This paper proves that positional weight systems under balanced binary factorization admit a unique center-dominant architecture for odd n, and that no such architecture exists for even n. The proof is elementary but the result appears to be new: the structure has a known physical realization (n = 3, k = 4) that has been studied for sixty years without this observation. The constrained Gray code framework extends beyond the conditions addressed by Killian-Savage (2004) and connects to the Mutze survey (2023, E-JC DS26). AI tools were used for editorial assistance; all proofs are the author's own.
Bernhard Pfennigschmidt (Fri,) studied this question.