The Solvability Complexity Index measures how many layers of approximation a computational problem requires — how many times you must take a limit to converge on the answer. A problem with SCI = 1 is directly computable. SCI = 2 requires one limit. SCI = 3 requires a limit of limits. The index classifies problems by their inherent computational depth.
But the original SCI framework allows unrestricted post-processing of oracle transcripts, and this paper shows that unrestricted post-processing can collapse the hierarchy — making the index incomparable with Weihrauch reducibility, the standard measure of computational difficulty for infinite-input problems.
The fix is to restrict post-processing to regularity classes: continuous, Borel, or Baire functions. Each restriction produces a genuine hierarchy. Borel towers compute exclusively Borel targets. Continuous-base towers yield results in a finite Baire class. The restrictions are not arbitrary — they enforce logical constraints that prevent the pathological collapses.
The resulting Weihrauch-SCI rank measures the minimum number of iterated limit-oracles needed to compute a function, bridging two previously separate theories of computational difficulty. The authors demonstrate the framework on a Koopman operator example, showing that spectral computation for infinite-dimensional dynamical systems fits naturally into the hierarchy.
The lesson: a complexity measure is only meaningful if its allowed operations are constrained enough to produce a non-trivial ordering. Unconstrained computation is too powerful — it trivializes distinctions that matter. The hierarchy is in the restrictions, not the resources.