Serial self-terminating search benchmark ($E[T_{\mathrm{present}}]=b+\tau (N+1)/2,E[T_{\mathrm{absent}}]=b+N\tau$)

Assume $N$ candidate items are examined once each, each costs time $\tau$, nonsearch processing costs $b$, and a uniformly positioned target ends search when encountered. The target rank has mean $N^{-1}{\sum }_{k=1}^{N}k=(N+1)/2$, proving the present-trial expression. Absent trials inspect all $N$ items. The resulting 2:1 slope ratio is a benchmark under these assumptions, not a unique diagnostic of a serial mechanism.