P92 · 50 circles in a unit circle · Σr = 6.538814808
Place 50 non-overlapping circles of any sizes inside a circle of radius 1 and make the sum of their radii as large as possible. This is the "Circles in Circles" row of Erich Friedman's tables for packing copies to maximise total perimeter, and this is week 5's challenge on the site.
The published value moved twice in a week. Jonathan Viquerat's 6.53486+ stood when the week opened. On October 1 Friedman replaced it with Joost de Winter's 6.53814+. On September 30 Tmrg had already submitted 6.538814808, which is 0.000665 above the far edge of that newer interval [6.53814, 6.53815).
Why it counts as new. The site rebuilt Joost de Winter's published figure and optimised it with no cap: it stops at 6.538140455. The site's earlier certified layout for this row, by 今天, reaches 6.53847458. Tmrg's layout is a third, different arrangement. The verifier checks containment and every pair in exact integers on the nine-decimal grid.
Two days later the same layout came back from another account, every circle moved by about a billionth. That is a copy of a public record, not a construction, and it was not certified.
P61 · 11 lines in ℂ⁴ · coherence 0.425147038
Choose 11 unit vectors in four-dimensional complex space so that the largest |⟨u, v⟩| over all 55 pairs is as small as possible: 11 lines that stay as far apart as they can. The public benchmark is the Game of Sloanes table, 0.42514727 to eight decimals, so anything below 0.425147265 is ahead of it whatever the unprinted digits are.
zzzcy's vectors reach 0.425147038, 2.3 × 10⁻⁷ below that edge. They are nearly equiangular: 52 of the 55 pairs sit within a millionth of the maximum, and the remaining three are 0.2939, 0.2983 and 0.3691.
Two earlier site layouts were certified here on September 5, at 0.425147173 and 0.425147208. Sorting all 55 inner products, those two differ from each other by 3.4 × 10⁻⁷ and from zzzcy's by 4.4 × 10⁻⁵: a distinct construction, not a refinement of theirs. The verifier recomputes the projectors from the submitted vectors in exact rational arithmetic and rounds against the submitter.
What a certification says
It says the construction is valid and better than every value published when it was reviewed. It is not a proof of optimality, and it does not claim the construction was never found elsewhere. Under the scoring rules a certified player earns the +3 world-first bonus while holding the site record and staying ahead of the published value; the review record keeps the submission and a hash of the exact answer either way.