleGasZ.get ⟨n.val, by simp [nobleGasZ]⟩ := by
  fin_cases n <;> rfl

/-- The noble gas differences recover the period lengths. -/
theorem period_lengths_from_noble_gaps :
    [2, 10 - 2, 18 - 10, 36 - 18, 54 - 36, 86 - 54] = [2, 8, 8, 18, 18, 32] := by
  native_decide

What this page does not claim

The block electron counts 2, 6, 10, 14 are derived from the framework's forcing chain; they are defined constants. The library proves that the noble gases are the only chemically inert elements; it only shows its own definition matches the known set. The framework predicts the existence of element 118 or any element beyond the current periodic table.

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

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