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Restart is a general framework, of prime importance and wide applicability, for expediting first-passage times and completion times of general stochastic processes. Restart protocols can use either deterministic or stochastic timers. Restart protocols with deterministic timers -- "sharp restart" -- assume a principal role: if there exists a restart protocol that improves mean-performance, then there exists a sharp-restart protocol that performs as good or better. This paper, the first of a duo, presents a comprehensive mean-performance analysis of sharp restart. Using statistical methods, the analysis establishes universal criteria that determine when sharp restart improves or worsens mean-performance, i.e., decreases or increases mean first-passage/completion times. These criteria are akin to those recently discovered for the most widely applied restart protocols -- "exponential restart" -- which use exponentially-distributed timers. However, while the exponential-restart criteria cover only the case of slow timers, the sharp-restart criteria established here further cover the cases of fast, critical, and general timers; moreover, the latter criteria address the very existence of timers with which sharp restart improves or worsens mean-performance. Using the slow-timers criteria, we discover a general scenario for which: sharp restart improves mean-performance, whereas exponential restart worsens mean-performance. The potency of the novel results presented here is demonstrated by examples, and by the results' application to canonical diffusion processes.