A Periodic Table of Physical Descriptions III - Systematic Gap Analysis and Predicted Descriptive Frameworks

This paper is the capstone of a three-paper series that treats physical theories as an atlas of degree-of-freedom (DOF) bases connected by a small set of recurring transformations (renormalization, coarse-graining, symmetry factoring, phase creation). Papers I–II introduced a periodic taxonomy of descriptions organized by (i) carrier (material, geometric, informational) and (ii) the “primary object” treated as fundamental (values, distributions, collective fields, invariants, correlations, relations, topology, phase classes). Here we push the periodic table into its predictive mode: systematic gap analysis. We identify six structurally implied but underdeveloped descriptive frameworks and develop them as standardized “missing cells” of the taxonomy: Geometric Kinetic Theory (geometric carrier × distributions) Geometric Hydrodynamics (geometric carrier × collective fields) Curvature Correlations (geometric carrier × correlations) Geometric Charges (geometric carrier × invariants) Matter Topology (material carrier × topology) Information Kinetic Theory / Information Networks (informational carrier × distributions/relations) A central methodological contribution is the Stability-First Discovery Criterion: when multiple reconstructions or representations exist for the same underlying data, robust discovery should prioritize observables that are invariant under representation change. In this framework, “new physics” is not established by a fragile anomaly in low-row, value-based statistics alone; it requires coherence in higher-row observables (relations/topology) that are empirically stable across pipelines. Empirically, we complete the Row V / Row R / Row T stability ladder using the same DES Y3 weak-lensing convergence maps analyzed in Paper II. Holding the data and mask fixed and varying only the reconstruction basis (KS, Wiener, NullB, GLIMPSE), the coefficient of variation (CV) across methods obeys a strict hierarchy: CV(Row T)