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 -- $\lfloor n/2\rfloor$ 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.