isMetal
plain-language theorem explainer
The definition isMetal(Z) classifies an atomic number Z as metallic precisely when it belongs to the alkali, alkaline earth, or transition metal families. Researchers deriving lattice conductivity or electron-sea properties within Recognition Science would reference this predicate to link low ionization energies to delocalized states. The definition is a direct disjunction over three fixed lists of atomic numbers, accompanied by a decidable instance.
Claim. A natural number $Z$ satisfies isMetal$(Z)$ if and only if $Z$ lies in the alkali-metal list, the alkaline-earth list, or the transition-metal list.
background
The module derives metallic bonding from the Recognition Science mechanism: valence electrons delocalize across a cation lattice to minimize J-cost, realizing an 8-tick collective rhythm with phi-scaling in conductivity. The predicate isMetal draws on three sibling definitions: alkaliMetalZ lists the six alkali elements with low ionization energy, alkalineEarthZ lists the six group-2 elements, and transitionMetalZ enumerates the d-block rows from Sc to Hg. Upstream, alkaliMetalZ appears in IonicBond and IonizationEnergy modules as the set of elements with minimum ionization proxy within each period.
proof idea
The definition is a direct disjunction of membership tests over the three precomputed lists. The accompanying DecidablePred instance is obtained by a single if-then-else that returns isTrue on the disjunction and isFalse otherwise.
why it matters
This predicate anchors the metallic-bonding derivation (CH-011) by identifying which atomic numbers participate in the delocalized electron sea. It instantiates the eight-tick coherence (T7) for many-body metallic states and supplies the input domain for downstream siblings such as freeElectrons and conductivityProxy. No parent theorems or external citations are recorded yet, leaving open its integration with the phi-ladder mass formula and the alpha band.
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