Exploring Collatz Dynamics with Human-LLM Collaboration

arXiv:2603.11066v4 Announce Type: replace-cross Abstract: We develop a structural framework for the Collatz map based on odd-to-odd dynamics, modular return structure, and a decomposition of trajectories into bursts and gaps. On the unconditional side, we prove several exact results. The fiber-57 branch q = 7 (mod 8) returns in exactly two odd-to-odd steps with uniform affine destination. The branch q = 3 (mod 8) cannot return within four steps (minimum gap five), and its earliest returns form an explicit dyadic cylinder family indexed by w = v_2(243m+119). The algebraic chain map on the five-element invariant core is a permutation at every depth, so any genuine contraction must come from return dynamics rather than core algebra. These yield an exact depth-2 known-gap partial return kernel with Perron root 129/1024 -- not asserted as the full bottleneck constant, since q = 3 (mod 8) returns with gap >= 6 are unresolved. The main body independently develops a conditional reduction via burst-gap decomposition, phantom-cycle gain analysis, and a weak-mixing hierarchy, establishing an exact geometric block law, exponential almost-all crossing bound, and per-orbit phantom gain within budget (4.65x margin). The framework reduces the convergence programme to a single orbitwise regularity statement, formulated either through the weak-mixing hierarchy or the fiber-57 anti-concentration conjecture. The remaining obstruction is to prove that no deterministic orbit can concentrate its fiber-57 returns on the sustaining core strongly enough to maintain indefinite non-termination. This work is not a complete proof of the Collatz conjecture. It is a sharpened reduction isolating the unresolved difficulty to a single orbitwise upgrade from ensemble behavior to pointwise control, concentrated in the q = 3 (mod 8) return channel.