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Abstract: We argue that for any single-trace operator in N=4 SYM theory, there is a large twist double-scaling limit in which the Feynman graphs have an iterative structure. Such structure can be recast using a graph-building operator. Generically, this operator mixes between single-trace operators with different scaling limits. The mixing captures both the finite coupling spectrum and corrections away from the large twist limit. The picture that emerges from this work opens the door to a systematic expansion of N=4 SYM theory around the large twist limit.