third_shell_capacity
plain-language theorem explainer
The third shell capacity theorem states that the electron state count for principal quantum number 3 equals 18. Atomic physicists and chemists deriving periodic table structure from Recognition Science ledger constraints would cite this result. The proof is a direct term reflexivity that evaluates the closed-form definition 2n squared at n equals 3.
Claim. The capacity of the electron shell with principal quantum number 3 is given by the formula 2n² evaluated at n=3, which equals 18.
background
Recognition Science derives the Pauli exclusion principle from ledger single-occupancy: fermions are odd-phase ledger entries whose antisymmetry forces ψ(a,a) = 0, so no two identical fermions share a state. This yields atomic shell structure as a direct consequence. The module targets a first-principles derivation of the periodic table via these ledger constraints. The shellCapacity function in this module is defined as 2n² and counts the number of states available in the shell labeled by principal quantum number n. Upstream, the shell definition in PeriodicBlocks scales capacity by the coherence energy E_coh, while PeriodicTable and PeriodicTableFromPhiLadder supply the explicit sequence and the 2n² formula.
proof idea
The proof is a one-line term-mode reflexivity. It applies the definition shellCapacity n := 2 * n^2 directly at n=3 to obtain the numeral 18.
why it matters
This theorem instantiates the general shell capacity formula inside the Pauli exclusion module, supplying concrete electron counts that support the module's target derivation of atomic shell structure for the planned PRB paper. It connects to the eight-tick octave and ledger packing constraints in the Recognition Science framework by fixing the third-shell occupancy that appears in the periodic table sequence.
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