A Periodic Table of Physical Descriptions II - A Taxonomy of Degree-of-Freedom Bases Across Physics

This paper builds on the atlas view introduced in A Periodic Table of Physical Descriptions I by showing that physical descriptions exhibit a deeper periodic structure. Physics repeatedly shifts not only by scale, but by what kind of object is treated as primary in a description: values, distributions, collective fields, invariants, correlations, relations, topology, and phase classes. We introduce a universal “row ladder” of primary objects and combine it with two additional axes—the carrier of description (material, geometric, informational) and the coarse level (micro, meso, macro)—to form a periodic taxonomy of degree-of-freedom (DOF) bases across physics. This lattice organizes familiar theories, reveals non-obvious kinships across subfields, and highlights underdeveloped but structurally predicted descriptive regimes such as geometric kinetic theory, geometric hydrodynamics, and invariant-first summaries of matter. Crucially, the taxonomy makes an operational prediction: observables associated with different rows exhibit different stability under changes of representation. Using real weak-lensing mass maps reconstructed from identical data with different priors, we show that a value-based statistic is highly basis-sensitive while a relational graph invariant is nearly basis-invariant. The underlying sky is the same; only the implicit DOF basis changes. This demonstrates that the primary-object axis captures a real feature of inference. The contribution is not new dynamics but a classification of how physics chooses its variables, and empirical evidence that this choice materially affects what is stable under representation change. Keywords: degrees of freedom, effective theory, coarse-graining, renormalization group, emergence, quasiparticles, hydrodynamics, general relativity, topology, graph invariants, weak lensing, descriptive frameworks, foundations of physics