The Gate of Balance, the Gold Gate, or the Fifth Gate is a gate that all Magisterium students must pass through in order to graduate. It is similar to The Gate of Control, or Prima Materia, that all students must pass through to finish their Iron year.
This gate is about doing good works in the world and genuinely intending to do good. The gate only opens to those who have the intention to do good in the world. Constantine Madden never graduated from the Magisterium and therefore never passed though this gate.
At the end of The Golden Tower Callum Hunt, Aaron Stewart in Alex Strike's body, and Tamara Rajavi pass through the gate and graduate from the Magisterium.
After Master Rufus finds out that Call and Aaron used their makar abilities to push Alex's soul out of his body and replace it with Aaron's, he is uncertain of their intentions. When an assembly meeting is call to decide whether Alex (currently Aaron's soul, but the assembly doesn't know this) should be allowed to go free, Rufus is the tie-breaking vote. He abstains and instead requests to see Call and Aaron go through the Fifth Gate.
Tamara insists that if Call and Aaron must go through this test to prove that they aren't evil, she must take the test too. So, similarly to the end of The Iron Trial, Call, Aaron, and Tamara all walk through the gate successfully at the very end of the book.