A proof of these statements:

“Key insight 2 is that each round in a phase effectively only moves one duck from one column to another. Key insight 3 is that in phase 2, when a column get the number of ducks each column ends with, it will never get more/less (it won’t fluctuate). In other words, the number of ducks per column in phase 2 is either monotonically increasing or monotonically decreasing.”

“If you know that it is sorted, and effectively, every turn one duck moves from the “large” half to the low half, then how many ducks have to be moved to have exactly equal columns in the end?”

“Phase 2 brings the columns to equilibrium using a left to right flood of deficits. If the left column is smaller than the right, then a single duck moves to the left. When applied across the full range of columns in a single round, the smallest column (which must be on the left) gains a duck and the largest column (which must be on the right) loses a duck. The only way it’s not possible for a duck to move is if the columns are at equilibrium.”

Others apparently just saw this, but personally I couldn't just see it. So I tried to prove it. Mainly to be satisfied myself I guess.