The term GEESE COMPLEMENT IN A CAROL is referring to the six-letter word 'swans'. Swans are often featured in carols, such as the traditional English carol 'The Twelve Days of Christmas', which includes the line 'five golden rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree'. The carol also mentions 'six geese a-laying', making the swans a complement to the geese.