安全数组分割的范围代数 (Range Algebra for Safe Array Splits)

from arxiv, After I compressed the previous, much longer version, this one became an array of tautologies, which is difficult to justify as an article in any sense. The value is no different from a Boolean algebra, which is nonsensical to restate these days. It was even difficult to actually write in LaTeX without repeating the same symbols. It is not working as an article

We present a language-agnostic range algebra that derives correct index intervals for splitting arrays and implementing binary search, eliminating off-by-one and empty-range bugs. From two primitives -- $\lfloor n/2\rfloor$ and $\lceil n/2\rceil = \lfloor (n + 1)/2\rfloor$ -- we obtain four canonical splits (Natural, Left+, Right+, Center-cut) with proofs of coverage and balance for all $n \ge 0$ and any base index $b \in \mathbb{Z}$. We record invariants (normalization to left-closed/right-open intervals $[u, w)$ with len $= w - u$), document cross-language division quirks, and give drop-in code patterns and property tests. The result is a small, memorable spec that can be copied verbatim into C/C++/Java/C\#/Go/Rust/Swift/JS/Python.

翻译：我们提出了一种与语言无关的范围代数，用于推导分割数组和实现二分查找的正确索引区间，从而消除差一错误和空范围错误。从两个基本操作——$\lfloor n/2\rfloor$ 和 $\lceil n/2\rceil = \lfloor (n + 1)/2\rfloor$——出发，我们推导出四种规范分割方式（自然分割、左偏分割、右偏分割、中心切割），并证明了对于所有 $n \ge 0$ 及任意基索引 $b \in \mathbb{Z}$ 的覆盖性和平衡性。我们记录了不变性（归一化为左闭右开区间 $[u, w)$，其长度 $= w - u$），整理了跨语言除法运算的特性差异，并提供了可直接使用的代码模式和属性测试。最终成果是一个简洁易记的规范，可直接复制到 C/C++/Java/C\#/Go/Rust/Swift/JS/Python 等语言中使用。