来自原始戴克单词的加法基：正则下近似、莫兹金编码和数字提升
Additive Bases from Primitive Dyck Words: Regular Underapproximations, Motzkin Coding, and Digit Lifting
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中文总结 AI 辅助
研究由规范二进制展开为原始戴克单词的整数构成的加法表示，通过配对连续位揭示关系，证明区间数字提升定理和算法，结合工具对正偶数分类，得出848是最终阈值且界限渐近最优。
中文摘要 AI 辅助
我们研究了由其规范二进制展开为原始戴克单词的整数构成的加法表示。将连续位配对可得到戴克路径与双色莫兹金路径之间经典关系的一种位置形式：除了10之外，原始戴克单词恰好是4进制单词3w0的二进制块图像，其中w是一个双色莫兹金单词。这揭示了正则下近似、数字闭包和精确的生成界限。我们证明了关于数字闭集的区间数字提升定理以及一种构造性的4进制传播算法，该算法能在对数级递归阶段中将有限和集证书提升到无限尾部。结合这些工具与精确的有限证书和生成间隙下界，我们对所有需要超过六个原始戴克求和项的正偶数进行了分类。整数46需要八个，34、44、98、154、198、202、206、838、842和846需要七个；其他正偶数最多需要六个。因此848是精确的最终阈值。该界限是渐近最优的，因为对于每个k >= 2，10 * 4 ^(k + 1)-6需要六个求和项。相关的减半族具有精确的渐近加法阶数五。补充程序使用精确整数运算重现所有有限证书。
英文摘要
We study additive representations by integers whose canonical binary expansions are primitive Dyck words. Pairing consecutive bits yields a positional form of the classical relation between Dyck paths and two-colored Motzkin paths: except for 10, primitive Dyck words are exactly the binary block images of base-4 words 3w0, where w is a two-colored Motzkin word. This exposes a regular underapproximation, digit closure, and sharp generation bounds. We prove an interval digit-lifting theorem for digitally closed sets and a constructive base-4 propagation algorithm that lifts finite sumset certificates to infinite tails in logarithmically many recursive stages. Combining these tools with exact finite certificates and generation-gap lower bounds, we classify all positive even integers requiring more than six primitive Dyck summands. The integer 46 requires eight, and 34, 44, 98, 154, 198, 202, 206, 838, 842, and 846 require seven; every other positive even integer requires at most six. Thus 848 is the sharp eventual threshold. The bound is asymptotically optimal because 10*4^(k+1)-6 requires six summands for every k >= 2. The associated halved family has exact asymptotic additive order five. Supplementary programs reproduce all finite certificates using exact integer arithmetic.
