Well, essentially anarcho-capitalism
is pure capitalism.
Capitalism = the private ownership of the means of production.
Socialism = the public ownership of the means of production.
Classical Liberalism (what I guess you associate as "pure capitalism", actually isn't.) And part of the reason for Capitalism's decline over the 20th Century / since the American Revolution, is because of democratic nature / tradition of the US (politicians / special interests) which even a Constitutional Republic, and Constitution - a MINIMAL STATE cannot protect against. It has inherent contradictions which itself cannot resolve.
Anarcho-capitalism is the logical extension of Libertarianism. The non aggression axiom (principle) and the Lockean / Rothbardian homesteading private property rights principle. This is applied to EVERYTHING. Others say: radical, I say: 'consistent'. It is the pure capitalism.
Libertarian socialist - is someone who is using the Libertarian in the European label sense of the world. When the socialists stole "Liberal" ie. morphed classical liberal to make liberal mean "socialist", in America - the free market folks where able to obtain the label "Libertarian".
A Libertarian socialist - is a Chomsky supporter. They are completely ignorant of economics, and think the State actually protects private property (lolz, it doesn't - it has to violate private property to exist). Confused and lost.
Hope that helps.
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