(Abridged) The explosion mechanism of electron-capture supernovae (ECSNe) remains equivocal. We attempt to constrain the explosion mechanism (neutron-star-forming implosion or thermonuclear explosion) and the frequency of occurrence of ECSNe using nucleosynthesis simulations of the latter scenario, population synthesis, the solar abundance distribution, pre-solar meteoritic oxide grain isotopic ratio measurements and the white dwarf mass-radius relation. Tracer particles from 3d hydrodynamic simulations were post-processed with a large nuclear reaction network in order to determine the complete compositional state of the bound ONeFe remnant and the ejecta, and population synthesis simulations were performed in order to estimate the ECSN rate with respect to the CCSN rate. The 3d deflagration simulations drastically overproduce the neutron-rich isotopes $^{48}$Ca, $^{50}$Ti, $^{54}$Cr, $^{60}$Fe and several of the Zn isotopes relative to their solar abundances. Using the solar abundance distribution as our constraint, we place an upper limit on the frequency of thermonuclear ECSNe as 1$-$3~% the frequency at which core-collapse supernovae (FeCCSNe) occur. This is on par with or 1~dex lower than the estimates for ECSNe from single stars. The upper limit from the yields is also in relatively good agreement with the predictions from our population synthesis simulations. The $^{54}$Cr/$^{52}$Cr and $^{50}$Ti/$^{48}$Ti isotopic ratios in the ejecta are a near-perfect match with recent measurements of extreme pre-solar meteoritc oxide grains, and $^{53}$Cr/$^{52}$Cr can also be matched if the ejecta condenses before mixing with the interstellar medium. Theoretical mass-radius relations for the bound ONeFe WD remnants of these explosions are apparently consistent with several observational WD candidates.
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