Why doesn't H₄CO₄ exist as a stable compound? Why doesn't $\ce{H4CO4}$ exist as a relatively stable compound? Its heavier homologue $\ce{H4SiO4}$ seems to exist although it is unstable to some extent but new methods have been able to stabilise orthosilicic acid using choline chloride.
Wikipedia reports that calculations show orthocarbonic acid spontaneously decomposes to the ordinary (meta)carbonic acid $\ce{H2CO3}$ plus water.  However, orthocarbonate esters $\ce{C(OR)4}$ do exist.

The orthoacid may, however, become stable at high pressure [1, with a nontechnical summary given in 2] and possibly exist within Uranus and Neptune.  It is further posited that $\ce{H2CO3}$, ordinarily stable only in aqueous solution, becomes a stable solid in the interiors of icy moons such as Europa, possibly impacting interactions thought to be necessary for the development of subterranean life.

A final note: the swastika-like shape of the orthocarbonic acid molecule has led unfortunately to the name "Hitler's acid".  Actually the swastika long predates the Nazis and retains non-Nazi usages to this day.