A Periodic Table of Machine Intelligence: Periods, Groups, and a Law of Shell Filling

Scaling laws relate compute to loss. They do not say which qualitative regime a model occupies, or when an auxiliary constraint will help. We formalize the Periodic Table of Machine Intelligence (PTMI) as a product of periods (interaction range on the computational graph: pointwise, local, global, plus iterative, routed, and autonomic lifts) and groups (functional valence: invariant, discriminative, predictive, generative, agentic, latent). Capability mass $M_c={\log }_{10}{C}_{\mathrm{train}}+\beta {\log }_{10}{C}_{\mathrm{infer}}$ tags isotopes of a cell; it is an estimator, not a thermodynamic potential. Two theorems make the original “shell filling” claim checkable. A linear residual whose stencil does not vanish on a representable truth strictly biases Tikhonov estimation; an under-receptive surrogate is a different operator, so a mismatched physics term can raise risk. Iteration lifts receptive field by composition, while width inside the pointwise period is a semigroup and plateaus on local tasks. I-Con already tabulated losses; PTMI's second axis is periods. We treat LIRN as a didactic occupant of the already-populated local-generative-iterative cell, not as a claim of first occupancy, and show on 1D/2D/3D Poisson and $k$NN graphs that compatible local iteration beats mismatched PINN-style MLPs and width scaling.