A geometric point of view on the synchronization of three Christoffel words

Let G = ( g 1 , … , g ℓ ) be a vector of positive integers and set n = ∑ i = 1 ℓ g i . Writing Ch ( a , b ) for the Christoffel word with parameters ( a , b ) , we study the following synchronization problem: choose one conjugate of each word Ch ( g i , n − g i ) so that, at every position 0 , … , n − 1 , exactly one of the chosen words contains the letter 1. This gives a gap-free form of superimposition, motivated by Fraenkel’s conjecture. We encode the choice of conjugates by a shift vector V = ( v 1 , … , v ℓ ) . On the cyclic Cayley graph of Z / n Z , this yields an orbital matrix O ( G , V ) , with entries o i , j = ( v i + j g i ) mod n , and a Christoffel–conjugate matrix C ( G , V ) , which records the columns where a wraparound occurs. The vector V is a synchronizing seed precisely when each column of C ( G , V ) contains exactly one nonzero entry. The main algebraic tool is a vertical invariant: the column sums of O ( G , V ) are constant if and only if this one-wraparound-per-column condition holds. Using this criterion, we give explicit synchronizing seeds for every pair of Christoffel words of common length, for triples in which two generators coincide, for triples with three equal generators, and for the all-distinct family proportional to ( 1,2,4 ) . Finally, we give a geometric interpretation: each synchronized row is the Freeman chain code of a standard 4-connected Réveillès segment, whose parameter μ i is the unique integer in { 1 − n , … , 0 } satisfying μ i ≡ − v i ( mod n ) .