There's also inherent tyranny-of-structurelessness issues that will arise from abolition of private property whereby certain members of an outwardly democratic cooperative will have more influence and significance than others, but whose de facto "ownership" will have no de jure significance, thus being easily denied and also inefficient since the law will not clearly delineate property lines when everything is meant to be "common".

So it's not the tenth amendment that conflicts with socialism? It's a variety of other laws and policies.

Yet you say that "there is absolutely zero compatibility between Socialism and the US Constitution", so perhaps you're thinking of one of the other amendments or some such? Perhaps something involving the protections on property.