Reverse Collatz Reachability, Modular Steering, and a Reduction to a Uniform Interval-Hit Problem

Disclosure. This document contains AI-assisted mathematical exploration. The research direction, hypotheses, and numerical experiments (if applicable) were generated and performed by the author. A large language model was used to assist with symbolic derivations and drafting text. Mathematical correctness is not guaranteed; this document represents exploratory AI-assisted research. This upload is part of an experiment on whether large language models can produce research-level mathematical content under guided direction. Expert feedback and verification (positive or negative) are welcome and will be incorporated into future revisions. We investigate the Collatz conjecture from the perspective of reverse iteration and reachability rather than forward dynamics. By introducing a bounded family of reverse Collatz hops with exponents restricted to a fixed finite set, we obtain explicit control of growth while preserving exponential branching in the reverse Collatz graph. An exact closed form for reverse iterates separates scale from offset contributions and reveals a layered (3)-adic structure that governs modular behavior. Within this framework, we prove full modular reachability in both the (2)-adic and (3)-adic components. These results combine via the Chinese Remainder Theorem to eliminate all qualitative modular obstructions to reverse reachability. We further show that the Collatz conjecture reduces to a single quantitative condition: a uniform interval-hit property asserting that every residue class modulo 3 · 2^k is attained by some reverse reachable integer whose magnitude is (O(2^k)). This reduction reframes the Collatz conjecture as a finite density and expansion problem in a bounded-hop residue graph, rather than a question of undecidability or halting behavior. All remaining difficulty is isolated to this explicit interval-hit condition.