Level, Before Shape: Landscape-Dependent Control of the Discovery Rate in Cuckoo Search

The discovery rate pₐ governs diversification in Cuckoo Search (CS) and is conventionally set to 0.25. Its optimal value depends on the landscape; its schedule shape does not. Holding shape constant and sweeping pₐ over 51 replications, median error rises monotonically with pₐ on smooth unimodal functions. Still, it falls monotonically on the rugged multimodal Michalewicz function, improving sixteen-fold across the range tested. Conversely, fixing the arithmetic run-average and varying only shape across six schedules separates them over twenty benchmark functions — but not because shape matters: rank tracks the geometric mean of the realised rate (r = 0.891) and time spent below 0.10 (r = −0.869), not direction of change. Renormalising on the geometric mean reverses the ordering, moving the cubic schedule from first of six to last with its shape unchanged. The discovery rate acts through where the search operates, not the trajectory taken, and no single-moment normalisation separates shape from level. These results reinterpret Mareli and Twala (2018), which concluded that an exponentially increasing schedule improves CS. We identify two defects: a criterion that ranks by dispersion rather than quality, and a specification that lets the exponential arm run at a mean pₐ of 0.430 against a nominal cap of 0.25. Under a corrected criterion, that arm ranks last, and a constant schedule at matched level exceeds it. We propose CSAD, a stagnation-adaptive scheme that raises pₐ when the incumbent stalls and decays it when refinement pays; it is competitive at d = 10 but the worst arm at higher dimensions, a failure we diagnose rather than omit. Instrumentation shows the natural explanation—that higher pₐ buys escapes from local optima—is only half correct: it accounts for precision lost on smooth landscapes but not the gain on rugged ones, where diversity and relocations both fall as pₐ rises. Fitness-distance correlation is likewise uncorrelated with its direction, so the effect is reported as a regularity whose mechanism remains open.