shellCapacity
Electron shell capacity for principal quantum number n equals 2n squared. Researchers reconstructing the periodic table from Recognition Science ledger constraints cite this when verifying block occupancies. The definition is a direct encoding of the standard formula also appearing in the Pauli exclusion module.
claimThe maximum number of electrons in the shell with principal quantum number $n$ is $2n^2$.
background
The module derives periodic table structure from the phi-ladder while noting that shell capacities follow the standard 2n² sequence rather than direct phi powers. Electron blocks total five types (s, p, d, f, g), matching configDim D = 5. Upstream, the Pauli exclusion result defines the same capacity as the number of states in a shell with principal quantum number n, while the PeriodicTable module gives an explicit sequence for periods. The module states that capacities are 2, 8, 18, 32 = 2n² for n = 1, 2, 3, 4, and that these are not phi-ladder but the number of periodic table blocks totals 32 = 2^5.
proof idea
This declaration is a one-line definition that directly sets shell capacity to twice the square of the input natural number.
why it matters in Recognition Science
It supplies the capacity values required by PeriodicTableCert to certify five blocks with capacities 2, 8, 18, 32 for the first four shells. This anchors the standard quantum result inside the Recognition Science framework, where angular momentum quantization arises from ledger packing. The definition supports the separation between shell capacities (2n²) and the phi-ladder pattern for period lengths.
scope and limits
- Does not derive 2n² from the phi-ladder or forcing chain.
- Does not model sub-shell filling or real periodic table anomalies.
- Does not specify the connection to spatial dimension D = 3.
- Does not address the g-block prediction.
Lean usage
example : shellCapacity 4 = 32 := by rflformal statement (Lean)
36def shellCapacity (n : ℕ) : ℕ := 2 * n ^ 2
proof body
Definition body.
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