Finiteness Theorems for Weak Chains in Syracuse Orbits

We study the Syracuse map and introduce a parameter-chain induction framework for analysing its orbits. We rigorously prove: (1) explicit formulas for the first m-1 iterates of a parameter-even integer (Lemma B); (2) weak chains cannot continue indefinitely (Proposition L), proved via an integrality constraint on the recurrence x_{k+1}=(3x_k+1)/2; (3) a sharp upper bound L <= delta(x)-1 for weak-chain length; (4) a complete mod 8 classification of the 2-adic valuation of 3^m q - 1; (5) no orbit stays forever in [n, 3n/2). We supplement these with extensive analysis: ergodic and additive-combinatorial perspectives, a study of the parameter-even network showing all tested chains eventually leave the network and descend, and an entropy-drift model revealing that delta-values experience strong negative drift with mean change -9.0 per cycle. These results partially address a question raised in recent work of Chang (2026). A precise remaining open problem (Proposition F) is identified. All theorems are verified computationally for m <= 20, odd q <= 200 (1,900 cases).