Inevitable Infinite Branching in the Multiplication of Singularities

TL;DR

This paper discusses the inherent infinite branching in the multiplication of singularities within nonlinear models in physics, emphasizing the lack of a unique method and the necessity of external criteria for such operations.

Contribution

It highlights the fundamental impossibility of defining a canonical multiplication of singularities, revealing the intrinsic complexity in nonlinear singularity theories.

Findings

Multiplication of singularities inevitably branches infinitely.

No unique natural way exists to perform nonlinear operations on singularities.

External considerations are necessary to choose among branches.

Abstract

Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of singularities have been developed. A critically important related feature is that, above certain levels in singularities, the operation of multiplication, and in general, nonlinear operations on such singularities do inevitably branch in infinitely many ways, without the possibility for the existence of some unique natural or canonical way such nonlinear operations may be performed. Consequently, the choice in such branchings has to come from extraneous considerations.