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The LiH$^{+}$ molecule is prototypical of the indirect dissociative recombination (DR) process, in which a colliding electron destroys the molecule through Rydberg capture pathways. This Letter develops the first quantitative test of the Siegert state multichannel quantum defect theory description of indirect DR for a diatomic molecular ion. The R-matrix approach is adopted to calculate ab-initio quantum defects, functions of the internuclear distance that characterize both Rydberg states and the zero-energy collisions of electrons with LiH$^{+}$ ions. The calculated DR rate coefficient agrees accurately with recent experimental data (S. Krohn et al, Phys. Rev. Lett. 86, 4005). We identify the doorways to fast indirect DR as complex resonance manifolds, which couple closed channels having both high and low principal quantum numbers. This sheds new light on the competition between direct and indirect DR pathways, and suggests the reason why previous theory underestimated the DR rate by an order of magnitude., Comment: Submitted to PRL