fourth_shell_capacity
The fourth principal shell accommodates 32 electrons via the capacity formula 2n². Atomic physicists and chemists building electron configurations from quantum first principles would cite this when tabulating n=4 occupancies. The proof is a direct reflexivity evaluation of the shellCapacity definition at n=4.
claimThe electron capacity of the shell with principal quantum number $n=4$ equals $2n^2=32$.
background
The module derives the Pauli exclusion principle from ledger single-occupancy: fermions carry odd phase through the 8-tick cycle, forcing antisymmetry that prohibits two identical entries in one state. This yields atomic shell structure and the periodic table. The local definition shellCapacity (n : ℕ) := 2 * n^2 counts states per principal shell. Upstream, PeriodicTableFromPhiLadder.shellCapacity states the same formula as 2n², while PeriodicTable.shellCapacity lists the sequence {2,8,8,18,18,32,...} forced by angular momentum quantization from ledger packing.
proof idea
One-line term proof that applies reflexivity to the definition shellCapacity n = 2 * n^2 evaluated at n=4, confirming the arithmetic result 32.
why it matters in Recognition Science
This populates the n=4 entry in the shell capacity sequence used for noble gas configurations and periodic table construction. It instantiates the angular momentum quantization step that follows from the Recognition Composition Law and the eight-tick octave, closing one concrete case in the derivation of atomic structure from ledger constraints. No open scaffolding remains for this specific capacity.
scope and limits
- Does not derive the 2n² formula from ledger axioms.
- Does not treat subshell ordering or filling exceptions.
- Does not address degeneracy pressure or white-dwarf stability.
- Does not connect the capacity count to explicit J-cost or defectDist calculations.
formal statement (Lean)
118theorem fourth_shell_capacity : shellCapacity 4 = 32 := rfl
proof body
Term-mode proof.
119
120/-! ## Noble Gas Configurations -/
121
122/-- Noble gas electron counts (cumulative filled shells). -/