shellNumber
plain-language theorem explainer
shellNumber maps atomic number Z to its 1-indexed shell by adding one to the period returned by periodOf. Researchers deriving atomic radii via phi-ladder scaling in Recognition Science cite this to set base radii that increase down groups. It is realized as a direct definition that invokes periodOf and increments the result.
Claim. For atomic number $Z$, the shell number is $s(Z) = p(Z) + 1$, where $p(Z)$ is the period of the element with atomic number $Z$ (1 for $Z≤2$, 2 for $Z≤10$, 3 for $Z≤18$, 4 for $Z≤36$, 5 for $Z≤54$).
background
The Atomic Radii module (CH-007) states that radii decrease across a period and increase down a group because new shells appear farther from the nucleus. Shell number is the 1-indexed index used for phi-ladder scaling of base radius. Upstream, periodOf from PeriodicTable supplies the period via the listed piecewise thresholds on Z. The shell definition from PeriodicBlocks supplies the coherence scale as E_coh times block capacity at each level.
proof idea
One-line definition that applies the periodOf function from PeriodicTable to the input Z and adds one.
why it matters
This definition supplies the exponent for shellRadiusProxy (phi raised to shellNumber) and is invoked by electronegativity proxies, group-17 ordering theorems, and alkali-halogen ionic-bond verification. It realizes the group-trend prediction of the module doc-comment and the phi-ladder mechanism for chemistry. It sits downstream of the periodOf and shell primitives and upstream of the ionic-bond and EN results.
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