How can "brink" be "eve" - even chronologically it doesn't really work - is there any chance it could be "e**e" in which case EDGE is much more likely ?
What about the record "Eve of Destruction" in that context it is on the Brink of Destruction. Brink very definitely can be eve. Marpycoo.
from Thesaurus.com
Main Entry: eve: Part of Speech: noun: Definition: night before: Synonyms: brink, evening, evening before, night, threshold, verge, vigil