归一化梅森基中位置数制完备性的代数证明
Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis
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中文总结 AI 辅助
本文通过分裂恒等式和贪心算法，代数证明了基于归一化梅森基的位置数制对任意自然数具有规范表示的存在性与绝对唯一性，形成封闭无歧义的代数空间。
中文摘要 AI 辅助
本文介绍并研究了一种基于归一化梅森基 $M_k = (2^k-1)/3$ 构造的非经典位置数制，该数制在结构上将经典梅森序列（OEIS \ extbf{A000225}）与交替二进制位掩码整数序列（OEIS \ extbf{A002450}）联系起来。本研究的主要目标是提供该数制完备性的严格代数证明——具体而言，即关于任意自然数规范表示的存在性和绝对唯一性的定理。我们表明，该基的内部动力学由分裂恒等式 $3M_k = M_{k+1} + 2M_{k-1}$ 确定性支配，该恒等式严格将系数的有效字母表限制为集合 $\{0, 1, 2\}$。基于关于低位数字上确界的引理以及构造性贪心算法，证明了所提出的基构成一个封闭且无歧义的代数空间，完全排除了位置质量的交叠。
英文摘要
This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis $M_k = (2^k-1)/3$, structurally linking the classical Mersenne sequence (OEIS \textbf{A000225}) with the integer sequence of alternating binary bitmasks (OEIS \textbf{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.
