The Covalent Alphabet Seven atoms from four counting conditions, and a measurement of what counting cannot decide

Life builds covalently from hydrogen, carbon, nitrogen, oxygen, phosphorus and sulfur, and that set is normally a fact about biology rather than a consequence of anything. Here it is nearly a consequence of counting. In a shell structure fixed by three spatial dimensions and a two-state spin, each element carries a closure share: the number of electron pairs it must share to reach a closed shell. Four conditions written only in those counts, mentioning no element, admit exactly seven atomic numbers: 1, 6, 7, 8, 14, 15, 16. Six are life's covalent set. The seventh is silicon, which the conditions cannot distinguish from carbon because every count they read agrees. Each condition is shown necessary by the exact set it would otherwise admit. Summed over a structure, the same closure requirement fixes every hydrogen count and nothing beyond it: exhaustive enumeration gives 230,577 admissible side chains against the 16 that occur.