divergentheorema

Divergentheorema is a fictional or hypothetical principle used in mathematical discourse to discuss divergence phenomena across different structures. The term is formed from the Latin root divergere (to diverge) and theorema (theorem). It is sometimes employed as a didactic device to compare how divergence of a sequence, a series, or a dynamical process manifests in related observable quantities.

Informal formulation: Consider a space X endowed with a family F of functionals. The divergentheorema asserts that divergence of a process {x_n} in X—meaning lack of convergence with respect to the natural topology—manifests as unbounded or non-stabilizing values under some f in F. Conversely, if every functional in F assigns bounded, stabilizing values to {x_n}, then the sequence has a convergent subsequence within X. In practice, the exact conditions depend on the context: analysis, topology, or dynamics.

Applications and interpretation: In analysis, it provides a heuristic link between non-convergence and observable unboundedness. In dynamical systems, it helps categorize divergent orbits. In computation, it can model non-termination as a divergence of state traces. Despite its didactic utility, the divergentheorema is not part of standard, peer-reviewed literature and should be treated as an illustrative concept rather than a formal theorem.

Relation to established results: It should not be confused with the divergence theorem of vector calculus (Gauss's theorem). The divergentheorema is distinct and informal, illustrating how different notions of divergence relate across mathematical contexts.