AI 中文总结
本文解决了7个四元勒让德对的未解决长度，采用确定性中途相遇等方法搜索，验证无需软件，还明确了各主张的支持程度。
AI 中文摘要
长度为ℓ的四元勒让德对是一对取值于四次单位根的序列，其周期自相关在所有非零移位处的和均为-2。每一对这样的序列可生成阶数为2ℓ+2的四元阿达马矩阵，以及阶数为4ℓ+4的二元阿达马矩阵。在Kotsireas、Koutschan和Winterhof以及Jedwab和Pender于2025年的表格中，100以内的14个偶数长度仍未解决。本文解决了其中7个——4个最小的未解决长度42、46、52、58，以及66、72、80——并完整列出每一对。验证一对仅需几行代码和高斯整数上的精确算术，因此无需运行任何软件即可确认主要结果。每次搜索将一对序列在Gray映射下表示为四个二元序列，限制为反射对称行，在中点频率施加精确频谱约束，并对其自相关向量采用中途相遇法匹配候选；从长度52开始，沿除子链的精确压缩会在提升任何候选前缩小搜索空间。所有过程均为确定性：接受决策中不涉及种子、启发式停止规则或浮点值。在长度64、70和76时，相同的机制穷尽运行后未找到任何结果：在每个长度构建的可逆四模块队列中不存在该对。这些是对特定记录队列的穷尽搜索，而非不存在性结果，且证书本身并不能证明该队列捕获了该长度下的所有可逆对象。最后一节按读者为相信论文中的每个主张必须接受的内容对其进行排序，并明确说明随附的人工制品对每个主张的支持程度。
英文摘要
A quaternary Legendre pair of length $\ell$ is a pair of sequences over the fourth roots of unity whose periodic autocorrelations sum to $-2$ at every nonzero shift. Each such pair yields a quaternary Hadamard matrix of order $2\ell+2$ and a binary Hadamard matrix of order $4\ell+4$. Fourteen even lengths up to 100 remained open in the 2025 tables of Kotsireas, Koutschan and Winterhof and of Jedwab and Pender. We settle seven of them --- the four smallest open lengths, 42, 46, 52, and 58, together with 66, 72, and 80 --- and print every pair in full. Checking one takes a few lines of code and exact arithmetic over the Gaussian integers, so the main results can be confirmed without running any of our software. Each search writes a pair as four binary sequences under a Gray map, restricts to reflection-symmetric rows, imposes an exact spectral constraint at the midpoint frequency, and matches candidates meet-in-the-middle on their autocorrelation vectors; from length 52 onward, exact compression along divisor chains shrinks the search before any candidate is lifted. Everything is deterministic: no seed, heuristic stopping rule, or floating-point value enters an acceptance decision. At lengths 64, 70 and 76 the same machinery ran to exhaustion and found nothing: the reversible four-block queue built at each length contains no pair. These are exhaustions of a specific recorded queue, not nonexistence results, and the certificates do not by themselves establish that the queue captures every reversible object of its length. A closing section sorts every claim in the paper by what a reader must accept in order to believe it, and states exactly how far the accompanying artifacts support each one.
Comments14 pages. Paper, source code, witness packages, and queue-exhaustion certificates available at https://doi.org/10.5281/zenodo.21776795
