Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis

This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis $M_k=(2^k-1)\mathrm{/}3$, structurally linking the classical Mersenne sequence (OEIS $\mathbf{\text{A000225}}$) with the integer sequence of alternating binary bitmasks (OEIS $\mathbf{\text{A002450}}$). The main objective of this study is to provide a rigorous algebraic proof of the system’s completeness—specifically, theorems on the existence and absolute uniqueness of a canonical representation for any natural number. We show that the internal dynamics of the basis are deterministically governed by the splitting identity $3M_k=M_{k+1}+2M_{k-1}$, which strictly restricts the valid alphabet of coefficients to the set $\{0,1,2\}$. Based on the lemma concerning the supremum of lower-order digits and a constructive greedy algorithm, it is proven that the proposed basis forms a closed and unambiguous algebraic space, entirely precluding the overlap of positional masses.