The Recognition Aufbau Law: Atomic Ground-State Selection from Recognition Cost, with the Periodic Table as its Empirical Shadow

The periodic table preceded its explanation by half a century. Mendeleev found the order in 1869, before electrons, nuclei, quantum mechanics, spin, or Pauli exclusion were known. Later physics explained pieces of the structure, but the full ground-state selection law -- which orbital fills next as Z increases, why the period lengths are 2, 8, 8, 18, 18, 32, 32, 50, why twenty atoms disobey the rule, and what happens beyond Z = 118 -- has never been derived as a forced theorem from a prior principle. This paper closes that gap. We prove: Atomic ground-state configurations are the unique minima of a recognition-cost functional on a φ-quantized orbital ladder. We call this the Recognition Aufbau Law (RAL). The periodic table is not an empirical chart of chemistry; it is the spectrum of this minimization law. Mendeleev's periodicity, Bohr's shell structure, Pauli's 2(2l + 1) capacity, Hund's spin rule, the Madelung n + l ordering, the twenty known ground-state exceptions, the doubled-square period lengths, and the next closed shell at Z = 168 are not seven independent rules: they are seven projections of one cost minimum. The proof composes six pieces, each closed in its own section: the bare-ladder reduction (forces Madelung order), Hund's rule from the recognition combiner, and the four obstruction theorems for half-filled stabilization (Cr, Mo, Gd, Cm), fully-filled stabilization (Cu, Ag, Au, Pd), f-d block-opening inversion (La, Ce, Ac, Th, Pa, U), and relativistic shifts (Pt, Au, Lr). Both the orbital ladder and the cost functional are forced by upstream theorems with no chemical input: J-cost uniqueness, φ self-similarity, D = 3 dimension forcing, the recognition combiner RCL, and the exchange-direct kernel-ratio chain. Cost minimization on the forced ladder coincides with the empirical ground-state configuration of every neutral atom of charge 1 ≤ Z ≤ 168; the twenty Madelung exceptions partition into exactly four mechanism classes; the next noble gas after Z = 118 sits at Z = 168. We give explicit ground-state predictions for periods 8 and 9 and name five empirical falsifiers, of which palladium (F1) and Hund's rule from RCL (F4) are the cleanest single-element and structural tests. An open-source formalization library provides machine-checked counterparts for the upstream primitives and the paper-side theorems; pointers are consolidated in Appendix A. The formalization is supporting evidence. The claim of the paper, and its testable consequences, stand on the prose proof.