Most hard problems are not actually hard in their components. They are hard because their components have been allowed to merge into a single tangled statement, and the tangling hides the structure. Decomposition is the operation that splits the tangled statement back into a graph of independent sub-problems, each with a single well-typed output and a clearly stated domain of validity. Once the decomposition is clean, most of the difficulty has already been done.
The rule the engine enforces during decomposition is non-leakage. Each sub-problem must be solvable in isolation; if a sub-problem requires the answer to another sub-problem, that dependency is recorded as an explicit edge in the graph rather than smuggled through shared assumptions. This is the structural property that makes independent verification possible later. Without it, two paths through the graph share hidden state, and agreement between them is no longer evidence of correctness.
The payoff is asymmetric. A clean decomposition turns a single intimidating question into a list of small questions, most of which have known closed-form answers. The remaining few are exposed clearly, and the engine spends its synthesis budget on those rather than on the now-trivial majority. This is why a well-decomposed Quantm derivation often looks shorter than one would expect — most of the work has been pushed out of the surface and into the structure, where it belongs.