World Solve

Some lower bounds on computational efficiency are proven for specific tasks, but a fully unified theory covering every possible algorithmic problem remains unresolved. This unresolved question shapes the outer limits of computer science.

Game theory has solved many two-player scenarios completely, but multiplayer games with shifting alliances and incomplete information remain far less understood in full generality. This unresolved question has real implications for economics and political strategy.

Gödel's theorems already limit what's possible here for sufficiently powerful systems, but the full landscape of alternative systems and their limitations remains an active and unresolved area of mathematical logic. This continues to shape our understanding of the foundations of mathematics itself.

This long-standing conjecture claims there's always at least one prime number between any two consecutive perfect squares, and while verified extensively by computer, no general proof exists. Solving it would sharpen our understanding of prime distribution.

Combinatorial geometry problems like this remain partially solved for specific shapes and configurations, but a fully general guarantee across all possible line arrangements is still unproven. This connects visual intuition with rigorous mathematical guarantees.

Certain classes of expressions can be checked for equivalence efficiently, but a fully general and guaranteed method covering every possible mathematical expression remains unresolved. This unresolved question has real consequences for automated theorem proving and symbolic computation.

Various conjectures link the number of prime factors a number has to specific patterns of behavior, but many of these connections remain only partially proven. This continues to be explored within analytic number theory.

Economic models often assume equilibrium is reachable under certain conditions, but proving this holds true across every possible complex real-world scenario remains mathematically incomplete. This connects pure mathematics directly to economic theory.

Modern cryptography largely depends on problems assumed, but not proven, to be computationally hard, leaving a theoretical gap in fully guaranteed security. This unresolved foundation underlies much of digital security today.

The Prime Number Theorem offers an excellent approximation, but the finer details of exactly how actual prime counts deviate from this smooth prediction remain tied to the still-unproven Riemann Hypothesis. Fully resolving this would refine one of number theory's most central results.

Fair division theory has strong results for two-party negotiations, but multi-party fairness guarantees become significantly harder to prove in full generality. This unresolved area connects mathematics with real-world conflict resolution.

Random graph theory offers probabilistic guarantees for many structures, but proving certainty, rather than high likelihood, for every possible case remains an unresolved and active research area. This affects our understanding of networks in biology, technology, and social systems.

Some optimization problems are proven solvable efficiently, others proven intractable, but a complete boundary map covering every conceivable variation remains incomplete. This unresolved question is central to computational complexity theory.

Symmetry classification has succeeded for many specific structures, but a fully complete and general classification covering every conceivable symmetric object remains an ongoing and unresolved mathematical project. This shapes foundational research in group theory and geometry.

Formal proof systems capture enormous portions of mathematical reasoning, but whether a truly complete and finite rule set exists for absolutely every valid form of proof remains philosophically and technically unresolved. This touches the very foundations of what mathematics is capable of expressing.

Many geometric constants have been linked to constants like pi through elegant proofs, but numerous others remain isolated without any confirmed deeper connection. This unresolved gap continues to intrigue mathematicians working in geometry and analysis.

Ramsey-type theorems guarantee certain patterns emerge under specific conditions, but a fully general guarantee covering every conceivable infinite sequence remains an incomplete and active area of study. This connects deeply to combinatorics and the philosophy of mathematical order.

Additive combinatorics explores these kinds of guaranteed substructure questions extensively, but many specific cases remain without a fully general proof. This continues to be an active and evolving research area.

Probability theory offers strong predictive tools for many games, but a fully complete and general model covering every conceivable game of chance remains theoretically unresolved. This touches both pure mathematics and the philosophy of randomness.

Many famous fractals are built from simple recursive rules, but whether every possible fractal pattern can always be reduced to such a finite description remains an open mathematical question. This connects geometry, computer science, and the study of natural patterns.