The council did not only define. Its twenty-eighth canon restated and extended the claim made at Constantinople in 381, giving the see of the imperial city equal privileges with Rome and second rank after it, and assigning it jurisdiction over three civil dioceses. The papal legates protested, having withdrawn before the session, and Leo refused to confirm it while confirming everything doctrinal the council had done.
What makes this more than an administrative squabble is the reasoning. The canon grounds ecclesiastical rank in the political standing of a city. Rome's own position, on that logic, derived from its having been the seat of empire, and would follow the empire when it moved. The Latin answer was that Roman primacy rests on Peter and is therefore independent of politics entirely. Both positions are visible in the record of this council, held by parties who had just agreed unanimously about Christ.
The pattern is worth naming because it recurs. Churches divide over authority at least as often as over doctrine, and disputes about who may decide are harder to settle than disputes about what is true, since there is no agreed forum in which to settle them. Chalcedon produced the Christology most of the world's Christians confess and, in the same weeks, sharpened the constitutional quarrel that eventually split the church that confessed it.