One pipeline was used for every family. A population stage runs many candidate layouts at once on a smooth stand-in for the objective (a soft-max or soft-min with a cyclically annealed temperature), so that every nearly-binding constraint contributes to the gradient. A finisher then takes exact trust-region steps: the branches that can become binding inside the trust radius are linearized and the step is the solution of a small linear program (HiGHS); after every step the branches are rediscovered from scratch and the step is accepted only on the rediscovered objective. Finally an independent certificate decides what counts: interval branch-and-bound over the whole square for the lights, exact rational distances for the ratio problems, whole-domain critical points and exact rational provers for the coverings, exact decimal predicates for the packings.
The lights problem is where the pipeline produced most of its gains (24 of the 35 open counts). The other families had been searched far harder by others, and there the pipeline mostly reproduces the listed values; the three exceptions are circles in the L-shaped container at n = 44, 45 and 46.
The paper with the mathematics, the algorithms, the negative results and full-precision coordinates is here as a PDF. Coordinates of every beaten case can be opened from its card.
The trailing plus on the page's values means "and more digits". A claim is only made when our certified value clears the displayed digits by at least one unit of the last decimal place, which is safe whether the display was truncated or rounded.
Data as of 2026-09-12 05:21 UTC. Every value is certified by an independent validator; none is a claim of optimality. Nothing has been submitted to the page.