The guard is surprised by the question—apparently most people just sort of assume it's in the tower. But they're happy to point at a building right by the plaza that's—not exactly unassuming, per se, but doesn't stand out overmuch, and is downright humble compared to the tower.

Yeah he had a feeling. Apparently most non-mages are under the misapprehension that "mage" and "wizard" are synonyms—Vallynn himself had been, when he arrived at the Academy—but it turns out that there's a formal difference, at least when it comes to the Guild and most importantly to the Eden Group classifications. Anyone who shows enough aptitude for and interest in magic can join the Mages' Guild, but being a wizard is more of a rank. You need to show skill with at least five out of the seven standard elements (Neutral, Fire, Earth, Wind, Water, Shadow, and Light) (Life and Death are typically not seen as "standard" though lots of spells taught by the Church also have those elements, and some people consider Poison an element), and you need to be proficient with at least two of them.

Eden in particular is also interested in that. In addition to their ranking system they have a horizontal progression system called "titles" or "specialisations", which are used for more tailored bounties and missions. Sometimes something needs a Fire Wizard specifically, so if you are a wizard and you have a proficiency in Fire with the Mages' Guild you can apply for the Fire Wizard spec with Eden.

In any case, since those two words aren't synonyms and the tower is always called the Wizard Tower, Vallynn concluded that probably the Mages' Guild might not use it as its main headquarters, or at least as the place they welcome new recruits at.

Into the building he goes.