The Hashtag and Hashtag* Framework: An Analytic–Topological Construction

We introduce two formal mathematical constructs denoted by # and #*. The operator # is defined as a nonlinear differential expression combining gradient and Laplacian interactions, while #* is defined as a topological constant associated with the structural identity of a topological shape. The purpose of this note is to formally define both objects, establish their invariance properties, and present a preliminary analytic–topological bridge theorem.

Originality and AI-use statement:
This work is an original research output by Begüm Yıldırım. AI tools, if used, were limited to language refinement, grammar correction, formatting, translation assistance, and clarity improvement. The conceptual framework, research direction, interpretation, models, and conclusions belong to the author. External sources, datasets, or prior works are cited where applicable.