OpenAI math · Research directory
Software ideas, with their evidence.
Explore what each finding could solve, what software it may enable, and what still stands between a theorem and a useful business.
Commercial value is still a hypothesis. All families have a first assessment; individual manuscript and proof review continues.
Opportunity directory
Open full comparison table →Proof artifact registry for research teams
The collection combines manuscripts, separate challenge statements, solution modules, checker configurations, and different coverage levels. The 416-configuration static preflight found all referenced modules and two…
Prototype now
Theorem dependency and correction alerts
The 7 October history records a sign error that invalidated one manuscript and two dependent papers. Other corrections changed hypotheses and companion citations. The release preserves prior versions, enabling a concrete…
Prototype now
Formal theorem coverage auditor
Several scope documents distinguish a paper's claim from the selected formal statement. Family 131 excludes the exact sampler; family 149's formal scope concerns an older boundedness result; family 276 excludes arbitrary…
Prototype now
Fixed-margin data sandbox and sampling-law audit
Family 115 claims exact uniform sampling of unbounded nonnegative integer tables with arbitrary equal-total margins in expected polynomial bit complexity, and bounded-time approximate sampling. A separate paper claims an FPRAS…
Prototype law audit; defer literal research backend
Graph sampling protocol and null-model calibration
Family 131 claims total-variation mixing time at distance one-quarter at most 2n^8 for every graphical labeled degree vector, for one precisely specified lazy chain on simple undirected graphs with the complete host. Half the…
Prototype bounded protocol audit; defer cheap general certified sampling
Perfect-pairing fragility and flexibility analyzer
Family 113 claims an FPRAS for the number of perfect matchings in every finite simple unweighted undirected graph. It returns zero with certainty on infeasible input and has worst-case polynomial bit time on every random tape.…
Prototype bounded exact diagnostics; defer literal FPRAS
General-graph allocation with independent certificates
Family 120 claims one randomized word-RAM program for explicit maximum-cardinality matching in general simple unweighted graphs, with every-path input-size time C p^(1+eta(p)), p=n+m, eta tending to zero, and success…
Prototype certificates; defer literal accelerated backend
Long-document edit comparison: source backend deferred
The literal approximation branch requires ell at least 2^1000 and falls back to exact distance below its guard. Established exact libraries already provide distance, threshold queries and alignments.
Defer direct backend; exact incumbent baseline established; low commercial confidence
Three-machine scheduling model and solver assurance
Uniform deterministic exact unit-job scheduling on exactly three identical machines; exponent 150020, no practical claim.
Prototype assurance adapter; defer direct scheduler
Metric movement evaluation and startup audit
The source separates existence of a causal squared-log competitive policy from its uniform finite-rational-metric implementation. The latter simulates a finite table constructor for floor(log2(t+1)) steps at request t and…
Prototype diagnostic
Optimization assurance workbench: facility planning
The source claims a fixed-epsilon deterministic 1+2/e+epsilon approximation for unweighted rational metric k-median. Accuracy-dependent history expansion and budget reduction leave practical runtime unestablished. A separate…
First product experiment; buyer validation pending; source backend deferred
Noncommutative formula testing and rewrite witnesses
Explicit rational hitting matrices admit a bounded structured first-column implementation. Nonzero gives a finite free-algebra counterexample; zero identity conclusions remain conditional on unaccepted universal source claims.
Prototype exact witness SDK; low commercial confidence; defer larger companion backends
Reaction network model assurance plugin
The newer paper states a common compact absorbing set for each positive compatibility class of a weakly reversible mass-action system with fixed positive rates. The listed formal scope supports older trajectory-dependent…
Prototype model checks
Polynomial-bound evidence for nonnormal matrix workflows
Source sharp complete constant two requires entire numerical range and Euclidean tensor model; practical enclosure and supremum are separate engineering.
Prototype bounded polynomial/model review; defer general library; low commercial confidence
Optimal transport sensitivity benchmark and audit
For a uniform compact convex source in dimension at least two, the source claims target-uniform one-third Holder stability of quadratic optimal maps. Its cube example uses a central atom of mass a, tilted by b=a/2: W2^2=ab^2…
Prototype diagnostic
Deterministic finite field factorization library
Family 142 claims complete deterministic polynomial bit-time factorization of dense polynomials over prime fields, with the prime supplied in binary. Its proof depends on a companion uniform Hecke zero-free result. The family…
Conditional research
Exact selection and reconciliation witness assurance
Separate two-sided .49-time and one-sided low-space paper-level decisions; known 64-bit main guards begin at 600000 and 36000000000 items, with unevaluated additional cutoffs.
Prototype assurance module; defer direct source backends
Mean payoff strategy synthesis plugin
The family claims quasipolynomial algorithms for several signed-weight mean-payoff and parity-game variants. Its scope notes formalize a randomized zero-threshold mean-payoff algorithm and a finite counterexample, rather than…
Conditional engineering
Thermal qubit channel capacity calculator
The paper reduces the unassisted classical capacity of generalized amplitude damping to a one-variable maximum and identifies a two-state ensemble. The selected formal statement excludes arbitrary-partner additivity and…
Conditional engineering
Research algorithm feasibility and semantic workbench
Existing dossiers document extreme finite constants behind headline complexity improvements. New reviews add delayed k-server activation at 2^T_P-1 and the exact logspace derandomization compiler's explicit resource-bound…
Prototype now
Matroid online allocation research simulator
The one-sample matroid prophet result guarantees only an absolute reward fraction 2^-310 and does not assert a polynomial-time implementation. Independence and matched sample/value distributions are part of the statement.
Defer direct product
Matrix tensor construction research toolkit
The square matrix result bounds the complex arithmetic exponent by nine fourths, with related rectangular and every-field bounds. Scope notes explicitly distinguish arithmetic complexity from bit complexity and practical…
Defer performance product
Byte-view literal packaging and build-cost benchmark
Full constructed two-approximation in symbol length; polynomial source dictionary/rule work and actual byte/API objectives need bridges.
Prototype bounded artifact comparison; defer source backend; low commercial confidence
All-cut tree assurance and rounding research toolkit
Family 174 claims a deterministic polynomial-bit constructor of a C/k-thin spanning tree in every finite loopless k-edge-connected multigraph, including binary multiplicities. A separate corollary gives simultaneous all-cut…
Prototype exact finite cut audit; defer literal constructor
Finite dataset embedding contract audit
Family 094 gives subpolynomial target dimension for fixed distortion greater than one in the same real Lp geometry, while exact embeddings have quadratic worst-case dimension for p other than two. The manuscript explicitly…
Prototype diagnostic
Inverse boundary measurement evidence workbench
The latest family 365 paper claims that ideal zero-frequency measurements, with inputs and observations on one arbitrary nonempty open boundary patch, determine a smooth metric and smooth unitary connection on a trivial…
Prototype evidence workflow
Elastic material identifiability experiment lab
Family 372 states global uniqueness of smooth isotropic Lamé parameters on a bounded connected smooth three-dimensional domain from the full static displacement-to-traction operator. The ellipticity assumptions are μ>0 and…
Conditional evidence workflow
Variational segmentation structure diagnostics
Family 366 classifies interior discontinuity structures of reduced planar absolute Mumford–Shah minimizers as smooth arcs, crack tips or three-way junctions meeting at 120 degrees, with local finiteness properties. Family 367…
Conditional diagnostic
Scientific prediction and solution-class contract audit
The PDE collection contains global classical existence for a specified relativistic Vlasov–Maxwell class, periodic hard-sphere Boltzmann nonuniqueness, stable blowup for selected high-power defocusing NLS equations on a…
Prototype evidence workflow
Tensor-network representation and compute feasibility audit
Family 265 claims an entropy area law from a full-system spectral gap for unique ground states on finite induced square lattices. A companion states existence of polynomial-bond PEPS approximations to uniformly gapped…
Conditional evidence workflow
Quantum optimization regime and resource audit
The QAOA manuscript claims SK limiting expected energy optimality with system size tending to infinity before depth. It explicitly provides neither a quantitative required depth nor an efficient angle-selection procedure. Its…
Prototype evidence workflow
Graph community recovery feasibility audit
Family 229 gives an exact threshold dλ²>1, with nonreconstruction at equality, for three-state symmetric tree broadcasting, and states the corresponding symmetric three-community sparse-SBM weak-recovery threshold using…
Conditional evidence workflow
Random SAT benchmark and transition calibration
The collection states limiting random-k-SAT thresholds, hitting-time variance Θk(n), and computability of the random 3-SAT threshold. Clauses contain distinct variables, independent uniform signs and sampling with replacement.…
Conditional benchmark workflow
Certified numerical computation reproducibility workbench
The dimension-six mutually unbiased bases paper claims exactly three, with exclusion of four arbitrary bases through a documented computation under specified binary64 and compiler conditions. A companion uses exact…
Prototype evidence workflow
Binary waveform spectrum and correlation contract audit
The latest family 076 manuscript claims uniformly ultraflat real sign polynomials through all sufficiently large lengths and unbounded binary merit factors. The selected formal scope supports asymptotically minimal upper…
Prototype diagnostics; defer new sequence generator
Exact spectral topology acceptance and construction research
Family 178 claims deterministic polynomial-bit-time output of simple fixed-degree nonbipartite Ramanujan graphs on every sufficiently large even size, with degree-dependent exponent. Its algorithm section supplies a rational…
Retain exact finite acceptance; defer literal full constructor
Periodic microstructure interface reference benchmark
Family 354 claims the exact isotropic perimeter profile of the unit cubic flat three-torus: with v=min(V,1−V), the minimum is min((36π)^(1/3)v^(2/3), 2sqrt(πv), 2). Minimizers are balls, tubes around shortest geodesics or…
Prototype formula; conditional simulation integration
Typed normalization assurance adapter
Family 245 claims that system-wide weak beta normalization implies system-wide strong beta normalization for an arbitrary pure type system. It permits arbitrary sorts and nonfunctional axiom/product relations, open valid…
Prototype evidence-platform module
Positive-basis algebra and witness audit
Family 169 claims a nonnegative q-polynomial elementary expansion of the chromatic quasisymmetric function of every natural unit interval graph. Its source witness assigns a partition to each eligible nondescent permutation,…
Prototype bounded CAS integration; low commercial confidence
Queryable permutation and sparse shuffle framework
Family 238's signed-tensor companion describes a decision forest that always outputs a permutation on n=2^d slots and computes each point using Ld adaptive queries to shared independent fair switch bits. For one absolute L,…
Prototype qualified sparse providers; defer source-guaranteed seeded loader and blanket memory claims
Bounded integer-flow scenario and flexibility analysis
Family 115's cell-bounded-table paper has a full bounded integral-flow corollary. It claims an FPRAS for integer arc-value vectors in explicitly listed finite directed multigraphs with finite binary integer lower/upper bounds…
Prototype finite model diagnostics; defer unrestricted counting/sampling backend