OpenAI Math

Collection

One article per result family from the OpenAI mathematics repository, with companion manuscripts discussed together. The notes introduce the central objects, state hypotheses and endpoints carefully, explain the claimed proof mechanisms, and work out selected elementary deductions.

🏷️ Scope and current coverage

The source repository contains 372 families in 722 manuscripts in the pinned release. This series selects 277 families after excluding the catalogue’s algebra, mathematical logic, group theory, and combined algebraic-and-complex-geometry sections, together with 21 cross-listed families whose central subject falls under those exclusions. Operator algebras are retained as analysis; topology and computational topics remain eligible unless specifically excluded in the register.

All 277 eligible families now have an article, covering 531 manuscripts. Every article includes an expanded key-steps section: the main estimates or lemmas, an outline of their proofs, and an explanation of how they combine. Dependency diagrams and worked calculations accompany selected arguments. The series includes analysis, probability, PDEs, geometry, dynamics, eligible number theory and combinatorics, theoretical computer science, mathematical physics, operator algebras, and topology. The coverage register records all 372 families, including the 95 excluded families and the reason for each exclusion.

🏷️ Prerequisites and reading sequence

The baseline is graduate real analysis: Lebesgue integration, Lᵖ spaces, basic functional analysis, and Fourier transforms. The conformal articles also use a first course in complex analysis. Each article defines the additional objects it needs before stating the research result.

🏷️ Partial differential equations

The PDE articles assume basic Sobolev spaces, weak formulations, and the divergence theorem. Each introduces its additional kinetic, geometric, or probabilistic objects. A useful starting sequence is kinetic transport (362–364), variational regularity (366–368), then optimal transport (373–374).

🏷️ Probability and statistical mechanics

These articles assume graduate probability: conditional expectation, weak convergence, Markov chains, and basic Gaussian variables. The planar scaling-limit articles introduce the required random curves and generalized fields. Start with 213–214 for percolation, 215–216 for lattice spin systems, or 217 and 221–222 for disordered mean-field models.

The probability and statistical-mechanics articles connect finite estimates to their limiting objects, with separate proof outlines for the companion manuscripts.

🏷️ Functional analysis

These articles assume a first graduate course in Banach and Hilbert spaces: duals, weak convergence, compact operators, and basic convexity. Each defines its metric invariants before stating the result. Start with sphere isometries and nonlinear equivalence (322, 324), then approximation and cotype (330, 326), followed by the random-walk tests (327, 332). The complete Crouzeix article also uses complex analysis.

Family 323 remains excluded because its central conclusion concerns logical independence. All other functional analysis families are now covered.

🏷️ Convex and metric geometry

These articles begin with support functions, polarity, covering density, or metric distortion as needed. Start with 087–088 for affine convex geometry, 089 and 094 for embeddings, or 096–097 for concrete Gaussian and vector examples. The hyperbolicity article uses convex optimization and linear algebra, without developing the excluded algebraic subjects.

🏷️ Dynamical systems and ergodic theory

These articles define their dynamical invariants before the research statements. Start with the explicit ODE calculations in 143 and 149, the spectral model in 144, or the elementary dimension upper bound in 148. The billiard pair treats distinct smooth and polygonal settings; the entropy articles keep measure, regularity, and invertibility assumptions separate.

🏷️ Number theory

Start with the elementary reductions in Egyptian fractions (025), prime-gap counting (026), or periodic Gaussian-prime obstructions (028). The analytic articles introduce their characters, Dirichlet series, and approximation invariants. Companion manuscripts remain together; the simultaneous primitive-root statement in 029 retains its four explicit hypotheses. Arithmetic field data are defined where needed without developing separate algebra or algebraic-geometry courses.

🏷️ Combinatorics

These articles introduce graph parameters, finite colorings, incidences, and distance multiplicities before their research statements. Start with the worked reductions in 157, 159, 162, 171, or 174. The geometric articles retain exact distance and periodicity conventions. Specialized algebraic machinery used in some proofs is identified as an input rather than developed as a separate subject. All thirty-five eligible combinatorics families are now covered. Family 187 includes execution of the two supplied finite Snaky certificate checkers; other verification boundaries are stated in each article.

🏷️ Theoretical computer science

These articles define approximation guarantees, constraint games, computational resources, and infinite-play objectives before the research statements. Basic finite probability and linear algebra suffice for the introductory calculations; the proof discussions identify the additional Fourier and analytic interfaces. Start with 102 for approximation gaps, 105–106 for perfect completeness and coloring promises, or 104 for potential methods in games. Family 103 explains why total recursion memory and termination on every enumerated choice matter in derandomization. All eligible families in this section are now covered. Family 140 compares six approaches to memory–sample lower bounds for exact Gaussian regression. Families 107–109 distinguish arithmetic exponents, determinant representations, and tape bit complexity; 110–111 introduce online and prophet guarantees; 113–115 develop approximate counting, entropy, and exact sampling.

🏷️ Differential geometry

These articles introduce induced metrics, curvature, and the global invariants used in their statements. Families 333 and 334 contrast smooth surface immersion in target dimensions four and three. Families 335 and 336 develop scalar curvature estimates for fundamental cycles and maps with small fibers, keeping the companion papers and their dependencies together. Family 337 compares smooth isoperimetry with singular CAT(0) fillings; 338 follows flow and shrinking charts to global complex coordinates; 339 explains how entropy equality forces smooth invariant distributions and metric rigidity. Families 340–342 introduce exact Lagrangian embeddings, taming and compatibility, and deformation of positive triples while preserving their periods. Read 342 before the current estimates in 341. Family 343 develops the higher-dimensional ball-packing criterion through Hamiltonian rearrangement and surface models; 344 explains the Blaschke volume comparison and its metric equality case. Family 345 treats infinitely many geodesic images through filtered loop topology; 346 keeps the almost-everywhere and sharp-dimensional varifold regularity manuscripts together. Family 347 compares critical-number, ordinary Morse-number, stable-Morse, and cyclic integral Hamiltonian fixed-point counts across five manuscripts. Family 348 joins Einstein metric classification, zero-plane product rigidity, and a topological energy gap obtained by Ricci-flow smoothing. Family 349 develops the Clifford least-volume threshold through immersion density, a detecting distance family, and a nested cone induction. Family 350 compares average-growth control of surface nodal length with smooth fixed-metric counterexamples and a quantitative power gain in higher dimensions. Family 351 compares four-dimensional smooth extension under bounded scalar curvature with higher-dimensional Ricci-flow counterexamples, including the static bubble-tree estimate used in the extension proof. Family 352 develops a finite-time Calabi-flow singularity through a certified shrinking profile, selected initial modes, and a compact parabolic correction. Family 353 joins affine Bernstein rigidity in dimensions three through nine to a smooth dimension-ten counterexample, with tangent-section geometry and radial ODE preliminaries. Family 354 classifies isoperimetric regions in the cubic three-torus through endpoint reductions, nested planar slices, and certified scalar inequalities. Family 355 explains fixed-center tangent-flow uniqueness at the first surface singularity through a cylindrical-opening obstruction, localized Gaussian gradient estimates, and finite rescaled length. Family 356 pairs the distributional characterization of Alexandrov lower curvature with its weak-Hessian companion, explaining weighted entropy, concentration on every geodesic, and distance comparison. Family 357 constructs open bi-Lipschitz charts at every regular noncollapsed RCD point through harmonic coordinates, directional Bochner estimates, and corrections with summable metric errors. Family 358 separates absence of conjugate points from the existence of nonpositive curvature on a closed three-manifold, using a concave inverse metric and whole-passage index estimates. Family 359 compares two-sided negative Kähler pinching without bounded coordinates with one-sided negative curvature and the holomorphic Liouville property, introducing potentials, representing probabilities, and holomorphic disk tests. Family 360 pairs convex injectivity domains under weak MTW with uniform bi-Hölder optimal transport, explaining global cost supports, conjugate endpoints, gap sections, and the determinant argument for rough densities. Family 361 pairs even-dimensional and three-dimensional harmonic-dimension counterexamples, explaining cone spectra, averaged growth, exact transmission, and the checked finite spectral selection.

🏷️ Mathematical physics

Family 260 introduces spacelike initial data, energy conditions, trapped boundaries, invariant mass, and full enclosing area. Its thirteen manuscripts are organized by the numerical comparison, deformation and rigidity arguments, charged branches, angular-momentum conventions, anti-de Sitter ends, and the CKS hyperboloidal replacement. Family 261 compares planar pure-point localization at each fixed positive disorder with higher-dimensional weak-disorder absolute continuity, introducing spectral measures, resolvents, conditional averaging, and logarithmic capacity. Family 262 develops the sharp Lieb–Thirring constant, the finite-interval action mechanism, noncommuting matrix potentials, and the classification of fixed-channel equality profiles. Family 263 introduces correlated Coulomb atoms, the uniform charge and first-ionization bounds, priced-sector integration for energy asymptotics, and the expected-tail argument for neutral radii. Family 264 explains the local near-Kerr cosmic-censorship results through curvature, signed holonomy, and averaged holonomy at the weak-connection threshold. Family 265 treats the two-dimensional global-gap area law and its polynomial-PEPS companion, with entropy cancellation, the extra compression inputs, and the same-grid routing argument. Family 266 compares the dimension-six mutually unbiased bases search with exact Fourier certificates, including a locally executed companion verifier and the distinct bounds of three and five. Family 267 groups five Bose-gas manuscripts, distinguishing positive fractions, positive temperature, the exact depletion coefficient, and the full momentum profile after the thermodynamic limit. Family 268 derives the coupled-purity mechanism behind the claimed periodic Haldane gap and distinguishes the additional boundary-overlap argument for endpoint fields. Family 269 connects an exact rational Fock-space certificate to the Laughlin gap and explains the additional local estimates required for uniform scalar-disorder stability. Family 270 distinguishes the unique zero-energy BFSS threshold state from positive eigenstates confined within selected rotation sectors, and explains the profile compactness behind the threshold argument. Family 271 groups spontaneous Heisenberg ferromagnetic order, Bloch’s law, its first lattice correction, and the fixed-radius spherical magnetization law. Family 272 separates entanglement from distillable secret key, derives the quantitative one-bit obstruction, and explains the PPT pair and channel-square counterexamples. Family 273 develops photon-number entropy, the thermal-defect cancellation argument, and the constrained output-entropy and degraded broadcast consequences. Family 274 compares uniform product-projection localization with regular-trajectory pruning in the measured-output quantum parity lower bound. Family 275 explains the binary-charge and unit-charge continuum Coulomb hardness reductions, including superexchange, positive-density quadrature, and the lower bound on all omitted continuum states. Family 276 derives the scalar classical-capacity formula for generalized amplitude damping from a rectangular entropy inequality, and separates repeated-use additivity from the stronger arbitrary-partner claims. Family 277 explains threshold concentration for entangled games through conditioning costs, common purifications and support components, distinguishing the manuscript and formal-scope exponents. Family 278 explains a Coulomb density with no Kohn–Sham ground-state ensemble representation in the stated local-potential class, through a full ensemble kinetic certificate and a planar trace obstruction. Family 279 explains exact quantum factoring over a fixed finite gate set, with dyadic probability corrections, verified auxiliary factorizations and deferred tests that agree on each recorded run. Family 281 links full support of the zero-temperature SK order parameter to the manuscript’s thermodynamic-first QAOA optimality theorem, preserving the gap between scalar formalized interfaces and circuit convergence. Family 282 explains the local scale-to-conformal enhancement theorem, with the operational hypotheses and the reconstruction of a physical improvement scalar. Family 283 explains polynomial-resource unitary synthesis from a target-dependent Boolean oracle, including coherent label erasure and the single-call recursion. Family 284 completes the eligible mathematical physics section with nearly quartic quantum query speedups for total Boolean functions, explaining addressed witnesses, probabilistic tournaments and adaptive classical lower bounds.

The family 260 scope page selects CKS end replacement and canonical Schwarzschild equality examples only. It explicitly leaves the general Bondi inequality unformalized; a conditional connected-boundary result is not a substitute for that theorem. Family 261 selects the planar operator and its spectral interval only, explicitly excluding the pure-point type conclusion; its delocalization companion is not listed. Family 262 selects the full scalar spectral-moment inequality and its sharp attaining profile throughout the open exponent interval; neither matrix companion nor the matrix equality classification is listed. Family 263 selects generalized ionization energies and the iterated neutral-radius limits. Its energy statement does not include fixed-number convergence or ground-state attainment; its radius statement takes ground states as inputs. The molecular excess-charge and uniform first-ionization conclusions are not listed. No family scope pages for 264 or 265 were present in the pinned snapshot; the notes do not infer complete absence of related code from those missing pages.

🏷️ Operator algebras

These articles treat operator algebras as analysis, beginning with regular representations, norm closure, spectral projections, and traces. Family 285 separates its three manuscript constructions and their hypotheses. Family 286 treats finite correspondences of lattice factors, including scalar twists, actual summands and stable recovery. Family 287 follows the free-group-factor manuscript from polynomial flows and a spectral coefficient construction to an adaptive generating limit. Family 288 explains the uniform commutator estimate behind Kadison similarity and universal hyperreflexivity. Family 289 separates two-sided von Neumann stability from one-sided embedding obstructions and the failure of spatial conjugacy for close norm-closed algebras. Family 290 develops bounded modular spectral recovery, its bicentralizer application and the relative core-commutant mechanism. Family 291 compares Cuntz-semigroup divisibility, two-piece nuclear approximation, uniform tracial splitting and equivariant Jiang–Su absorption, with their distinct hypotheses. Family 292 explains the explicit nuclear norm-ultrapower embedding obstruction through spectral finiteness, varying matrix dimensions and a finite predecessor chain. Family 293 distinguishes ordinary and hyperinvariant subspaces through backward transport between weighted-shift fibers, and explains the conditional convolution obstruction for prescribed irrational rotations. Family 294 connects continuous matrix scaling and finite bundle tests to a fixed quasitrace additivity defect, while distinguishing its separate stable-finiteness and tensor-product consequences. Family 295 uses a random-walk Liouville theorem and free-unitary rigidity to explain bounded Hochschild primitives. Family 296 explains generic relative generators through uniform word perturbations and a Baire argument, then gives the corner matrix encoding for a single generator. Family 297 develops pure-state excision, selective extensions and countable determination in the manuscript counterexample to Naimark’s problem. Family 298 compares microstate volumes and free Fisher information using two noise scales and a persistent matrix-block event. Family 299 converts absence of central-sequence characters into Jiang–Su absorption through full matrix cones and an exact dimension-drop defect comparison. Family 300 distinguishes approximation paving from strong-operator paving, and explains how a common partition preserves quadratic order. Family 301 explains why extended trace cones preserve ideal information and how local models lead to classification after Razak–Jacelon stabilization. Family 302 connects mean dimension to radius of comparison through boundary-preserving compression and exact orbit-cover reconstruction. Family 303 completes this section by separating individual-to-strong pure infiniteness, exact amplification removal, and the separable nuclear absorption consequence.

🏷️ Attribution and proof status

The repository describes manuscripts produced by an internal OpenAI model and explicitly notes that their verification stages differ. These notes therefore label the research statements as manuscript claims. Worked elementary deductions are distinguished from the new estimates on which the manuscripts depend.

Reading an introduction, inspecting a formalization catalogue entry, checking a rendered formula, and independently verifying a full proof are different activities. The present articles have a selected-section source review. They do not report independent full-proof verification, local Lean certification, or peer-review acceptance.

Some family-level formalization headings are broader than their documented comparator lists. In particular, the scope pages for 076, 082, 084, 363, 365, 367, and 373 describe narrower results than the strongest manuscripts in those families. Family 371 requires care with the quantifier on admissible powers, and 376 has distinct selected statements for different flow constructions. Probability families 211, 212, 215, 218, 220, and 222 likewise have narrower documented selections than their full manuscript families. In particular, the Ising-perceptron selection in 222 proves finiteness of a variational value rather than convergence of the pressure. Further examples include 227 (selected cutoff and critical mixing statements), 229 (supercritical reconstruction only), 230 (positivity rather than the complete exact measure), 235 (the earlier three-clause variance bound), and 237–238 (selected supporting results). Family 331 distinguishes unconditional diamond distortion from further graph deductions with 22 companion inputs, and retains support and tail hypotheses in several auxiliary estimates. No scope page was present for 332 in the pinned release. Combinatorics scope pages also require care: 157 documents the list-coloring result but not the two counterexample manuscripts, 159 selects the reciprocal-sum consequence rather than the quantitative bound, and 161 selects the Sidorenko counterexample rather than its forcing consequence. Families 167 and 174 each have separate comparator entries for their two main results. No scope pages were present for 164, 166, or 171 in the pinned snapshot. The later combinatorics scope pages also omit the graph-decomposition companion in 175, the two broader polynomial-difference papers in 182, the correspondence-coloring paper in 184, and the hypergraph and nonmonotone influence results in 186. Family 191 documents only an unbounded sequence of sizes. Family 187 has a separate checked finite-certificate report in its article. Family 352 likewise records successful runs of its finite shooting certificate and two supplementary rational checks, with their analytic scope stated separately. Family 354 reproduces its finite scalar tables through an independent transcription of the manuscript’s outward arithmetic prescription. No scope page was present for 178. The individual articles state these limits explicitly.

In theoretical computer science, family 104’s scope page lists a randomized mean-payoff algorithm and a finite counterexample to a particular optimization rule. It does not list the deterministic, stochastic, and parity extensions covered by the family article as formalized headline algorithms. No scope pages were present for 103, 109, or 120 in the pinned snapshot. Family 107 documents the stronger complex 9/4 bound despite listing the September manuscripts. Family 113 selects a triangle-expansion face correspondence rather than its full global face-dimension theorem, and 114 selects the matroid FPRAS rather than the later binary-capacity polymatroid extension. Family 117 selects the SDP-gap sequence rather than the hardness companion, and 119 selects information contraction rather than the separate Hellinger theorem. Family 122 selects the sample lower bounds despite a broader heading mentioning a uniform decoder. Family 126 records superpolynomial unshifted matching bounds rather than the stronger exponential shifted-matrix theorem.

For differential geometry, 333’s documented selection is the immersion theorem, without an embedding conclusion. Family 334 documents the local obstruction but does not explicitly select its additional Euclidean Taylor-jet strengthening. Family 337 selects the integral-filling theorem and sharpness of its coefficient, rather than the generalized smooth comparison or Euclidean equality-region classification. Family 342 selects fixed-structure compatibility without a prescribed class; its scope page does not explicitly select the cone identity or positive-index-one same-class corollary. No scope pages were present for 335, 336, 338, 339, 340, 341, 344, 345, 346, 349, 351, 352, 355, or 357 in the pinned release. Family 343 selects the full ball-packing criterion, including its closed-source neighborhood convention. Despite its broad heading, the scope page for 347 selects only the degenerate three-fixed-point quadric example, not the nondegenerate Morse-number or stable-Morse constructions. Family 348 likewise documents only the strictly positive sectional-curvature classification; its nonnegative-curvature and L²-gap companions are outside that documented selection. Family 353 selects Euclidean-complete graph rigidity in dimensions three through nine; its immersed consequence and smooth dimension-ten counterexample are outside the documented selection despite the broader heading. Family 354 documents the full isoperimetric profile and all minimizers, including both transition equality cases. Despite its characterization heading, family 356 selects only the weak-Hessian-to-every-geodesic companion theorem, not the full Alexandrov characterization. Family 358 selects the main three-manifold existence theorem, including constant geodesics and unbounded time intervals, but excludes the separate no-focal-points and CAT(0) consequences. Family 359 selects the negatively pinched threefold and its bounded-coordinate obstruction; its one-sided negative-curvature Liouville companion is outside the documented selection. Family 360 documents both convexity and uniform bi-Hölder transport, but its extra local-injectivity selection covers only finite target families and explicitly excludes the manuscript’s arbitrary-potential assertion. Family 361 selects the dimension-16, degree-50000 counterexample, excluding the separate tangent-cone and cone-spectrum conclusions; its three-dimensional companion is not listed in that scope page. Family 350 documents the surface upper bound and both qualitative counterexamples; for the power-law manuscript, its selected statement records only the unbounded square-root ratio on S⁴ × S¹, without the fixed positive power excess.

The source snapshot is adc7f1241b42, accessed October 6, 2026. Every manuscript link in an article uses that commit, so later upstream revisions do not silently change the source being explained.

See Also

• Coverage register provides the complete family inventory and exclusions.
• Latent Seminar is the wider collection of mathematical notes.

🏷️ Topology: actions, manifolds, and homotopy

Family 304 explains the claimed obstruction to faithful p-adic actions through a sphere invariant with uniformly bounded denominators. It distinguishes the real coefficient plane, integral Witt classes, rational degree cancellation, and deep categorical inputs. Family 305 distinguishes two disk-embedding obstructions and the companion manifold-realization claim, explaining the mixed-parameter tensor calculation and the role of marked boundaries. Family 306 separates the imported cosmetic-surgery reduction, finite-lattice nonvanishing, doubled projective-link calculation, and the gauge-theoretic compactness and gluing interfaces. Family 307 compares the reduced disk-winding construction with the maximal torus construction, including integral persistence, averaging transfers and universal-norm control. Family 308 contrasts varying-prime finite descent with the explicit prime-1009 obstruction calculation for Smith–Toda complexes. Family 309 separates the prime-three Kervaire detector exclusion from the Moore-spectrum construction of an actual order-three representative. Family 310 explains Quillen homology through augmented chains, frame coefficient bounds, fixed-complement restriction and propagation. Family 311 connects stabilizer-orbit rigidity with the finite chain of chromatic supports and thick tensor ideals. Family 312 develops the homotopy hypothesis through elementary expansions, tree comparison and partial cylinders. Family 313 treats finite generation for the local sphere: transfers and compactness. Family 314 treats chromatic Smith theory: cyclic length and fixed-point loss. Family 315 treats the four-dimensional Singer conjecture: finite energy and boundary paths. Family 316 treats the stable Hurewicz image: last suspensions and cone detection. Family 317 treats Thomason structures: strict higher categories and stationary prisms. Family 318 treats chromatic splitting: filtrations, attachments and sphere kernels. Family 319 treats Hahn–Wilson finite generation: a gluing obstruction. Family 320 treats Borel rigidity in dimension four: cyclic covers and realized symmetries. Family 321 treats Wall’s finite D(2) problem: cyclic towers and twisted homology. All eighteen topology families are covered.