# Geometric split-count research benchmark Family 183: Power savings for planar halving lines and k-sets. First application triage, 9 October 2026 Australia/Brisbane. ## Problem and potential new use Geometry and learning-theory researchers can compare planar halving pairs and linearly separable k-subsets to new count bounds. ## Applicability and commercial boundary The constants and saving exponents are nonquantitative; no practical classifier runtime improvement follows. The selected scope covers halving pairs and specific level switches rather than all later k-set claims. ## Initial business decision Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company. ## Next verification action Extract a finite counting benchmark without presenting the asymptotic exponent as a measured speedup. ## Evidence scope The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scopes were additionally inspected; explicit effectiveness passages were inspected for 076, 143 and 178. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json). [A power saving for planar halving lines](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf).