halogen_dist_one
plain-language theorem explainer
Halogens occupy atomic numbers immediately preceding noble gas closures. Researchers modeling ionic bonding within Recognition Science would reference this to verify the single-electron acceptor property of halogens. The argument exhausts the five cases in the halogen list and resolves each equality by decidable computation.
Claim. For every atomic number $z$ in the halogen set $H = [9, 17, 35, 53, 85]$, the distance to the next noble-gas shell closure satisfies $d(z) = 1$, where $d(z)$ denotes the steps from $z$ to the subsequent closed shell.
background
The IonicBond module derives ionic compound formation from electron transfer driven by shell closure. HalogenZ supplies the concrete list of atomic numbers for fluorine through astatine. The function distToNextClosure computes the gap from an element's atomic number to the next noble gas, which equals one precisely for these species. This supports the eight-tick closure drive in the module documentation, where halogens accept one electron to reach noble-gas configuration.
proof idea
The term proof specializes the universal quantifier to a concrete atomic number z together with its membership hypothesis. It rewrites membership via the definition of halogenZ, producing a five-way disjunction of equalities. Case analysis on the disjunction followed by native_decide resolves each instance of the distance equaling one.
why it matters
This helper is invoked inside alkali_halogen_stable_1_1 to establish the closure condition for halogen partners in stable 1:1 ionic compounds. It instantiates the RS prediction that ionic bonds form between elements one electron from closure, aligning with the eight-tick octave landmark. The result closes a small gap in the chemistry derivation without new hypotheses.
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