Periodic system of mathematical models (PSMM): classification, navigation, and the solution of fundamental problems (Part IV)

The present work completes a cycle of studies devoted to the principle of geometric polyvariance. In previous papers, it was shown that the universal coefficient ring (and its infinite-dimensional generalizations) is canonically isomorphic to the invariant rings of twenty-one distinct mathematical structures — ranging from elementary geometric objects to spectral graphs, hyperelliptic curves, integrable systems, quantum groups, free probability, and motive theory. A matrix bridge connecting metric and affine realizations was constructed, and a universal dynamics with a parameter unifying classical, quantum, and geometric evolution was introduced. In this fourth paper, we move from a catalog to a system: a multidimensional classification space is introduced — the Periodic System of Mathematical Models (PSMM). Its coordinates are given by eighteen independent axes, grouped into five blocks (nuclear, algebraic, geometric, dynamic, and categorical characteristics). Each PSMM cell, corresponding to a unique combination of coordinates, is equipped with a standardized passport including a canonical example, realizing models, principal invariants, a dynamical version, transitions to neighboring cells, special modifications, and open problems. The operation of the system is demonstrated using examples from previous papers and new cells (elementary and special functions, general relativity, quantum mechanics). A five-stage navigation algorithm through the PSMM is formulated, allowing one to localize an arbitrary mathematical problem, investigate its cell and adjacent areas, transfer methods, and synthesize a solution. A running example — the twin prime conjecture — illustrates the generation of new research directions. The culmination of the work is the application of the algorithm to fundamental problems of modern science. For quantum gravity, the static and dynamic bridges between the general relativity and quantum mechanics cells lead to a hypothetical cell in which the Wheeler-DeWitt equation takes an explicit form amenable to numerical simulation. For the Riemann hypothesis, the PSMM offers a map for the search for a Hilbert–Pólya operator through connections with modular forms, random matrices, and spectral geometry. Approaches to the problems of P vs NP and time travel are also discussed. In conclusion, prospects for the development of the PSMM are outlined: expansion of the axis system, filling empty cells (prediction of new models), categorification, creation of an interactive database "PSMM-online", and establishing connections with other classifications (Lie algebras, Mendeleev table). The work invites the scientific community to collaboratively construct a unified map of mathematical reality.