Constraints on the threshold K^- nuclear potential from FINUDA ^AZ(K^-stop,Π^-)ZλA spectra

1sλ hypernuclear formation rates in stopped K- reactions on several p-shell targets are derived from hypernuclear formation spectra measured recently by the FINUDA Collaboration and are compared with calculated 1sλ formation rates based on a chirally motivated coupled channel model. The calculated rates are about 15% of the derived rates, and in contrast with previous calculations depend weakly on the depth of the threshold K- nuclear potential. The A dependence of the calculated 1sΛ rates is in fair agreement with that of the derived 1sΛ rates, showing a slight preference for a deep density dependent potential, ReV_K-(Ρ0)∼-(150-200) MeV, over a shallow potential, ReV_k-(Ρ0)∼-50 MeV. These new features originate from a substantial energy and density dependence found for the in-medium subthreshold K^-n→Π^-Λ branching ratio that enters the hypernuclear formation rate calculations.