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When every attempt of the structurizer ladder refuses a graph, the title aborts. Rebuild such a graph from a hierarchical decomposition into strongly connected components instead: each original block keeps one copy and runs behind a guard on its own reach flag, and each component becomes a loop with a single latch that repeats it while a back edge set its again flag. Every edge sets the reach flag of its target, so exactly one flag names the next original block and the blocks run in the input's order for every branch decision. Irreducible and non-terminating graphs are handled the same way, and no block is cloned. The ladder runs first and its output is unchanged. The fallback only runs when both attempt series refuse; if it cannot structure the graph, the ladder's error is reported as before. Blocks that start the top-level order unconditionally run unguarded, so the entry block's results reach later blocks without a stale path. APS5_FORCE_STRUCTURIZER_FALLBACK=1 uses the construction for every supported graph, for tests. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>