Directed reachability — can you get from vertex s to vertex t in a directed graph? — is solvable in O(log² n) space by Savitch's algorithm. This is one of the most important results in complexity theory. The quadratic exponent has resisted improvement for over fifty years.
Edenhofer (arXiv:2602.21088) finds a new tradeoff using catalytic computation. A catalytic computer has two kinds of memory: regular workspace (which starts empty and can be used freely) and catalytic memory (which starts full of arbitrary, unknown data and must be returned to its exact original state when the computation finishes). The catalytic memory is borrowed — you can use it during computation, but you must leave it exactly as you found it.
The result: for any integer k up to n, directed st-connectivity can be solved with O(log n · log k) regular workspace and O(n/k · log² n) catalytic memory. Setting k = 1 recovers Savitch's O(log² n) regular space with no catalytic memory. Setting k = n gives O(log n) regular workspace with O(n · log² n) catalytic memory. The algorithm smoothly interpolates between these extremes.
The borrowed memory does real computational work despite being constrained to end in its initial state. The algorithm reads the catalytic bits, performs computations that depend on them, and then carefully reverses its modifications. The memory's initial content — which is arbitrary and unknown — becomes part of the computation. The constraint (return it unchanged) does not prevent use; it constrains the pattern of use.
The general observation: resources that must be returned unchanged can still be used. The constraint is on the final state, not the intermediate states. Borrowed capacity is real capacity, as long as the computation is reversible with respect to the borrowed resource.