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A complete statistic T (in some statistical model) is such that for all functions g, if E g(T)=0 for all parameter values, then g is identically zero.

For more information see https://en.wikipedia.org/wiki/Completeness_(statistics) and for some intuition and examples see What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it? and links therein