en_increases_across_period
plain-language theorem explainer
Electronegativity increases across a period because valence electron count grows toward the next noble gas closure. Chemists deriving periodic trends from RS shell structure would cite this to justify the Mulliken-scale prediction. The argument reduces the claim directly to the definition of valence electrons and applies arithmetic comparison.
Claim. Let $Z_1,Z_2$ be natural numbers. If both exceed the same previous noble gas closure and $Z_1<Z_2$, then the valence electron count of $Z_1$ is strictly less than that of $Z_2$.
background
The module derives electronegativity from φ-ladder scaling, where EN measures an atom's tendency to attract electrons and follows distToNextClosure^{-1} modulated by shell number. Key definitions are prevClosure(Z), the atomic number of the preceding noble gas (0 for Z≤2, 2 for Z≤10, 10 for Z≤18, etc.), and valenceElectrons(Z) := Z - prevClosure(Z), counting electrons beyond the closed shell. The local setting states that EN increases across a period as atoms approach closure, consistent with the classical relation EN ~ √(IE × EA). Upstream results supply the closure function from PeriodicTable and the gap function F from AnchorPolicy, which aligns with RSBridge.gap for scaling.
proof idea
The proof applies simplification to unfold the valenceElectrons definition, reducing the inequality to a direct comparison of Z1 and Z2 under the shared closure hypothesis, then discharges it via the omega tactic for linear arithmetic.
why it matters
This result fills the explicit prediction that EN increases across a period in the CH-008 framework, grounding classical periodic trends in RS shell structure. It connects to the phi-ladder and eight-tick octave by supplying chemical periodicity from the same forcing chain. No immediate downstream uses appear, but the theorem underpins sibling results on EN ranking and group-17 ordering.
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