Status of famous open problems
Where each problem stands, in one line: what is proved, the best bounds, and the latest change. Every problem page has the full statement, the history of the bounds and primary sources. We re-check the list against the literature and correct it when something moves — tell us if we missed a result.
Mathematics
| Problem | Where it stands | Latest change |
|---|---|---|
| Bounds on the de Bruijn–Newman constant Λ Analysis | Open. 0 ≤ Λ ≤ 0.2 is published (Rodgers–Tao 2020; Platt–Trudgian 2021); unrefereed 2026 computer-assisted claims go below 0.18. | 2026: Λ ≤ 0.1787854 (unrefereed) Gomila, with Romik and Atkin |
| The Birch and Swinnerton-Dyer conjecture Number theory | Open (Clay Millennium Problem). The rank part is known for analytic rank 0 and 1 (Gross–Zagier, Kolyvagin) and for a positive proportion of curves; no case of analytic rank ≥ 2 is proven. | — |
| The Hodge conjecture Geometry | Open (Clay Millennium Problem). The main recent progress is Markman's 2025 proof for abelian fourfolds; no general proof or disproof exists. | — |
| The Riemann Hypothesis Number theory | Open (Clay Millennium Problem). Verified for zeros up to height 3·10^12; the best claimed proportion of zeros simple and on the critical line is more than 2/3 (August 2026, Anthropic/Claude, independent proof by Lamzouri), not yet refereed. | 2026: > 2/3, simple and on the line (unrefereed) Anthropic research model; Lamzouri |
| Computer-assisted proofs of finite-time singularities in 3D Euler Analysis | Open (claims under review). Chen–Hou proved blowup for 3D axisymmetric Euler with a boundary; September 2026 claims of blowup on R³ (OpenAI; Alpöge–Buckmaster, with forcing) are not yet peer reviewed. | — |
| Crossing numbers of complete and complete bipartite graphs Graph theory | Open. Hill's conjecture is proven for n ≤ 14 (cr(K₁₃) = 225, cr(K₁₄) = 315; Aichholzer et al. 2021); Zarankiewicz's conjecture is proven for min(m,n) ≤ 6 and some m = 7, 8 cases, and cr(K_{n,n}) ≥ 0.9118·Z(n,n) asymptotically. | 2021: cr(K₁₃) = 225, cr(K₁₄) = 315 Aichholzer et al. |
| Do odd perfect numbers exist? Number theory | Open. An odd perfect number would exceed 10^1500, have at least 101 prime factors (Ochem–Rao 2012) and at least 10 distinct ones (Nielsen 2015). | 2015: ω(N) ≥ 10 Nielsen |
| Frankl's union-closed sets conjecture Combinatorics | Open. The best refereed constant is about 0.38271 (Liu 2023); a 2026 preprint claims about 0.38289, far from the conjectured 1/2. | 2026: ≈ 0.38289 (preprint) Costa & Sadhu |
| Legendre's conjecture Number theory | Open. Verified computationally to about 4·10^18; a prime between consecutive cubes is known for n > exp(exp(32.76)) (Mossinghoff–Trudgian–Yang 2024). | 2024: cubes: n ≥ exp(exp(32.76)) Mossinghoff, Trudgian, Yang |
| The (binary) Goldbach conjecture Number theory | Open. Verified for all even n ≤ 4·10^18 (Oliveira e Silva, Herzog, Pardi 2014); the ternary version was proved by Helfgott (2013). | 2014: verified for even n ≤ 4·10^18 Oliveira e Silva, Herzog, Pardi |
| The 1/3–2/3 conjecture for balanced pairs in posets Combinatorics | Open. The best published general bound is (5−√5)/10 ≈ 0.2764 (Brightwell–Felsner–Trotter 1995); a 2026 preprint claims a tiny improvement, and all posets with up to 14 elements have been verified. | 2026: verified for all posets with ≤ 14 elements (preprint) A. Gupta |
| The chromatic number of the plane (Hadwiger–Nelson problem) Combinatorics | Open. 5 ≤ χ(ℝ²) ≤ 7; the smallest known 5-chromatic unit-distance graph has 509 vertices (Parts 2020), and no 6-chromatic one is known. | 2020: 509 vertices Parts (Polymath16) |
| The Collatz (3n + 1) conjecture Number theory | Open. Convergence is verified for all n < 2^71 (Barina 2025), and Tao (2019) showed that almost all orbits attain almost bounded values. | 2025: verified for all n < 2^71 D. Barina |
| The Erdős unit distance problem in the plane Geometry | Open (the exponent). Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026 (an OpenAI model; human-verified exposition arXiv 2605.20695); the exponent lies between 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter). | 2026: α > 1 (conjecture disproved) OpenAI model; Alon, Bloom, Gowers et al. |
| The Erdős–Rado sunflower conjecture Combinatorics | Open for every k ≥ 3. The best bound is (Ck log n)^n (Alweiss–Lovett–Wu–Zhang 2019 and refinements); a 2026 preprint proves the conjecture for families of bounded VC-dimension. | — |
| The Erdős–Straus conjecture Number theory | Open. Verified for all n ≤ 10^18 (Mihnea & Dumitru 2025); all n outside finitely many residue classes are covered by explicit identities. | 2025: verified for n ≤ 10^18 Mihnea & Dumitru |
| The Erdős–Szekeres happy ending problem Combinatorics | Open. ES(n) is known only up to ES(6) = 17 (Szekeres–Peters 2006); ES(7) = 33 is unproven, and ES(n) = 2^{n+o(n)} (Suk 2017). | 2017: ES(n) = 2^{n+o(n)} Suk |
| The graceful tree conjecture (Ringel–Kotzig) Graph theory | Open. Verified by computer for all trees with up to 35 vertices (Fang 2010); Ringel's weaker packing conjecture was proved for large n in 2020. | 2010: ≤ 35 vertices Fang |
| The graph reconstruction conjecture Graph theory | Open. Verified by computer for all graphs with up to 13 vertices (McKay); trees, regular and disconnected graphs, among other classes, are reconstructible. | — |
| The invariant subspace problem for Hilbert spaces Analysis | Open for separable Hilbert spaces; Enflo's 2023 preprint has not been accepted. | — |
| The inverse Galois problem over Q Algebra | Open in general. M23 was realised over Q in August 2026 (Huang, Jackson, Lee, Poonen, Pries, Zhang), so all 26 sporadic groups are now realised. | — |
| The Jacobian conjecture in two variables Algebra | Open for n = 2. A degree-7 counterexample C³ → C³ (Alpöge, July 2026, found with a Claude model) disproved the conjecture for all n ≥ 3. | — |
| The Kakeya conjecture in dimensions n ≥ 4 Analysis | Open for n ≥ 4. Wang and Zahl proved the case of R³ in 2025. | 2025: Kakeya sets in R³ have dimension 3 Wang & Zahl |
| The kissing number in dimension 5 Geometry | Open. 40 ≤ τ₅ ≤ 44 (lower bound from the D5 lattice; upper bound Mittelmann–Vallentin 2010). | 2010: τ₅ ≤ 44 Mittelmann & Vallentin |
| The lonely runner conjecture Combinatorics | Open. Proved for up to 13 runners (computer-assisted, 2025–2026); a September 2026 preprint claims 14 and 15 runners and has not yet been independently checked. | 2026: 11–13 runners Sungkawichai & Trakulthongchai |
| The perfect cuboid problem Number theory | Open. Computer searches show any perfect cuboid would need an odd edge above 2.5·10^13 and a space diagonal above 9·10^15. | — |
| The twin prime conjecture and bounded prime gaps Number theory | Open. The refereed record is H₁ ≤ 246 (Polymath 8b, 2014); unrefereed 2026 claims reach 240, 212 and 186. | 2026: H₁ ≤ 240 (preprint) J. Stadlmann |
| Cap sets in dimension 7 Combinatorics | Open. 236 ≤ r₃(7) ≤ 291 (lower bound: Edel's construction; a 2022 preprint claims ≤ 288). Exact values are known only for n ≤ 6. | — |
| Cap sets in dimension 8 Combinatorics | Open. The best known cap in F₃⁸ has 512 points (FunSearch, Romera-Paredes et al., Nature 2023/24); no larger construction has been reported since. | 2023: r₃(8) ≥ 512 Romera-Paredes et al. (FunSearch) |
| Costas arrays of order 32 and 33 Combinatorics | Open. Costas arrays are fully enumerated only up to order 29, and no Costas array of order 32 or 33 is known. | 2012: complete enumeration to order 29 Drakakis, Rickard et al. |
| Erdős minimum overlap problem Combinatorics | Open. 0.379005 ≤ lim M(n)/n ≤ 0.380868 (lower: White 2022; upper: SimpleTES 2026, after AlphaEvolve's 0.380924 in 2025). | 2026: M ≤ 0.380868 SimpleTES |
| Formalised Erdős problems (Lean 4) Number theory | Ongoing. Many erdosproblems.com problems are stated in Lean in Formal Conjectures; most remain open, and a growing number have been settled recently, often with AI assistance. | — |
| Hadamard matrices of open orders Combinatorics | The Hadamard conjecture is open. Matrices for all 12 previously open orders below 2000 (668, 716, …, 1964) were announced on 12 August 2026 (Alpöge, Voinov, Reynolds-Haertle, with Claude), in encoded form and without a paper yet; the smallest open order is now above 2000. | 2026: all 12 open orders below 2000 constructed (announced) Alpöge, Voinov, Reynolds-Haertle, with Claude |
| Kissing configurations in dimensions 10–31 Geometry | Open. 2025–2026 brought new lower bounds in several dimensions (e.g. τ₁₁ ≥ 604, τ₁₂ ≥ 841, τ₁₉ ≥ 11,948); a September 2026 preprint claims small further gains in dimensions 25–31. | 2026: τ₁₉ ≥ 11,948 B. S. Ho |
| Large cap sets in F_3^n Combinatorics | Open. Exact maxima are known for n ≤ 6; the asymptotic capacity lies between 2.2203 (Zhai et al. 2025) and 2.756 (Ellenberg–Gijswijt 2016). | 2025: capacity ≥ 2.2203 Zhai et al. (X-evolve) |
| Lean formalisation of Busy Beaver deciders and results Logic & formalisation | Partly done. BB(5) = 47,176,870 and BB(2,4) = 3,932,964 are proved in Coq/Rocq; a community Lean 4 port claims BB(5) using native_decide, while a proof checked by the Lean kernel alone is still missing. | — |
| Optimal Golomb rulers Combinatorics | Open. Optimal rulers are proven for up to 28 marks (OGR-28, length 585, distributed.net 2022); no optimality search for 29 marks is running. | 2022: OGR-28 optimal (585) distributed.net |
| Small van der Waerden numbers Combinatorics | Open. Only seven non-trivial values are known, the latest W(3,4) = 293 (Kouril 2012). | 2012: W(3,4) = 293 Kouril |
| Smaller covering designs Combinatorics | Open (records table). Best known upper bounds on C(v,k,t) are maintained by the La Jolla Covering Repository. | — |
| Snake-in-the-box — longest induced paths in hypercubes Combinatorics | Open. Optimal snakes are known only for n ≤ 8; the best known lengths for n = 9–13 are 191, 379, 746, 1476 and 2924 (Orland et al., 2026 preprint). | 2026: n = 9–13: 191, 379, 746, 1476, 2924 Orland, Fagan et al. |
| The degree–diameter problem for graphs Graph theory | Open. Only a few n(d,k) are known exactly, the existence of a degree-57 Moore graph is unresolved, and new record graphs keep appearing (e.g. 2026 preprints for diameter 3 and 5). | — |
| The Ramsey number R(4,6) Graph theory | Open. 36 ≤ R(4,6) ≤ 40 (lower bound: Exoo 2012; upper bound: Angeltveit–McKay, listed in Radziszowski's survey). | 2024: R(4,6) ≤ 40 Angeltveit & McKay |
| The Ramsey number R(5,5) Graph theory | Open. 43 ≤ R(5,5) ≤ 46 (lower bound: Exoo 1989; upper bound: Angeltveit–McKay, arXiv 2024, J. Graph Theory 2026). | 2024: R(5,5) ≤ 46 Angeltveit & McKay |
| The sixth Schur number S(6) Combinatorics | Open. S(6) ≥ 536 (Fredricksen–Sweet 2000), unchanged since; S(5) = 160 was settled by SAT in 2017. | 2017: S(5) = 160 (SAT proof) Heule |
Computer science
| Problem | Where it stands | Latest change |
|---|---|---|
| Explicit Boolean circuit lower bounds Complexity | Open. The best lower bound for general circuits computing an explicit function is 3.1n − o(n) (Li & Yang 2022); no superlinear bound is known. | 2022: 3.1n − o(n) Li & Yang |
| P versus NP Complexity | Open (Clay Millennium Problem). No accepted proof either way; any proof must get past the relativization, natural-proofs and algebrization barriers. | — |
| Approximation ratio and integrality gap for metric TSP Algorithms | Open. The best approximation ratio is 3/2 − ε with ε ≈ 10⁻³⁶ (Karlin–Klein–Oveis Gharan); the subtour LP gap is at most 4/3 for all instances with up to 15 vertices (Cook–Hougardy–Petrich 2026). | 2026: gap ≤ 4/3 verified for n ≤ 15 Cook, Hougardy, Petrich |
| Explicit rigid matrices (Valiant's rigidity problem) Complexity | Open. No explicit matrices reach Valiant's rigidity parameters; constructions with an NP oracle give only sub-polynomial rank. | — |
| The Černý conjecture on synchronizing automata Computability | Open. The best general upper bound on the reset threshold is about 0.1654·n³ (Shitov 2019), against the conjectured (n−1)². | 2019: ≈ 0.1654·n³ Shitov |
| The exponent ω of matrix multiplication Algorithms | Open. The best published bound is ω < 2.371339 (Alman et al. 2024); an August 2026 preprint claims ω < 2.371177. | 2026: ω < 2.371177 (preprint) Dupont et al. |
| The log-rank conjecture in communication complexity Complexity | Open. The best upper bound is O(√rank) (Sudakov–Tomon 2023); the largest known separation is Ω̃(log² rank) (Göös–Pitassi–Watson 2015). | 2023: D ≤ O(√r) Sudakov & Tomon |
| The Unique Games Conjecture Complexity | Open. The 2-to-2 Games Conjecture was proven (Khot–Minzer–Safra 2018), but the full Unique Games Conjecture remains open. | — |
| A theory of neural scaling laws Machine learning | Open. Empirical power laws are well documented and solvable models explain some features, but no theory predicts the exponents for realistic architectures and data. | — |
| Deciding hard small Turing machines (Busy Beaver) Computability | Open. BB(5) = 47,176,870 was proved in Rocq (2024). BB(6) is unknown: Σ(6) > 10↑↑10↑↑10↑↑8 (mxdys, June 2025), and about 815 six-state machines were still undecided in late September 2026, including Collatz-like cryptids such as Antihydra. | 2025: Σ(6) > 10↑↑10↑↑10↑↑8 mxdys |
| Mechanisms of grokking (delayed generalisation) Machine learning | Open. Several partial mechanistic accounts exist (Fourier circuits, weight norm, circuit efficiency, lazy-to-rich dynamics); none predicts when grokking happens in general. | — |
| Packing equal circles in a unit square Optimisation | Open. Optimal packings are proven for n ≤ 33 (n = 31–33 by Markót 2021) and n = 36; best known packings are tabulated up to n = 10,000 on Packomania. | 2021: optimality proven for n = 31, 32, 33 M. C. Markót |
| Proof-producing SAT solving of open combinatorial instances Algorithms | Open-ended. SAT with verified proofs has settled Pythagorean triples, Schur number five, Keller dimension 7 and the empty hexagon number; S(6) and small Ramsey or van der Waerden values remain open. | — |
| Smallest sorting networks for 13+ inputs Algorithms | Open for n ≥ 13. Minimum sizes are proven for n ≤ 12; the best known sizes for n = 13–16 are 45, 51, 56 and 60 comparators. | 2020: n = 11: 35 and n = 12: 39 proven optimal Harder |
| Tensor rank of 3×3 matrix multiplication Algorithms | Open. The rank is at most 23 (Laderman 1976) over every field; lower bounds are 19 in general (Bläser 2003) and 21 over F₂ (2026 preprints). | 2026: rank ≥ 20 over F₂ C. Wang |
| Tensor rank of 4×4 matrix multiplication Algorithms | Open. The best ranks are 47 over GF(2) (AlphaTensor 2022) and 48 in characteristic ≠ 2 (AlphaEvolve 2025 over C; Dumas–Pernet–Sedoglavic over Q). | 2025: 48 over Q Dumas, Pernet, Sedoglavic |
| The Thomson problem (minimum-energy charges on a sphere) Optimisation | Open in general. Global minimisers are proven for n = 2–6 and 12, and in 2026 for n = 8 (Lean-verified preprint) and n = 7 (numerical global optimisation, plus an unreviewed Lean proof). | 2026: n = 7, numerically certified Kuznetsov & Sahinidis |
Physics & astronomy
| Problem | Where it stands | Latest change |
|---|---|---|
| Yang–Mills existence and mass gap Theoretical physics | Open (Clay Millennium Problem). No construction of a non-trivial 4D quantum Yang–Mills theory with a mass gap exists. | — |
| Existence of SIC-POVMs (Zauner's conjecture) Quantum information | Open. SICs are known numerically in all dimensions up to 151 and in some larger ones, and exactly in a few dozen; a construction for all dimensions via Stark units is conditional on unproven conjectures. | — |
| Ground-state phase diagram of the doped 2D Hubbard model Physics | Open. Controlled numerics find no superconductivity in the pure t′ = 0 model near optimal doping, but find d-wave superconductivity with t′ included; the full phase diagram is unsettled. | — |
| Hadronic vacuum polarisation in the muon g−2 Physics | Open. The 2025 Theory Initiative prediction, based on lattice QCD for the leading hadronic term, agrees with experiment; the conflict between lattice and e⁺e⁻ data-driven values is unexplained. | — |
| Mechanism of high-temperature superconductivity in the cuprates Theoretical physics | Open. There is no consensus on the cuprate pairing mechanism; the ambient-pressure Tc record is 151 K in pressure-quenched Hg-1223 (PNAS 2026). | — |
| Mutually unbiased bases in dimension 6 Quantum information | Open. Only three mutually unbiased bases are known in dimension 6; numerical evidence suggests a complete set of seven does not exist. | — |
| Small-scale problems of cold dark matter Astrophysics & cosmology | Open. Baryonic feedback, self-interacting dark matter and observational systematics all remain viable, partial explanations of the cusp–core, too-big-to-fail and diversity problems. | — |
| The cosmological lithium problem Astrophysics & cosmology | Open. Big Bang nucleosynthesis with CMB parameters predicts 7Li/H ≈ 5×10⁻¹⁰, about three times the observed Spite plateau; proposed resolutions are not confirmed. | — |
| The Hubble tension Astrophysics & cosmology | Open. Distance-ladder values of about 73 km/s/Mpc disagree with 67.4 ± 0.5 from Planck under ΛCDM; the TRGB calibration (about 70) keeps the ladder itself disputed. | — |
| The solar coronal heating problem Astrophysics & cosmology | Open. Waves, turbulence and nanoflare reconnection are all observed, but no mechanism has been shown to supply the full coronal energy budget. | — |
| The strong CP problem Theoretical physics | Open. The neutron EDM bound implies θ̄ ≲ 10⁻¹⁰; the axion and other solutions remain unconfirmed. | — |
| Theory of the glass transition Physics | Open. Swap Monte Carlo now equilibrates model liquids down to and below the experimental glass transition, but whether an ideal glass transition exists is unresolved. | — |
| Classical simulation of random circuit sampling experiments Quantum information | Open. The 2019 Sycamore circuits have been sampled classically (Pan–Chen–Zhang 2022); for the newest experiments no classical spoof at matching fidelity is known. | — |
| Periodic orbits of the Newtonian three-body problem Physics | Open-ended. The largest catalogue lists over 10,000 three-dimensional periodic orbits (Li & Liao 2025), on top of thousands of planar families. | — |
| Precision critical exponents of the 3D Ising universality class Theoretical physics | Open. The most precise values come from the conformal bootstrap: Δσ = 0.518148806(24), Δε = 1.41262528(29) (Chang et al. 2025). | 2025: Δσ = 0.518148806(24), Δε = 1.41262528(29) Chang et al. |
| The dense-matter equation of state from neutron-star observations Astrophysics & cosmology | Open. Gravitational-wave and NICER measurements put radii around 11–13.5 km, and the analyses do not fully agree. | — |
| The origin of fast radio bursts Astrophysics & cosmology | Open. Magnetars are an established source (SGR 1935+2154, 2020), but whether all FRBs share one origin is unresolved. | — |
| Thresholds and decoders for quantum error-correcting codes under circuit-level noise Quantum information | Open frontier. Surface codes now run below threshold in hardware (Google 2024/25), and qLDPC codes reach about 0.8% circuit-level thresholds. | — |
Chemistry & materials
| Problem | Where it stands | Latest change |
|---|---|---|
| Density functional approximations with chemical accuracy Chemistry | Open. Learned functionals such as Skala approach hybrid-level accuracy at semi-local cost, but no functional is uniformly chemically accurate. | — |
| Electrochemical ammonia synthesis under ambient conditions Chemistry | Open. Careful 15N2-controlled studies (Andersen et al., Nature 2019) found no verifiable ammonia from aqueous pure-metal catalysts; only lithium-mediated, non-aqueous synthesis is robustly confirmed. | — |
| High-pressure hydride superconductors — prediction and verification Materials | Open. H₃S and LaH₁₀ are the best-established hydride superconductors; the CSH (2020) and Lu–N–H (2023) claims were retracted, and newer room-temperature claims await reproduction. | — |
| Prebiotic routes to nucleotides and the RNA world Chemistry | Open. There is no demonstrated end-to-end, geochemically consistent route from simple feedstocks to self-replicating RNA, although key steps have been shown. | — |
| Computational discovery of high-zT thermoelectrics Materials | Open. The record bulk zT is about 3.1 (purified polycrystalline SnSe, 2021); predicting zT before synthesis remains unreliable. | — |
| Crystal structure prediction of molecular solids Chemistry | Open. In the seventh CCDC blind test (2024), periodic DFT-D ranking matched experiment for most targets, but generating and reliably ranking structures for complex targets remains unsolved. | — |
| Long-term operational stability of perovskite solar cells Materials | Open. Single-junction efficiencies are near 27% and perovskite–silicon tandems near 35%, but long-term operational stability remains the main barrier. | — |
| Predicting and discovering fast lithium solid electrolytes Materials | Open. The sulfide record is about 25 mS/cm (Kato et al. 2016); predicting conductivity from structure is still unreliable. | — |
| Predicting glass-forming ability of metallic alloys Materials | Open. Machine-learning models help find new glass-forming systems, but predicting glass-forming ability from composition alone remains unreliable. | — |
| Predicting protein-ligand binding affinity Chemistry | Open. Reported accuracy on CASF-2016 is inflated by train–test leakage; on leakage-filtered splits the best models reach Pearson R ≈ 0.8. | — |
| Rare-earth-free permanent magnets ("gap magnets") Materials | Open. No commercially viable rare-earth-free magnet fills the gap between ferrites (about 48 kJ/m³) and Nd-Fe-B (about 515 kJ/m³); screened candidates are unconfirmed experimentally. | — |
| Selectivity in CO2 electroreduction to multicarbon products Chemistry | Open. Copper remains the only elemental catalyst making appreciable C2+ products, and selectivity is not yet predictable from computed descriptors. | — |
| Visible-light photocatalysts for overall water splitting Chemistry | Open. Near-unity quantum efficiency has been shown only in the UV (Al:SrTiO₃, Takata et al. 2020); the largest panel demonstration reached at most 0.76% solar-to-hydrogen efficiency. | — |
Life sciences
| Problem | Where it stands | Latest change |
|---|---|---|
| A whole-nervous-system model of C. elegans that reproduces behaviour Neuroscience | Open. Connectomes across development exist (Witvliet et al. 2021), but the measured signal-propagation atlas (Randi et al. 2023) departs from wiring-based predictions, and no model reproduces measured whole-brain activity and behaviour. | — |
| RNA 3D structure prediction Biology | Open. In CASP16 accuracy still depended on templates, with no significant gain over earlier rounds. | — |
| The paradox of the plankton Ecology | Open. Several mechanisms suffice in theory for coexistence (e.g. oscillations and chaos), but their relative contribution in real plankton communities is unresolved. | — |
| Computational design of enzymes for chosen chemical reactions Biology | Open. Fully computational designs now reach Kemp eliminase efficiencies above 10⁴ M⁻¹s⁻¹ (Listov et al., Nature 2025) and serine hydrolases around 2×10⁵ (Lauko et al., Science 2025), but high hit rates for complex reactions are not routine. | — |
| Genes of unknown function in a minimal cell Biology | Open. JCVI-syn3A still has roughly 90 protein-coding genes of unknown function, about 30 of them essential. | — |
| Predicting protein conformational ensembles Biology | Open. In CASP16 only half of the ensemble targets were modelled well; generative emulators such as BioEmu are promising but not yet validated on populations in general. | — |
Earth & society
| Problem | Where it stands | Latest change |
|---|---|---|
| Antarctic marine ice-sheet and ice-cliff instability Earth science | Open. ISMIP6 models without ice-cliff instability project −7.8 to +30 cm of Antarctic sea-level contribution by 2100 under RCP8.5 (Seroussi et al. 2020); whether cliff instability matters this century is still contested. | — |
| Narrowing equilibrium climate sensitivity Climate | Open. IPCC AR6 assesses ECS at 3 °C (likely 2.5–4 °C, very likely 2–5 °C), consistent with Sherwood et al. 2020. | — |
| Testable earthquake forecasting Earth science | Open. Deterministic prediction has no demonstrated skill; probabilistic models are evaluated prospectively (CSEP), with modest gains over simple baselines. | — |
| The equity premium puzzle Economics | Open. No consensus explanation exists; habit formation, rare disasters, long-run risk and behavioural accounts all remain contenders. | — |
| Attributing the renewed growth of atmospheric methane Climate | Open. Atmospheric methane keeps rising (about 1939 ppb in 2026); how much of the post-2007 rise comes from emissions versus a weaker OH sink is disputed. | — |
| ENSO prediction beyond one year Climate | Open. Operational skill drops with lead time and across the spring barrier; deep-learning claims of skill beyond a year are still debated under strict hindcast protocols. | — |
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