Frontiers

For each open problem we follow, the dated bounds: lower bound, upper bound, published points and proposals.

• Planar unit distances: the final grid of OpenAI's argument
  • Upper bound : 32 (Oct 10, 2026)
• 21 circles in a rectangle: lower bound on the maximal sum of radii
  • Lower bound : ≈ 2.36583 (Aug 30, 2026)
• Edges versus triangles: lower bound on the envelope-curve score
  • Lower bound : ≈ -0.711709 (Jun 27, 2026)
• Erdős minimum overlap problem: upper bound on the constant C5
  • Upper bound : ≈ 0.380859 (Aug 15, 2026)
• First autocorrelation inequality: upper bound on the constant C1
  • Upper bound : ≈ 1.50274 (Oct 3, 2026)
• Flat polynomials of degree 69: upper bound on the minimum of C+
  • Upper bound : ≈ 1.28073 (Aug 10, 2026)
• Heilbronn problem for convex regions, 14 points: lower bound on the minimum triangle area
  • Lower bound : 0.0278355805 (Apr 2, 2026)
• 12 unit hexagons in a regular hexagon: upper bound on the minimal side
  • Upper bound : 3.9416523 (Apr 2, 2026)
• Discretised Kakeya needle (n=128): upper bound on the minimal area
  • Upper bound : ≈ 0.106754 (Oct 6, 2026)
• Max/min distance ratio of 16 points in the plane: upper bound on the minimum of R
  • Upper bound : 12.889230 (Mar 19, 2026)
• Prime number theorem: score S(f) of the constructive certificate
  • Lower bound : ≈ 0.997657 (Aug 15, 2026)
• Ring loading, 15 pairs: lower bound on the score A(u,v)
  • Lower bound : 1.125 (Sep 12, 2026)
• Second autocorrelation inequality: lower bound on the constant C2
  • Lower bound : ≈ 0.963588 (Jul 22, 2026)
• Tammes problem, 50 points on the sphere: lower bound on the maximal minimum distance
  • Lower bound : 0.5134721 (Apr 1, 2026)
• Third autocorrelation inequality (signed function): upper bound on the constant C3
  • Upper bound : ≈ 1.44912 (Oct 6, 2026)
• Binary two-deletion-correcting code, length 16: number of codewords
  • Lower bound : 219 (Oct 5, 2026)
  • Upper bound : 487 (Jun 5, 2026)
• Exact discrete Fourier transform below n log n: the exponent saving δ
  • Lower bound : 0.0012331685 (Oct 12, 2026)
• Kissing number in dimension 11
  • Lower bound : 604 (Apr 13, 2026)
  • Upper bound : 868 (Feb 24, 2022)
• Kissing number in dimension 18
  • Lower bound : 8358 (Sep 16, 2026)
  • Upper bound : 16,406 (Feb 24, 2022)
• Kissing number in dimension 19
  • Lower bound : 12,276 (Oct 10, 2026)
  • Upper bound : 24,417 (Feb 24, 2022)
• Kissing number in dimension 21
  • Lower bound : 31,288 (Oct 10, 2026)
  • Upper bound : 53,524 (Feb 24, 2022)
• Kissing number in dimension 25
  • Lower bound : 197,583 (Oct 5, 2026)
  • Upper bound : 265,006 (Jan 1, 2024)
• Kissing number in dimension 26
  • Lower bound : 199,868 (Oct 5, 2026)
  • Upper bound : 367,775 (Jan 1, 2024)
• Kissing number in dimension 27
  • Lower bound : 201,589 (Oct 5, 2026)
  • Upper bound : 522,212 (Jan 1, 2024)
• Kissing number in dimension 28
  • Lower bound : 205,172 (Oct 10, 2026)
  • Upper bound : 752,292 (Jan 1, 2024)
• Kissing number in dimension 29
  • Lower bound : 210,473 (Oct 10, 2026)
  • Upper bound : 1,075,991 (Jan 1, 2024)
• Kissing number in dimension 30
  • Lower bound : 222,072 (Oct 10, 2026)
  • Upper bound : 1,537,707 (Jan 1, 2024)
• Kissing number in dimension 31
  • Lower bound : 241,077 (Oct 10, 2026)
  • Upper bound : 2,213,487 (Jan 1, 2024)
• Kissing number in dimension 32
  • Lower bound : 347,584 (Sep 28, 2026)
  • Upper bound : 3,162,316 (Jan 1, 2024)
• Kissing number in dimension 33
  • Lower bound : 364,352 (Oct 10, 2026)
  • Upper bound : 4,494,570 (Jan 1, 2024)
• Kissing number in dimension 34
  • Lower bound : 384,196 (Sep 28, 2026)
  • Upper bound : 6,422,593 (Jan 1, 2024)
• Kissing number in dimension 35
  • Lower bound : 409,676 (Sep 28, 2026)
  • Upper bound : 9,162,403 (Jan 1, 2024)
• Kissing number in dimension 36
  • Lower bound : 484,760 (Sep 28, 2026)
  • Upper bound : 13,017,098 (Jan 1, 2024)
• Kissing number in dimension 37
  • Lower bound : 498,024 (Sep 28, 2026)
  • Upper bound : 18,498,316 (Jan 1, 2024)
• Kissing number in dimension 38
  • Lower bound : 591,900 (Sep 28, 2026)
  • Upper bound : 26,496,684 (Jan 1, 2024)
• Kissing number in dimension 39
  • Lower bound : 763,668 (Sep 28, 2026)
  • Upper bound : 37,826,766 (Jan 1, 2024)
• Kissing number in dimension 43
  • Lower bound : 2,553,792 (Sep 28, 2026)
  • Upper bound : 153,813,582 (Jan 1, 2024)
• Kissing number in dimension 45
  • Lower bound : 7,380,090 (Sep 28, 2026)
  • Upper bound : 316,735,249 (Jan 1, 2024)
• Integer multiplication below n log n: the exponent κ
  • Lower bound : 1/10 (Oct 17, 2026)
• Matrix multiplication: the barrier of the 9/4 method
  • Upper bound : 9/4 (Oct 2, 2026)
• Quasi-Riemann hypothesis: uniform zero-free half-plane, the threshold θ
  • Upper bound : 87483/100000 (Oct 18, 2026)
• OpenAI's Sidorenko counterexample: the remainder constant of one counting step
  • Upper bound : 189 (Oct 10, 2026)
• van der Waerden numbers: the coefficient c in the superexponential lower bound
  • Lower bound : 1/9 (Oct 14, 2026)
• Walsh–Hadamard and Fourier below n log n: Sussman's Lean proofs
  • Lower bound : ≈ 0.000876248 (Oct 10, 2026)