Surprisingly, there exists a quantum flux-side time-correlation function which has a non-zero t → 0+ limit and thus yields a rigorous quantum generalization of classical transition-state theory (TST). In this Part I of two articles, we introduce the new time-correlation function and derive its t → 0+ limit. The new ingredient is a generalized Kubo transform which allows the flux and side dividing surfaces to be the same function of path-integral space. Choosing this function to be a single point gives a t → 0+ limit which is identical to an expression introduced on heuristic grounds by Wigner in 1932; however, this expression does not give positive-definite quantum statistics, causing it to fail while still in the shallow-tunnelling regime. Positive-definite quantum statistics is obtained only if the dividing surface is invariant to imaginary-time translation, in which case the t → 0+ limit is identical to ring-polymer molecular dynamics (RPMD) TST. The RPMD-TST rate is not a strict upper bound to the exact quantum rate, but is a good approximation to one if real-time coherence effects are small. Part II will show that the RPMD-TST rate is equal to the exact quantum rate in the absence of recrossing.