P67 variable circles in a fixed-perimeter rectangle: n = 3–9
A complete interval branch-and-exclude over every contact order leaves a single contact structure, fixed by interval Newton, which gives the continuous optimum; n = 3, 4, 5, 7 and 8 have closed forms in radicals. A branch tree over integer offsets shows the current records are the largest nine-decimal sums of radii.
P63 optimal quadrature on the torus: n = 8, 13, 21, 34
An auxiliary function with nonnegative coefficients gives the continuous lower bound, attained by a Fibonacci-type lattice. At n = 8 the lattice lies on the grid; for the other three it does not, and localisation plus a complete integer enumeration show the current records are the grid minima.
P65 erasure-robust measurement directions: four rows
For d = 3 the icosahedron’s and dodecahedron’s diagonals are optimal at n = 6 and 10; at d = 4, n = 8 the optimum is the square of a root of an octic. These three are continuous optima. For d = 3, n = 8 the proof is about the nine-decimal grid: the current record is the largest score.
P60 sixteen lines in five dimensions
Rankin’s bound gives coherence at least 1/√5, which real configurations attain; the proof shows rational coordinates cannot, so the current record is the optimal grid score.
P61 five complex lines in three dimensions
Five complex lines have coherence at least (√13 − 1)/6, attained by the canonical code in the Game of Sloanes archive; the core of the proof is a two-million-node Bernstein certificate covering the whole phase domain.
Submitting a proof
Post a proof in the problem’s discussion, or email a larger computer-assisted one. Each adopted new argument earns its contributor one +2 proof award.