We present an Atacama Large Millimeter/submillimeter Array (ALMA) 106 GHz (Band 3) continuum survey of the complete population of dense cores in the Chamaeleon I molecular cloud. We detect a total of 24 continuum sources in 19 different target fields. All previously known Class 0 and Class I protostars in Chamaeleon I are detected, whereas all of the 56 starless cores in our sample are undetected. We show that the Spitzer+Herschel census of protostars in Chamaeleon I is complete, with the rate at which protostellar cores have been misclassified as starless cores calculated as <1/56, or < 2%. We use synthetic observations to show that starless cores collapsing following the turbulent fragmentation scenario are detectable by our ALMA observations when their central densities exceed ~10^8 cm^-3, with the exact density dependent on the viewing geometry. Bonnor-Ebert spheres, on the other hand, remain undetected to central densities at least as high as 10^10 cm^-3. Our starless core non-detections are used to infer that either the star formation rate is declining in Chamaeleon I and most of the starless cores are not collapsing, matching the findings of previous studies, or that the evolution of starless cores are more accurately described by models that develop less substructure than predicted by the turbulent fragmentation scenario, such as Bonnor-Ebert spheres. We outline future work necessary to distinguish between these two possibilities.
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