Discrete Invertible Transformation Geometry (DITG): A New Mathematical Structure with Applications to the Collatz Conjecture

This work introduces Discrete Invertible Transformation Geometry (DITG), a new mathematical framework for recursively mapping and transforming the positive integers N+\mathbb{N}^+N+ through invertible modular functions. DITG establishes a novel structure within discrete mathematics, providing a constructive resolution of the Collatz conjecture and opening new directions in number theory, cryptography, compression, and computational structures. The presented theory defines a class of fully surjective, invertible mappings based on recursive modular conditions. It outlines general conditions for constructing DITG-functions and explores their modular properties, solvability, and growth behavior. This release marks the foundational publication of DITG and lays the groundwork for future research, new cryptographic primitives, and computational innovations. Author: Ivar Olsson (ORCID: 0009-0006-3773-4034).