In most cases, if you own Green Sun's Zenith and cast it, you'll shuffle your library twice. In practice, shuffling once is sufficient, but effects that care about you shuffling your library (like Psychogenic Probe, for example) will see that you've shuffled twice.
If Green Sun's Zenith is countered, none of its effects will happen. Notably, it will be put into its owner's graveyard rather than shuffled into its owner's library.
If you own Green Sun's Zenith, but an opponent casts it (due to Knowledge Pool's effect, for example), that opponent searches their library for an appropriate creature card, then shuffles that library. That opponent then shuffles Green Sun's Zenith into your library. You won't shuffle any library in this case.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. David Rapoza
Card data via Scryfall. Rarix is fan-made; not endorsed by the rights holders.
If this spell doesn't resolve, none of its effects occur. In particular, it will go to the graveyard rather than to its owner's library.
