AI 中文总结
该研究提出镜像定理，定义基思镜像子类，证明中心 deficit 与区间距离的关系，将分类简化为有限情况，精确计数得35个高阶镜像、2个无限族，还分类8个跨基三重对齐并记录条件重构性质。
AI 中文摘要
基思数是指那些会在由自身数字生成的滑动窗口递推中重现的整数。我们引入基思镜像这一子类，其定义为在递推的入射轮廓中心处达到饱和。对于偶数索引m，令C_d(m)表示两个中心入射计数的和。我们证明中心 deficit m-C_d(m)恰好是m到区间[2d,2d+2]的距离，这使得基B≥3下阶数d≥3的每个基思镜像都在2d+2处返回。一个数字恒等式和解析容量界随后将分类问题简化为有限多种情况。通过精确计数，经与原始递推和入射定义独立验证，恰好得到35个阶数至少为3的镜像，而阶数2包含两个明确的无限族。我们还精确分类了8个满足R=C=x₁+x_d的跨基三重对齐，并记录了由入射对称性诱导的条件重构性质。
英文摘要
Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.
Comments20 pages. Exact finite verifier and captured output included as ancillary files
