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Version 3 - April 2026

This educational work presents a deterministic arithmetic model for consecutive homologous elements (Z₁ < Z₂ < Z₃) in the periodic table. The relationship Z₂ = (Z₁ + Z₃)/2 − K is derived, where the correction parameter K = (Δ₂ − Δ₁)/2 assumes only four integer values: 0, 3, 5, and 7. These correspond to half the electron capacities of the s, p, d, and f subshells newly introduced between Z₂ and Z₃. The model is validated across all 77 consecutive triads from hydrogen (Z=1) to oganesson (Z=118) and demonstrates perfect agreement with period-length predictions based on the Madelung rule. Intended strictly as a pedagogical scaffold, the work reduces cognitive load by linking observable period lengths to subshell capacities without requiring full electronic configurations. An interactive calculator tool is provided as supplementary material to facilitate classroom exploration.*