Proof of the TuDeng Conjecture

Abstract

We give a complete proof of the 2011 Tu--Deng conjecture. We begin from its original modular pair-count formulation, prove an equivalent cyclic Hamming weight-drop formulation, and establish the exact transfer identity that connects this count with a two-variable matrix polynomial. The proof then reduces the conjecture to normalized inequalities for the coefficients of that polynomial. A 2011 conjecture by the author which came to be called the Cusick Conjecture (it is a consequence of the Tu--Deng Conjecture) was proved by K. Cheng in 2026. The proof in the present paper extends the cyclic deletion ideas of Cheng. The new ideas might be applicable to other problems.