"Brinking on" implies moving towards an event or a situation, often with a sense of anticipation. Antonyms for this phrase could include "retracting from," "avoiding," or "discouraging." To retract from something would mean to move away from it or to decide against pursuing it further. Choosing to avoid something altogether would mean not engaging in it at all. Additionally, discouraging would entail actively working against something or someone, dissuading any potential for the situation even arising. Overall, while "brinking on" carries a certain energy of excitement and progress, its antonyms suggest a more reserved, cautious, or even oppositional mindset.