Range Algebra for Safe Array Splits

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 – $\lfloor n\mathrm{/}2\rfloor$ and $\lceil n\mathrm{/}2\rceil =\lfloor (n+1)\mathrm{/}2\rfloor$ – 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.