OpenAI Math Digest

Accessible guides to the mathematical result families in OpenAI’s math collection. Each article explains the question, its history, previous partial results, the new claims, and the ideas behind them. The intended reader knows undergraduate mathematics but need not specialize in the subject.

AI declaration

These digests are written by GPT-6-Astra. They have not been verified or checked by a human. Mathematical results are presented as claimed in the cited manuscripts. The digest process does not verify proofs or certify the correctness of results.

AI-generated explanations may contain errors. Readers can consult the linked manuscripts for the original statements and arguments. Importance scores reflect editorial judgments about the significance of the stated claims.

Family index

Family numbers follow the source catalogue. Open a category to browse its digests. Each list starts in index order; choose importance to see higher scores first, with equal scores kept in index order.

373 families digested · 17 categories

Number theory 31 families

Number theory digests

Index	Title	Importance
001	Milne’s rationality conjecture and geometry after reduction	4/5
002	The full BSD formula from a small Selmer group	5/5
003	A fixed zero-free region for zeta and Dirichlet L-functions	5/5
004	Hilbert’s tenth problem over the rationals: a claimed limit of algorithms	5/5
005	Catalan’s constant: a determinant approach to irrationality	4/5
006	Goldfeld’s conjecture: half the twists have rank zero, half rank one	5/5
007	Two-point Chowla: do neighboring integers forget prime-factor parity?	5/5
008	Deligne–Drinfeld: all associator symmetries from odd-weight generators	4/5
009	Reconstructing a function field from small arithmetic invariants	4/5
010	Modularity at two: geometric representations and completed Hecke families	5/5
011	Factoring one less than a prime: a Poisson–Dirichlet law	4/5
012	Largest prime factors of consecutive integers: a joint Dickman law	4/5
013	Inverse Goldbach: why the primes cannot split into a sumset	4/5
014	Arthur parameters and geometric Langlands over finite fields	5/5
015	How arithmetic torus packets spread through spaces of lattices	4/5
016	Zilber–Pink: when too many special intersections force structure	5/5
017	How well can fractions approximate pi?	4/5
018	Margulis–Platonov: local origins of abstract normal subgroups	4/5
019	Recovering p-adic points from fundamental-group sections	5/5
020	Squarefree quartic values and the large-prime obstacle	4/5
021	Jacobsthal’s function: how long can primes cover an interval?	3/5
022	Rational approximation with any fixed shift	4/5
023	The small positive bias in cubic Gauss sums	3/5
024	Counting distinct values of Euler’s totient function	3/5
025	Short Egyptian fractions and doubly exponential families	3/5
026	Large prime gaps occur with positive frequency	3/5
027	Integral points throughout character varieties of curves	4/5
028	The Gaussian moat: why bounded steps cannot go forever	4/5
029	Artin’s primitive roots for every admissible integer base	5/5
030	Modularity for every elliptic curve over an imaginary quadratic field	5/5
031	Uchida’s conjecture: recovering field embeddings from Galois maps	4/5

Algebraic and complex geometry 36 families

Algebraic and complex geometry digests

Index	Title	Importance
032	Hodge classes, complex multiplication, and the search for algebraic cycles	5/5
033	Adding the complexity of a base and its fibers	5/5
034	From nef canonical bundles to maps: abundance and uniform degrees	5/5
035	When one numerical direction becomes a fibration in positive characteristic	4/5
036	Numerical semiampleness and the existence of minimal models	5/5
037	The ordinary double point as the largest singular volume	4/5
038	Fujita freeness: how much positivity removes every base point?	5/5
039	Nagata’s conjecture and the cost of imposing many multiple points	5/5
040	Family 040 — Bloch’s conjecture: when points on a surface have no hidden differences	5/5
041	Family 041 — Hyperkähler fibrations: turning a boundary class into a map	5/5
042	Family 042 — Oka K3 surfaces: extending holomorphic maps with freedom	4/5
043	Family 043 — P = W: two ways to organize the same cohomology	4/5
044	Family 044 — Hikita’s conjecture: cohomology hidden inside a fixed scheme	4/5
046	Family 046 — Shafarevich’s conjecture: two claimed failures of holomorphic convexity	5/5
047	Family 047 — Affine cancellation: when adding one coordinate hides the difference	5/5
048	Family 048 — Free vector fields on a singular surface	5/5
049	Family 049 — Affine space embedded in a way coordinates cannot straighten	4/5
050	Family 050 — Ample bundles without Griffiths-positive metrics on the quadric	4/5
051	Family 051 — Hyperbolicity forces an ample canonical bundle	5/5
052	Family 052 — From a split tangent bundle to a product of manifolds	4/5
053	Family 053 — A relation beyond Pixton’s original system	4/5
054	Family 054 — K3 structures without rationality for cubic fourfolds	5/5
055	Family 055 — Stability at the Gepner point and at large volume	5/5
056	Family 056 — Why four-dimensional minimal model programs should stop	5/5
057	Family 057 — Special geometry and almost commutative fundamental groups	5/5
058	Family 058 — Universal covers with a finite algebraic description	5/5
059	Family 059 — The same local topology with different multiplicities	5/5
060	Family 060 — Finding a spherical shell inside every class VII surface	5/5
062	Family 062 — Contact geometry forces a homogeneous model	5/5
063	Family 063 — The sharp tradeoff between Fano curves and divisor directions	4/5
064	Family 064 — Constant Milnor number and an unchanged surface singularity	5/5
065	Virasoro constraints for projective bundles and complete intersections	5/5
066	Bounded klt complements and uniformly mild hypersurfaces	5/5
067	Nef tangent bundles force homogeneity in dimension six	4/5
068	From semipositive anticanonical curvature to algebraic sections	5/5
069	Quantum geometric Langlands at every irrational level	5/5

Real and complex analysis 16 families

Real and complex analysis digests

Index	Title	Importance
071	Round holes for every planar domain	5/5
072	How strongly can a conformal map distort area	5/5
073	A large fractal must determine many distances	5/5
074	Kakeya tube averages in three dimensions and sets in four	5/5
075	Fourier convergence at the L log L threshold	5/5
076	Nearly constant waves made only from plus and minus signs	5/5
077	Fourier extension from positively curved surfaces	5/5
078	How much smoothing makes spherical Fourier sums bounded	5/5
079	Time averaging smooths waves in every dimension	5/5
080	Schrödinger evolution returns to rough data at the exact threshold	5/5
081	Riesz transform bounds reveal hidden surface geometry	5/5
082	Controlling the triangular Hilbert transform across all scales	5/5
083	Hilbert transforms along a changing direction field	5/5
084	A nearly full interval can avoid every scaled geometric sequence	5/5
085	Taking the largest disk average without creating too much variation	4/5
086	Three interacting signals and the trilinear Hilbert transform	5/5

Convex and metric geometry 15 families

Convex and metric geometry digests

Index	Title	Importance
087	How small can a convex body and its polar be together	5/5
088	The smallest and unexpectedly large bodies made from shadows	5/5
089	Representing planar and bounded-treewidth distances by cuts	5/5
090	Why the triangular lattice minimizes so many planar energies	5/5
091	A geometric-mean law for symmetric convex bodies	5/5
092	The optimal order of overlap needed to cover high-dimensional space	5/5
093	When Gaussian tails in every direction control nonlinear fluctuations	5/5
094	Compressing finite Lp data into subpolynomially many coordinates	5/5
095	Where hyperbolic optimization goes beyond semidefinite representations	5/5
096	Why three Gaussian sectors can beat arbitrarily many cells	4/5
097	Keeping every partial sum of a vector sequence under control	4/5
098	A compact metric set that still needs infinitely many dimensions	5/5
099	The unavoidable distortion when edit distance is represented in L1	4/5
100	Covering a tetrahedron with less cylinder area than expected	4/5
101	Why simplices might be the most spread-out convex bodies	5/5

Theoretical computer science 41 families

Theoretical computer science digests

Index	Title	Importance
102	The Unique Games Conjecture and optimal approximation thresholds	5/5
103	Derandomization of logarithmic space — L = RL = BPL	5/5
104	Quasipolynomial algorithms for mean-payoff, stochastic and parity games	4/5
105	The 2-to-1 Games Conjecture with perfect completeness	3/5
106	Hardness of coloring three-colorable graphs	4/5
107	Matrix multiplication with exponent at most nine quarters	5/5
108	A cubic permanent–determinant lower bound	4/5
109	Integer multiplication below n log n	5/5
110	Optimal-order randomized k-server on arbitrary metrics	4/5
111	One-sample matroid prophet inequalities against an almighty adversary	3/5
112	Beyond the square-root exponent for depth-three circuits	4/5
113	Approximate counting and entropy of perfect matchings	4/5
114	Approximate counting of common integer polymatroid bases	3/5
115	Sampling and counting contingency tables with arbitrary margins	3/5
116	Uniform identity testing for noncommutative formulas	4/5
117	Uniform sparsest cut hardness and semidefinite gaps	4/5
118	Unbounded bin-packing gaps and the modified integer round-up conjecture	3/5
119	The Courtade–Kumar and Hellinger conjectures	4/5
120	Almost-linear-time exact matching in general graphs	4/5
121	Almost-linear expected-time approximation of edit distance	4/5
122	Trace reconstruction: how many incomplete copies reveal a string?	4/5
124	Three-machine scheduling: exact optimization with precedence constraints	4/5
125	Metric k-median: approaching the sharp approximation threshold	4/5
126	Perfect matching: why exact semidefinite descriptions must be large	4/5
127	Polynomial threshold functions: how often can one flipped bit change a decision?	3/5
128	Shortest common superstring: assembling fragments within a factor of two	3/5
129	Two-way finite automata: exponential costs of determinism and complement	3/5
130	Exact Fourier transforms below n log n: the role of the arithmetic model	4/5
131	Random graphs with prescribed degrees: a universal switch-chain mixing bound	3/5
132	Sensitivity versus block sensitivity: beyond every quadratic bound	3/5
133	Weisfeiler–Leman refinement: the cost of deciding what it can distinguish	4/5
134	Generalized star height: three nested repetitions for every regular language	4/5
135	Iterated matrix multiplication: the gate cost of five homogeneous layers	2/5
136	PCP for PPAD: preserving a search problem despite many failed constraints	5/5
137	One-tape computation: trading unrestricted time for two-fifths-power space	2/5
138	Subset Sum: a smaller time exponent and a separate low-memory algorithm	4/5
139	Log-concave sampling: a zero dimension exponent for exact oracle queries	2/5
140	Noiseless Gaussian regression: the sample price of finite memory and precision	1/5
141	Exact real feasibility through a fixed number of counting layers	5/5
142	Factoring polynomials over prime fields without random choices	5/5
378	Counting solutions faster than the Exponential Time Hypothesis allows	5/5

Dynamical systems and ergodic theory 12 families

Dynamical systems and ergodic theory digests

Index	Title	Importance
143	How many isolated oscillations can a polynomial system sustain	5/5
144	One observable whose time shifts describe an entire smooth system	5/5
145	Does mixing two observations force mixing any finite number	5/5
146	Area-visible chaos in the standard map at every large parameter	5/5
147	A full collar of billiard caustics forces an ellipse	5/5
148	Fractal dimension from the information that exact overlaps leave behind	5/5
149	A common safe region for every trajectory of a weakly reversible reaction network	5/5
150	An irrational angle makes a triangular billiard weakly mixing	5/5
151	Topological growth without orbit entropy at one derivative	5/5
152	Zero entropy does not guarantee a smooth model with smooth volume	5/5
153	When a random geometric series has a density	5/5
154	Arithmetic-progression observations converge along almost every mixing orbit	5/5

Combinatorics 37 families

Combinatorics digests

Index	Title	Importance
155	A translational tile that cannot repeat periodically in three dimensions	4/5
156	Why ten smaller pieces may not cover a set in nine dimensions	4/5
157	Graph coloring and clique minors after the coefficient-one conjectures	5/5
158	Why five colors would not suffice for the Euclidean plane	4/5
159	Arithmetic progressions from a divergent reciprocal sum	5/5
160	How long can a coloring avoid a long arithmetic progression	4/5
161	When a bipartite pattern is rarer than the random-graph benchmark	5/5
162	Intersecting hypergraphs that need one more covering vertex	5/5
164	One color for every finite subset sum and product	5/5
165	Exact crossing counts for complete and complete bipartite graphs	5/5
166	How many different distances must occur in higher dimensions	5/5
167	Distances from one point and limits on repeated unit distances	4/5
168	Can an unlabeled order determine a Kazhdan–Lusztig polynomial	5/5
169	Positive elementary coefficients for refined graph colorings	4/5
170	The logarithmic scale of off-diagonal Ramsey numbers	4/5
171	A monochromatic hypercube in a network only a constant times larger	5/5
172	Which finite shapes must appear in one color	5/5
173	Finding a vertex whose second neighborhood is at least as large as its first	5/5
174	A spanning tree that stays thin across every cut	5/5
175	Fractional covers, discrete convexity, and random-graph thresholds	5/5
176	Predicting graph appearance from the expected counts of all subgraphs	5/5
177	Sparse higher-dimensional spaces with no hidden parity obstructions	5/5
178	Constructing exact nonbipartite Ramanujan graphs at prescribed sizes	4/5
179	Why a perfectly orthogonal cyclic sign pattern would stop at length four	4/5
180	One cycle through every vertex of a cubic bipartite polyhedron	4/5
181	Partitioning every graph into only linearly many cycles and edges	4/5
182	A power saving for sets avoiding polynomial differences	5/5
183	Fewer balanced lines through a planar point set	4/5
184	A full logarithmic gain from excluding a fixed clique	4/5
185	Where unrestricted infinite matroid packing and covering break down	4/5
186	How narrow must a symmetric random-graph transition be	4/5
187	Making Snaky in at most twenty-one moves	1/5
188	The precise residue of random triangle removal	3/5
189	The exact threshold for a red cycle or a blue clique	3/5
190	Why rare ordered binary patterns can be hard to remove	2/5
191	Keeping every triangle larger than the predicted scale	4/5
192	A Boolean function can beat the square-root degree bound by any factor	3/5

Algebra 18 families

Algebra digests

Index	Title	Importance
193	Why a proper algebraic intersection should have positive multiplicity	5/5
194	Flat local maps cannot lower multiplicity	5/5
195	A three-dimensional singularity that no finite module can make Cohen–Macaulay	5/5
196	Zero divisors without finite-order group elements	5/5
197	A one-sided inverse that cannot be reversed in a group algebra	5/5
198	One finite algebra with arbitrarily long terminating resolutions	5/5
199	Vanishing self-extensions need not make a module projective	5/5
200	The largest relation counts come from a lex-plus-powers ideal	5/5
201	Two algebraic generators can produce an infinite-dimensional division ring	5/5
202	Counting modular representations from local weights	5/5
203	Bounded local defect leaves only finitely many block theories	5/5
204	Tensor invariants for even spin groups need no extra dilation	4/5
205	One irreducible representation whose square contains every other	5/5
206	Finite lattices that no finite algebra can realize	5/5
207	A group-algebra trace can remember only finitely many torsion classes	5/5
208	A common home for finite symmetric tensor categories	4/5
209	A regular local ring can lose integral K-theory at its fraction field	5/5
210	Swapping symmetric powers: the sixth case and a quadratic bound	4/5

Probability and statistical mechanics 29 families

Probability and statistical mechanics digests

Index	Title	Importance
211	When a random surface becomes a tree and how a particle moves on it	5/5
212	The shape of random travel times and the absence of infinite shortest lines	4/5
213	Why an infinite percolation cluster would disappear exactly at criticality	5/5
214	When infinite percolation clusters coexist	4/5
215	How two-dimensional spins acquire a mass	5/5
216	The XY transition: a quarter-power law, logarithms, and an exponential length scale	5/5
217	The one-sixth fluctuation scale of the SK spin glass	5/5
218	Why weakly perturbed Ising models would keep the same large-scale geometry	5/5
219	Gaussian-matrix statistics inside sparse random graphs	4/5
220	When escape through a random environment forces a definite direction and speed	5/5
221	A hierarchical formula for the free energy of diluted spin glasses	4/5
222	Random perceptrons: free energy, correlated patterns, and microscopic jamming	5/5
223	Random-cluster interfaces: conformal shapes, natural length, and perturbations	5/5
224	Voronoi percolation: Cardy crossings and the near-critical window	5/5
225	Six-vertex heights and the Gaussian free field across the balanced critical range	5/5
226	Double-dimer loops: from topological records to nested CLE₄ curves	4/5
227	The dynamical phase transition in the Sherrington–Kirkpatrick model	4/5
228	Engineering a temperature transition with radial pair interactions	4/5
229	When noisy descendants remember their ancestor: Potts reconstruction thresholds	4/5
230	Measuring SLE curves beyond their Hausdorff dimension	4/5
231	Building the free uniform spanning forest from independent randomness	4/5
232	Lipschitz random surfaces: Gaussian fields and a tuned SLE₄ interface	5/5
233	The joint scaling limit of critical Ashkin–Teller currents	4/5
234	All-temperature pressure for orthogonally invariant Ising spin glasses	4/5
235	Random SAT thresholds, fluctuations, and computability	4/5
236	The factor-of-IID threshold for the free Ising model on regular trees	4/5
237	The 3/4 size exponent for honeycomb self-avoiding walks	5/5
238	Optimal-order mixing of the Thorp shuffle	4/5
239	Sharp singularity rates for symmetric random sign matrices	4/5

Mathematical logic 6 families

Mathematical logic digests

Index	Title	Importance
240	Eventual uniqueness of large models, without a prescribed threshold	5/5
241	The ordering of computational difficulty has no hidden symmetries	5/5
242	Encoding every accepting computation by exactly one Diophantine witness	5/5
243	Counting without choices cannot express every polynomial-time query	5/5
244	Comparing a partition’s size does not recover the Axiom of Choice	5/5
245	If every legal expression can normalize, every reduction strategy terminates	4/5

Group theory 14 families

Group theory digests

Index	Title	Importance
246	A sphere at infinity determines hyperbolic three-dimensional geometry	5/5
247	An infinite group described by finitely many rules, with only finite-order elements	5/5
248	Why Thompson’s group F cannot have almost invariant finite sets	5/5
249	A group whose algebra needs two dimensions but whose space needs three	5/5
250	When an algorithmic group fits inside a simple group with finite structure in every dimension	5/5
251	Amenability decides when bounded actions can be made unitary and approximate actions repaired	5/5
252	Negative curvature does not guarantee detection by finite groups	5/5
253	An infinite simple amenable group with a finite presentation	5/5
254	Artin groups: complete topological models, controlled subgroups, and a curvature obstruction	5/5
255	Recognizing polycyclic groups from their large-scale shape	5/5
256	Solving independent equations without collapsing the coefficient group	5/5
257	A hyperbolic group that cannot act geometrically on a CAT(0) space	5/5
258	One relation, negative curvature, and finite special covers	5/5
259	The same group can have actions with different costs	5/5

Mathematical physics 25 families

Mathematical physics digests

Index	Title	Importance
260	How much mass does a black hole require? Penrose inequalities beyond time symmetry	5/5
261	When disorder traps a quantum particle—and when extended states survive	5/5
262	The sharp cost of binding: Lieb–Thirring inequalities and their equality cases	5/5
263	The outer electrons of large atoms: ionization bounds and universal scales	5/5
264	Predictability beyond a rotating black hole: three levels of strong cosmic censorship	5/5
265	Why a gapped quantum sheet should have only boundary-sized entanglement	5/5
266	Three complementary measurements in dimension six—and why a fourth is excluded	4/5
267	Condensation in an interacting Bose gas—and the particles left outside it	5/5
268	The spin-one Haldane gap: rings, boundary fields and a topological index	5/5
269	The Laughlin gap: a quantum Hall state that resists weak disorder	5/5
270	Bound states in BFSS matrix mechanics: one at zero, infinitely many above it	5/5
271	How a quantum ferromagnet orders: Bloch’s law and the magnetization sphere	5/5
272	Entanglement without a secret key	5/5
273	How much uncertainty survives when two quantum light beams mix?	5/5
274	Why shallow quantum circuits cannot compute parity	5/5
275	The computational difficulty of Coulomb energy in the full continuum	5/5
276	The classical capacity of a relaxing qubit	4/5
277	Repeating an entangled game: why unusually many wins become exponentially rare	4/5
278	A molecular density that no ordinary Kohn–Sham ensemble reproduces	5/5
279	Quantum factoring with certainty and a fixed finite gate set	4/5
280	Two languages for conformal field theory, with matching particles and fusion	5/5
281	QAOA reaches the spin-glass optimum—when system size goes first	4/5
282	When scale symmetry becomes local conformal symmetry in four dimensions	5/5
283	A Boolean oracle for every quantum unitary	4/5
284	How large can a quantum query advantage be? The quartic boundary	4/5

Operator algebras 19 families

Operator algebras digests

Index	Title	Importance
285	When geometry and operator indices part company: three group-algebra counterexamples	5/5
286	Recovering arithmetic groups from their operator algebras	5/5
287	Can an operator algebra forget how many free generators it had?	5/5
288	Can every bounded representation be made to respect adjoints?	5/5
289	When nearby operator algebras are the same—and when they are not	5/5
290	Simpler subalgebras that preserve every modular commutation constraint	5/5
291	Three forms of regularity become one: comparison, dimension and Jiang–Su absorption	5/5
292	A concrete algebra that cannot fit inside any nuclear norm ultrapower	5/5
293	Operators with invariant subspaces but no hyperinvariant ones	5/5
294	A quasitrace that cannot be a trace—and finite algebras with an infinite tensor product	5/5
295	Every bounded cocycle has a primitive: the Kadison–Ringrose cohomology claim	5/5
296	One operator can generate a whole von Neumann algebra	5/5
297	One irreducible representation does not determine the compact operators	4/5
298	Two free entropies disagree even when both are finite	5/5
299	No scalar shadows in central sequences: a criterion for Jiang–Su stability	5/5
300	Paving every maximal abelian subalgebra with a uniform number of pieces	5/5
301	When every extended trace determines a stabilized operator algebra	5/5
302	Half the mean dimension: an exact bridge from dynamics to operator comparison	5/5
303	From local infinite size to Cuntz-algebra absorption	5/5

Topology 18 families

Topology digests

Index	Title	Importance
304	Continuous symmetries of manifolds must form Lie groups	5/5
305	Four-dimensional disk embedding and a duality group with no manifold model	5/5
306	A nontrivial knot remembers its surgery slope	5/5
307	Large-scale topology can disappear under coarse assembly	5/5
308	Finite spectra that remove the first chromatic parameters	5/5
309	Exactly three survivors in the prime-three Kervaire family	5/5
310	The topology of elementary subgroups detects a normal p-subgroup	5/5
311	A finite chain of building rules for local stable homotopy	5/5
312	Recovering spaces from paths and all their higher homotopies	5/5
313	Finite arithmetic in every degree of the local sphere	5/5
314	Counting the chromatic layers lost when taking fixed points	5/5
315	Why square-summable topology should concentrate in dimension two	5/5
316	Which homology classes really come from stable maps of spheres?	5/5
317	A model of spaces inside every strict higher category dimension	5/5
318	Chromatic splitting: the pieces can fit without splitting apart	5/5
319	Local finiteness does not guarantee a finite global construction	5/5
320	The same homotopy type can hide different aspherical four-manifolds	5/5
321	A finite complex whose third dimension cannot be removed	5/5

Functional analysis 11 families

Functional analysis digests

Index	Title	Importance
322	Does the unit sphere remember the whole Banach space?	5/5
323	Can every Banach space have a separable infinite-dimensional quotient?	5/5
324	The same metric shape can hide different linear Banach spaces	5/5
325	A sharp numerical-range bound for matrix functions	5/5
326	When cotype on both sides forces nontrivial type	5/5
327	Random walks detect superreflexive Banach spaces	5/5
328	Fixed points for nonexpansive maps in reflexive spaces	5/5
329	A claimed failure of universal entropy duality	5/5
330	Discrete distances do not guarantee bounded approximation	5/5
331	Midpoint convexity without asymptotically convex renormings	5/5
332	Markov cotype of ℓ1 and Lipschitz extension	5/5

Differential geometry 29 families

Differential geometry digests

Index	Title	Importance
333	Every closed surface metric in four-dimensional Euclidean space	5/5
334	A smooth surface metric that cannot fit locally in three-space	5/5
335	How negative scalar curvature pays for topological complexity	5/5
336	Positive spectral curvature and a controlled loss of two dimensions	5/5
337	Sharp isoperimetry and fillings in nonpositive curvature	5/5
338	Positive bisectional curvature and global complex coordinates	5/5
339	When geodesic entropy forces local symmetry	5/5
340	An exact Lagrangian that cannot be untied by Hamiltonian motion	5/5
341	Deforming positive triples into hyperkähler geometry	5/5
342	From taming to compatibility in four dimensions	5/5
343	Two inequalities for packing symplectic balls	5/5
344	When every geodesic minimizes for the same distance	5/5
345	Infinitely many distinct closed geodesics on every sphere	5/5
346	How large can the singular set of a stationary surface be?	5/5
347	Which topological counts constrain Hamiltonian fixed points?	5/5
348	Einstein four-manifolds from positive curvature to a topological gap	5/5
349	The first minimal-hypersurface volume above an equator	5/5
350	Nodal size: a sharp surface bound and higher-dimensional failures	5/5
351	Can scalar curvature detect a Ricci-flow singularity?	5/5
352	A Calabi flow that becomes singular before reaching a canonical metric	5/5
353	When affine maximal graphs must be paraboloids—and when they need not be	5/5
354	Balls, tubes and slabs: the least-area shapes in a periodic cube	5/5
355	Does every magnification see the same mean-curvature singularity?	5/5
356	Recognizing Alexandrov curvature without a smooth metric	5/5
357	From a Euclidean tangent to coordinates with controlled distortion	5/5
358	Geodesics without focusing on a manifold that forbids nonpositive curvature	5/5
359	Negative Kähler curvature does not guarantee bounded holomorphic coordinates	5/5
360	Weak transport curvature, convex geodesic domains and continuous optimal maps	5/5
361	More polynomial-growth harmonic functions than Euclidean space	5/5

Partial differential equations 16 families

Partial differential equations digests

Index	Title	Importance
362	Can a collisionless relativistic plasma remain smooth for all time?	5/5
363	Two Boltzmann evolutions with the same data and exact local conservation	5/5
364	From Newtonian particles to Boltzmann evolution and Gaussian fluctuations	5/5
365	What one boundary patch can reveal about an unknown interior	5/5
366	Why optimal image edges should consist of arcs, tips and triple junctions	5/5
367	Why seven is the first dimension for singular Bernoulli minimizers	5/5
368	Smoothing three-dimensional deformations without folding or changing the target	5/5
369	Where the slowest heat mode reaches its hottest and coldest values	5/5
370	The critical boundary between existence and nonexistence for Lane–Emden systems	5/5
371	Stable blowup in a defocusing Schrödinger equation	5/5
372	Can static boundary measurements determine an elastic body’s two moduli?	5/5
373	When optimal Coulomb transport must split mass	5/5
374	How sensitively does an optimal transport map depend on its target?	4/5
375	Why monotone Allen–Cahn transitions must be planar through dimension eight	5/5
376	Programming a viscous fluid to detect whether a computation halts	4/5
377	How smooth is the least-steep extension of a function?	5/5

How to read the ratings

Importance measures relative mathematical significance across the collection: 1 is a focused contribution, 2 a useful specialized advance, 3 a substantial advance within a field, 4 a major advance with broad consequences or methods, and 5 an exceptional advance on a central problem. Ratings use the full scale as coverage grows and may be recalibrated. Each digest explains its score.

An importance rating assesses the significance of the stated result if correct. A completed digest means its exposition is ready to read; it does not indicate expert verification.

Sources

The overview introduces the families, and the manuscript map links their papers. Articles cite fixed repository revisions and record when those sources were consulted. Any discussion of formalization or expert review is attributed to its source.

Essential definitions are included in each article, with optional background links for further reading.

Recent additions

• 2026-10-27 — Family 378: exact counting for fixed-width satisfiability in subexponential time and polynomial space. All 373 families in the October 26 catalogue now have digests.

• 2026-10-26 — All 29 Differential geometry families (333–361) and all 16 Partial differential equations families (362–377), covering 78 manuscripts together.

• 2026-10-25 — All 25 Mathematical physics families (260–284), all 19 Operator algebras families (285–303), and all 11 Functional analysis families (322–332), covering 109 manuscripts together.

• 2026-10-22 — All 14 Group theory families (246–259) and all 18 Topology families (304–321), covering 48 manuscripts together.

• 2026-10-21 — All 12 Dynamical systems and ergodic theory families (143–154), all 18 Algebra families (193–210), and all six Mathematical logic families (240–245), covering 56 manuscripts together.

• 2026-10-14 — All 16 Real and complex analysis families, from 071: circle domains through 086: the trilinear Hilbert transform, and all 15 Convex and metric geometry families, from 087: Mahler volume products through 101: the sharp simplex inequality, covering 56 manuscripts together.

• 2026-10-12 — All 36 listed Algebraic and complex geometry families, from 032: Hodge classes and algebraic cycles through 069: quantum geometric Langlands, covering 86 current manuscripts.

• 2026-10-11 — The remaining 28 Probability and statistical mechanics families, from 211: critical random planar maps through 239: symmetric random sign matrices, covering 97 manuscripts. All 29 listed families in this category are now covered.

• 2026-10-11 — All 37 listed Combinatorics families, from 155: aperiodic tiling in three dimensions through 192: the square-root degree conjecture, covering 50 manuscripts.

• 2026-10-10 — All 31 Number theory families, from 001: Milne’s rationality conjecture through 031: Uchida’s homomorphism conjecture, covering 58 manuscripts.

• 2026-10-09 — 35 more Theoretical computer science families, from 107: matrix multiplication through 142: deterministic polynomial factorization.

• 2026-10-08 — Family 102: Unique Games and approximation thresholds.

• 2026-10-08 — Family 103: derandomization of logarithmic space.

• 2026-10-08 — Family 104: algorithms for mean-payoff, stochastic, and parity games.

• 2026-10-08 — Family 105: perfect completeness for two-to-one games.

• 2026-10-08 — Family 106: three-colorable graphs and independent-set hardness.

• 2026-10-08 — Family 227: first digest, introducing the claims and main ideas of eight constituent manuscripts.