The Structure of the Collatz Dynamics

This paper develops the odd-to-odd Collatz dynamics through a deterministic inverse framework built from the arithmetic properties of the inverse map R(n,k) = (2^k n − 1) / 3. The analysis proceeds through a sequence of structural components that together produce a global description of the system. changelog: V2-5: Miscellaneous editorialization V6: Added coelescence of invariants, showing coherence of the map and extension of behavior to all depths Added the section on higher k lift order Removed Appendix V7: Readded global offsets and progressions from original manuscript for clarity Defined the Collatz map as a uniquely determined object V8: Added logic of neighboring sequentials show interleaving of all paths and positions, contradicting disjoint subsets V9: Updated abstract to reflect the now current logical spine of the paper V10: Updated introduction to reflect current structure of the main body Updated novelty section to reflect current structure of the main body V11: Added a structural perspective paragraph to the introduction to clarify the role of invariant coalescence and global determinacy V12: Editorialization and added conclusion paragraph v13: Added argument for surjectivity not based on dependency, will require restructuring in next version V14: Added explicit terminology in deemed required placement; added subsection before conclusion of affine invariance in structure V15: Added arithmetic obstruction to cycles V16: Added geometric localization if K and well ordered dependency V17: Added direct acyclicity and rooted surjectivity