periodOf
plain-language theorem explainer
Atomic number Z maps to its periodic table period through boundaries set by noble gas closures at 2, 10, 18, 36, 54 and 86. Modelers building fit-free chemistry predictions from the recognition framework cite it when deriving shell-dependent radii and ionization scalings. The implementation is a direct piecewise conditional that encodes the eight-window neutrality points.
Claim. $p(Z) = 1$ if $Z ≤ 2$, $2$ if $Z ≤ 10$, $3$ if $Z ≤ 18$, $4$ if $Z ≤ 36$, $5$ if $Z ≤ 54$, $6$ if $Z ≤ 86$, and $7$ otherwise, where $p(Z)$ is the period index for atomic number $Z$.
background
The Periodic Table Engine supplies a fit-free scaffold that maps the eight-tick octave to chemistry via φ-tier rails and fixed block offsets. Noble gases arise exactly where cumulative valence cost meets the eight-window neutrality predicate, forcing the closure set {2, 10, 18, 36, 54, 86} as the chemical expression of ledger balance. This rests on the integer Z anchor maps from the masses and physics modules together with the meta-realization structure that records required self-reference properties.
proof idea
The definition is realized as a direct if-then-else chain that tests Z against the successive noble-gas thresholds and returns the matching period index.
why it matters
It supplies the period index required by shellNumber for radius proxies and by scaledIonization for φ-scaled energies. The definition realizes the Noble Gas Closure Theorem by encoding the deterministic eight-window neutrality condition. It anchors the chemistry layer to the T7 eight-tick octave of the forcing chain without free parameters.
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