fourth_shell_capacity
plain-language theorem explainer
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.
Claim. The 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
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.
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