Complete family assessment directory

Edition 2026-10-09-v13. Every one of the 372 active result families has its own source subdirectory and individual first-pass application assessment. All 719 catalog manuscripts are inventoried; they have not all received individual main-paper or proof review. Source revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.

Use the product catalog for 41 developed software/business hypotheses and the commercial priorities for the initial implementation shortlist. A family without a developed dossier still has an assessment explaining its possible use, material limit and next verification action. Academic reference uses are not presented as demonstrated lucrative businesses.

Discipline coverage

Discipline	Assessed families
Algebra	18
Algebraic and complex geometry	36
Combinatorics	37
Convex and metric geometry	15
Differential geometry	29
Dynamical systems and ergodic theory	12
Functional analysis	11
Group theory	14
Mathematical logic	6
Mathematical physics	25
Number theory	31
Operator algebras	19
Partial differential equations	16
Probability and statistical mechanics	29
Real and complex analysis	16
Theoretical computer science	40
Topology	18

Algebra

Family	Result and source directory	Latest application assessment	Dossier IDs
193	Serre’s intersection-multiplicity conjecture	Intersection multiplicity consistency library	First-pass assessment only
194	Lech’s multiplicity conjecture	Multiplicity monotonicity formal adapter	First-pass assessment only
195	A counterexample to the small Cohen–Macaulay module conjecture	Cohen–Macaulay approximation limitation ledger	First-pass assessment only
196	A counterexample to Kaplansky’s zero-divisor conjecture	Group-algebra zero-divisor counterexample fixture	First-pass assessment only
197	A torsion-free group algebra that is not directly finite	Algebraic inverse and cellular-automaton scope audit	First-pass assessment only
198	A counterexample to finitistic-dimension finiteness	Homological dimension counterexample library	First-pass assessment only
199	Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures	Homological conjecture dependency and counterexample graph	First-pass assessment only
200	Eisenbud–Green–Harris and lex-plus-powers	Hilbert function and Betti bound research adapter	First-pass assessment only
201	A counterexample to Kurosh’s division-ring problem	Division-ring local-finiteness limitation reference	First-pass assessment only
202	The blockwise Alperin weight conjecture	Brauer character and weight counting cross-check	First-pass assessment only
203	Donovan's conjecture over fields and complete mixed-characteristic DVRs	Block classification finiteness provenance	First-pass assessment only
204	Tensor saturation for even spin groups	Spin-group tensor invariant scale reduction	First-pass assessment only
205	Saxl’s conjecture and universal tensor squares	Symmetric representation tensor-square reference	First-pass assessment only
206	Finite lattice representation and undecidability	Finite algebra representation solver completeness contract	First-pass assessment only
207	The ℓ¹-Bass conjecture for all discrete groups	Idempotent trace support and coefficient-regime audit	First-pass assessment only
208	Finite symmetric tensor categories and the Verlinde tower	Finite symmetric tensor category fiber-functor adapter	First-pass assessment only
209	Integral counterexamples to Gersten’s conjecture	K-theory localization information-loss fixture	First-pass assessment only
210	Foulkes' conjecture for sixth powers and quadratic stabilization	Symmetric power map stabilization benchmark	First-pass assessment only

Algebraic and complex geometry

Family	Result and source directory	Latest application assessment	Dossier IDs
032	The rational Hodge conjecture for CM abelian varieties	Hodge–Tate proof dependency adapter	002
033	Iitaka subadditivity, variation, and logarithmic additivity	Orbifold subadditivity hypothesis ledger	First-pass assessment only
034	Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity	Abundance and Iitaka dependency evidence	First-pass assessment only
035	Log-canonical threefold abundance in numerical dimension one	Positive-characteristic threefold adjoint contract	First-pass assessment only
036	Numerical semiampleness and generalized minimal models	Numerical semiampleness model contract	First-pass assessment only
037	The ordinary-double-point volume gap	Normalized-volume singularity reference	First-pass assessment only
038	Fujita’s freeness conjecture	Adjoint linear-system generation contract	First-pass assessment only
039	Nagata’s conjecture and maximal Seshadri constants	Very-general-point interpolation boundary	First-pass assessment only
040	Bloch’s conjecture for complex surfaces	Surface cycle-class assumption reference	First-pass assessment only
041	Hyperkähler SYZ and projective-space bases	Hyperkähler fibration existence ledger	First-pass assessment only
042	Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII	Complex-manifold Oka classification reference	First-pass assessment only
043	<i>P</i> = <i>W</i> for fixed-determinant SL<sub><i>n</i></sub> moduli spaces	P=W cohomology computation cross-check	First-pass assessment only
044	The equivariant cohomological Hikita conjecture	Quiver cohomology correspondence adapter	First-pass assessment only
046	Shafarevich counterexamples in dimension two and with large fundamental group	Universal-cover convexity counterexample library	First-pass assessment only
047	Zariski cancellation and affine fibrations over the complex numbers	Algebraic cancellation counterexample fixture	First-pass assessment only
048	A characteristic-zero counterexample to Lipman–Zariski	Tangent-sheaf smoothness contract counterexample	First-pass assessment only
049	A stable-coordinate counterexample in four variables	Stable polynomial coordinate counterexample	First-pass assessment only
050	A counterexample to Griffiths’ positivity conjecture	Bundle positivity assumption audit	First-pass assessment only
051	Kobayashi’s canonical-ampleness conjecture	Hyperbolicity-to-projectivity evidence ledger	First-pass assessment only
052	Tangent splittings and product decompositions	Holomorphic splitting integrability contract	First-pass assessment only
053	A counterexample to Pixton completeness in Chow	Tautological relation completeness counterexample	First-pass assessment only
054	Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations	Cubic-fourfold rationality scope audit	First-pass assessment only
055	Gepner symmetry and large-volume stability on threefolds	Stability-condition research adapter	First-pass assessment only
056	Termination of projective and Kähler fourfold minimal model programs	Termination-versus-runtime solver contract	First-pass assessment only
057	Fundamental groups of special complex varieties and root orbifolds	Special-manifold group implication ledger	First-pass assessment only
058	Semialgebraic universal covers and bounded domains	Semialgebraic universal-cover classification reference	First-pass assessment only
059	Counterexamples to Zariski’s multiplicity conjecture	Topology-versus-multiplicity counterexample	First-pass assessment only
060	The Global Spherical Shell conjecture	Class-VII surface classification evidence	First-pass assessment only
062	Projective contact classification and the LeBrun–Salamon conjecture	Quaternionic and contact classification ledger	First-pass assessment only
063	The generalized Mukai conjecture	Fano invariant inequality reference	First-pass assessment only
064	Topological triviality of <i>μ</i>-constant surface singularities	Singularity-family topological triviality contract	First-pass assessment only
065	Virasoro constraints for complete intersections and projective-bundle towers	Projectivization transfer is a conditional research computation lead	First-pass assessment only
066	Bounded klt complements for Fano contractions	Complement-index boundedness contract	First-pass assessment only
067	The Campana–Peternell conjecture in dimension six	Six-dimensional Fano classification reference	First-pass assessment only
068	Anticanonical nonvanishing in every dimension	Semipositive anticanonical section contract	First-pass assessment only
069	Global quantum geometric Langlands at irrational level	Geometric Langlands correspondence provenance	First-pass assessment only

Combinatorics

Family	Result and source directory	Latest application assessment	Dossier IDs
155	A counterexample to periodic tiling in dimension three	Three-dimensional tiling feasibility and periodicity audit	First-pass assessment only
156	Borsuk's conjecture fails in dimension nine	Diameter partition counterexample library	First-pass assessment only
157	Graph coloring, clique minors, and Colin de Verdière invariants	Graph coloring bound assumption auditor	First-pass assessment only
158	The Euclidean plane cannot be colored with five colors	Plane-coloring boundary and finite certificate audit	First-pass assessment only
159	Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds	Arithmetic progression density reference	First-pass assessment only
160	Superexponential van der Waerden numbers	Ramsey forcing-scale reference	First-pass assessment only
161	Counterexamples to Sidorenko’s conjecture and the forcing conjecture	Graph null-model motif counterexample audit	First-pass assessment only
162	Counterexamples to Ryser’s covering conjecture	Hypergraph cover relaxation limitation audit	First-pass assessment only
164	Hindman’s finite sums and products conjecture	Finite sums/products Ramsey research explorer	First-pass assessment only
165	The Harary–Hill and Zarankiewicz crossing-number formulas	Complete-graph drawing crossing reference	First-pass assessment only
166	The higher-dimensional Erdős distinct-distances conjecture	Distinct-distance point-set reference	First-pass assessment only
167	Planar distinct distances and unit-distance bounds	Planar geometric incidence benchmark	First-pass assessment only
168	Combinatorial invariance of Kazhdan–Lusztig polynomials	Bruhat interval polynomial caching library	First-pass assessment only
169	Shareshian–Wachs elementary positivity	Elementary-basis witness and algebra audit	039
170	Sharp logarithmic exponents for off-diagonal Ramsey numbers	Off-diagonal Ramsey exponent reference	First-pass assessment only
171	The hypercube Ramsey conjecture	Hypercube Ramsey forcing reference	First-pass assessment only
172	Classification of finite Euclidean Ramsey configurations	Euclidean Ramsey configuration classifier research	First-pass assessment only
173	Seymour’s second-neighborhood conjecture	Oriented graph two-hop reach benchmark	First-pass assessment only
174	Deterministic construction of strong thin spanning trees	Thin trees: all-cut checking and conservative signing constants	024, 036
175	Talagrand’s expectation thresholds, discrete convexity, and graph decompositions	Expectation-threshold and decomposition reference	First-pass assessment only
176	The second Kahn–Kalai conjecture with an edge-count bound	Random-graph motif appearance planner	First-pass assessment only
177	Bounded-degree coboundary expanders	High-dimensional expander research generator	First-pass assessment only
178	Deterministic nonbipartite Ramanujan graphs in every fixed degree	Spectral construction: millions of tracked parameters and uninstantiated thresholds	036
179	The circulant Hadamard and Barker-sequence conjectures	Binary waveform constraint feasibility audit	035
180	Barnette’s Hamiltonian-cycle conjecture	Hamiltonian graph-class feasibility reference	First-pass assessment only
181	The Erdős–Gallai cycle-decomposition conjecture	Graph cycle decomposition research toolkit	First-pass assessment only
182	Power savings for intersective polynomial differences and prime arguments	Polynomial-difference avoidance reference	First-pass assessment only
183	Power savings for planar halving lines and <i>k</i>-sets	Geometric split-count research benchmark	First-pass assessment only
184	Correspondence coloring with a fixed forbidden subgraph	Conflict coloring class assumption audit	First-pass assessment only
185	Counterexamples to infinite matroid intersection and packing/covering	Infinite matroid guarantee limitation map	First-pass assessment only
186	Uniform influence and sharp thresholds for graph and hypergraph properties	Relabeling-invariant threshold research reference	First-pass assessment only
187	Snaky in 21 Maker moves	Finite strategy certificate teaching engine	First-pass assessment only
188	The sharp terminal leave in random triangle removal	Random triangle-removal process reference	First-pass assessment only
189	Cycle–clique Ramsey numbers	Cycle–clique Ramsey exact reference library	First-pass assessment only
190	Polynomial removal fails for ordered binary matrices	Ordered matrix pattern-sampling limitation audit	First-pass assessment only
191	A power improvement in the Heilbronn triangle lower bound	Nondegenerate planar point-set research design	First-pass assessment only
192	Boolean functions violate the square-root degree bound by arbitrary factors	Boolean Fourier degree limitation reference	First-pass assessment only

Convex and metric geometry

Family	Result and source directory	Latest application assessment	Dossier IDs
087	The Mahler conjectures, functional inequalities and polar-product symplectic width	Convex geometry inequality and equality-case workbench	First-pass assessment only
088	Sharp projection-body inequalities and a counterexample to simplex maximization	Projection volume model validation toolkit	First-pass assessment only
089	Bounded-distortion <i>L</i><sub>1</sub> embeddings of planar and bounded-treewidth graphs	Planar metric embedding and flow-cut review	024
090	Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy	Triangular lattice energy benchmark	First-pass assessment only
091	Logarithmic and <i>L</i><sub><i>p</i></sub> Brunn–Minkowski inequalities and the B-conjecture	Symmetric convex-body interpolation audit	First-pass assessment only
092	The optimal order of convex-body covering density	Covering-density research planner	First-pass assessment only
093	Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures	Log-concave sampling assumption audit	First-pass assessment only
094	Subpolynomial dimension reduction in <i>L</i><sub><i>p</i></sub>	Finite Lp embedding contract audit	025
095	Hyperbolicity cones without semidefinite lifts	Conic formulation scope checker	First-pass assessment only
096	The Gaussian propeller conjecture in every dimension	Kernel clustering hardness and Gaussian partition benchmark	First-pass assessment only
097	The Euclidean Steinitz–Bergström bound	Vector prefix balancing research simulator	First-pass assessment only
098	Compact counterexamples to bi-Lipschitz dimension reduction	Metric compression impossibility demonstrator	First-pass assessment only
099	The sharp exponential scale of edit-distance distortion	Edit metric embedding limits audit	008, 025
100	Cylinder coverings below the half-area bound	Cylinder covering counterexample visualizer	First-pass assessment only
101	The sharp simplex conjecture for isotropic constants	Log-concave entropy and isotropic geometry benchmark	First-pass assessment only

Differential geometry

Family	Result and source directory	Latest application assessment	Dossier IDs
333	Smooth isometric immersions of surfaces into ℝ<sup>4</sup>	Metric surface realization research library	First-pass assessment only
334	A smooth surface metric with no local isometric immersion in ℝ<sup>3</sup>	Surface realization limitation audit	First-pass assessment only
335	Gromov’s integral scalar-curvature bound for simplicial volume	Scalar curvature and topology assumption ledger	First-pass assessment only
336	Spectral scalar curvature, Urysohn width, and macroscopic dimension	Spectral geometry width research planner	First-pass assessment only
337	Sharp Cartan–Hadamard isoperimetry and rigidity	Curved-space isoperimetric and filling reference	First-pass assessment only
338	Yau's uniformization conjecture	Kähler uniformization provenance module	First-pass assessment only
339	Katok's entropy rigidity conjecture	Geodesic entropy rigidity research reference	First-pass assessment only
340	A counterexample to the nearby Lagrangian conjecture	Lagrangian equivalence limitation ledger	First-pass assessment only
341	Donaldson's hypersymplectic deformation conjecture	Hypersymplectic deformation research toolkit	First-pass assessment only
342	Donaldson's tamed-to-compatible conjecture	Almost-complex compatibility assumption map	First-pass assessment only
343	Symplectic ball packing in higher dimensions	Symplectic capacity packing research calculator	First-pass assessment only
344	The metric Blaschke conjecture	Injectivity-radius rigidity research reference	First-pass assessment only
345	Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds	Closed geodesic research benchmark library	First-pass assessment only
346	Sharp singular-set bounds for stationary integral varifolds	Varifold regularity class audit	First-pass assessment only
347	Counterexamples to stable-Morse and strong Arnold fixed-point bounds	Hamiltonian fixed-point lower-bound limitation library	First-pass assessment only
348	Nonnegative-curvature Einstein classification and an <i>L</i><sup>2</sup> topological gap	Einstein four-manifold classification reference	First-pass assessment only
349	The Solomon–Yau least-volume conjecture	Spherical minimal-surface volume reference	First-pass assessment only
350	Yau’s nodal bounds: surfaces and higher dimensions	Spectral nodal-set scope and mesh benchmark	First-pass assessment only
351	Scalar curvature and finite-time Ricci-flow singularities	Ricci flow singularity evidence audit	First-pass assessment only
352	A finite-time singularity of Calabi flow	Calabi flow failure reference	First-pass assessment only
353	Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample	Affine maximal graph rigidity reference	First-pass assessment only
354	The isoperimetric profile of the cubic three-torus	Periodic microstructure interface reference calculator	037
355	Unique tangent flows at the first surface singularity	Mean curvature first-singularity comparison reference	First-pass assessment only
356	Gigli’s characterization of Alexandrov curvature	Metric-measure curvature definition audit	First-pass assessment only
357	Bi-Lipschitz coordinates at every regular RCD point	RCD local coordinate research toolkit	First-pass assessment only
358	A three-manifold without conjugate points or nonpositive curvature	Metric existence implication limitation map	First-pass assessment only
359	Negative Kähler curvature without bounded holomorphic coordinates	Holomorphic coordinate limitation reference	First-pass assessment only
360	Weak MTW curvature gives convexity and regular optimal transport	Curved optimal-transport regularity contract	First-pass assessment only
361	Failure of integer-degree harmonic dimension comparison	Harmonic growth dimension comparison limitation	First-pass assessment only

Dynamical systems and ergodic theory

Family	Result and source directory	Latest application assessment	Dossier IDs
143	Hilbert's sixteenth problem: uniform bounds for limit cycles	Planar oscillator cycle-count assurance	First-pass assessment only
144	Banach’s simple Lebesgue-spectrum problem	Smooth dynamics spectral-model research library	First-pass assessment only
145	Rokhlin’s multiple-mixing problem	Single-action mixing claim contract	First-pass assessment only
146	Positive metric entropy for the standard map	Standard-map chaos research benchmark	First-pass assessment only
147	The near-boundary Birkhoff conjecture	Billiard integrability assumption audit	First-pass assessment only
148	The entropy-rate dimension formula for self-similar measures	Self-similar measure dimension workbench	First-pass assessment only
149	Classwise permanence for weakly reversible mass-action systems	Reaction network model assurance plugin	002, 003, 013
150	Weak mixing of triangular billiards with an irrational angle	Triangular billiard mixing reference	First-pass assessment only
151	A <i>C</i><sup>1</sup> counterexample to the entropy conjecture	Dynamical entropy counterexample provenance	First-pass assessment only
152	Zero entropy does not guarantee a smooth positive-volume model	Smooth-model representation limitation ledger	First-pass assessment only
153	Arithmetic classification and non-Pisot singularity for Bernoulli convolutions	Bernoulli convolution parameter classification research	First-pass assessment only
154	Pointwise multiple ergodic averages for mixing transformations	Multiple ergodic average convergence reference	First-pass assessment only

Functional analysis

Family	Result and source directory	Latest application assessment	Dossier IDs
322	Tingley’s sphere-isometry problem	Banach sphere isometry teaching and formalization module	First-pass assessment only
323	Independence of the separable quotient problem	Set-theoretic assumption provenance viewer	First-pass assessment only
324	Lipschitz equivalent Banach spaces need not be linearly isomorphic	Metric versus linear representation counterexample explorer	First-pass assessment only
325	The complete Crouzeix conjecture	Numerical range polynomial certificate toolkit	001, 014
326	The cotype–cotype conjecture under the approximation property	Banach geometry property implication map	First-pass assessment only
327	Markov type characterizes superreflexivity	Metric Markov type renorming research explorer	First-pass assessment only
328	Nonexpansive fixed points in reflexive Banach spaces	Fixed-point existence assumption checker	First-pass assessment only
329	A counterexample to metric-entropy duality	Covering-number duality counterexample benchmark	First-pass assessment only
330	A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces	Approximation property distinction library	First-pass assessment only
331	Reflexive midpoint convexity and diamond distortion	Convexity and diamond embedding benchmark	First-pass assessment only
332	Metric Markov cotype of <i>ℓ</i><sub>1</sub> and Hilbert-space Lipschitz extension	Lipschitz extension research toolkit	First-pass assessment only

Group theory

Family	Result and source directory	Latest application assessment	Dossier IDs
246	Cannon's conjecture	Hyperbolic boundary recognition research contract	First-pass assessment only
247	An infinite finitely presented residually finite 2-group and a finitely presented nil algebra	Residual-finiteness versus torsion fixture	First-pass assessment only
248	Thompson's group <i>F</i> is nonamenable	Amenability assumption evidence	First-pass assessment only
249	A finitely generated Eilenberg–Ganea counterexample	Cohomological versus geometric dimension fixture	First-pass assessment only
250	Boone–Higman embeddings with higher finiteness	Decidable word-problem embedding contract	First-pass assessment only
251	Amenability, unitarizability, and strong Ulam stability	Representation stability hypothesis adapter	First-pass assessment only
252	A torsion-free hyperbolic group that is neither residually finite nor linear over any field	Finite-representation separation counterexample	First-pass assessment only
253	An infinite finitely presented simple amenable group	Simple amenable group research fixture	First-pass assessment only
254	Classifying spaces and geometric obstructions for Artin groups	Artin membership is effective but has no claimed cost bound	First-pass assessment only
255	Quasi-isometric recognition of virtually polycyclic groups	Coarse-group recognition contract	First-pass assessment only
256	Nonsingular systems of equations over arbitrary groups	Overgroup equation solvability contract	First-pass assessment only
257	A hyperbolic group without a geometric CAT(0) action	CAT(0) action counterexample library	First-pass assessment only
258	Gersten’s conjecture and virtual compact specialness of one-relator groups	One-relator hyperbolicity conditional adapter	First-pass assessment only
259	A group without fixed price	Measured-action cost comparison reference	First-pass assessment only

Mathematical logic

Family	Result and source directory	Latest application assessment	Dossier IDs
240	Shelah's eventual categoricity and the prescribed-threshold obstruction	Model-theory threshold and axiom provenance	First-pass assessment only
241	Rigidity of the Turing degrees	Computability-degree rigidity reference	First-pass assessment only
242	Single-fold Diophantine representations and undecidability under an at-most-one-solution promise	Unique-solution promise solver limitation audit	First-pass assessment only
243	Separating choiceless counting from polynomial time and witnessed choice	Symmetry-preserving query language expressiveness audit	First-pass assessment only
244	The Partition Principle does not imply Choice	Set-theoretic assumption dependency viewer	First-pass assessment only
245	Weak normalization implies strong normalization in pure type systems	Pure-beta normalization assurance has a global premise	038

Mathematical physics

Family	Result and source directory	Latest application assessment	Dossier IDs
260	Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions	Relativistic initial-data inequality audit	First-pass assessment only
261	Localization and delocalization in the Anderson model	Disordered spectral model benchmark	First-pass assessment only
262	Sharp finite-matrix Lieb–Thirring inequalities and all equality cases	Matrix-valued spectral bound library	First-pass assessment only
263	The ionization and generalized ionization conjectures	Atomic binding and asymptotic plausibility checks	First-pass assessment only
264	Strong cosmic censorship near two-ended Kerr data	Relativity extension-class evidence map	First-pass assessment only
265	Area laws and tensor networks for two-dimensional gapped systems	Tensor-network representation feasibility audit	030
266	Exactly three mutually unbiased bases in dimension six	Certified computation reproducibility workbench	034
267	Positive-temperature Bose–Einstein condensation and exact quantum depletion	Dilute Bose gas reference regime planner	First-pass assessment only
268	The spin-one Haldane gap	Spin-chain gap reference suite	First-pass assessment only
269	Uniform Laughlin gap and stability under bounded scalar disorder	Laughlin model disorder robustness audit	First-pass assessment only
270	Threshold and positive-energy bound states of the BFSS matrix model	BFSS spectral conjecture provenance library	First-pass assessment only
271	Bloch's law, its lattice correction, and the spherical magnetization law	Ferromagnet asymptotic reference fitter	First-pass assessment only
272	Entanglement without distillable secret key	Quantum security resource assumption audit	First-pass assessment only
273	The entropy photon-number inequality	Bosonic entropy and channel-bound calculator	First-pass assessment only
274	Parity is not in QAC<sup>0</sup>	Quantum circuit architecture impossibility ledger	First-pass assessment only
275	QMA-hardness of continuum Coulomb energy	Electronic-structure complexity scope audit	First-pass assessment only
276	Classical capacity of generalized amplitude damping	Thermal qubit channel capacity calculator	003, 019
277	Threshold repetition for entangled games	Entangled-game soundness planning reference	First-pass assessment only
278	Failure of Kohn–Sham ensemble representation	Density-functional representation limitation audit	First-pass assessment only
279	Exact quantum factoring over a fixed finite gate set	Exact quantum factoring excludes a practical resource-improvement claim	First-pass assessment only
280	Unitary vertex operator algebras and conformal nets	VOA-to-conformal-net theorem and structure bridge	First-pass assessment only
281	QAOA attains the SK optimum in the thermodynamic-first limit	Quantum optimization regime and resource audit	031
282	From scale symmetry to local conformal symmetry in four-dimensional QFT	Quantum-field assumption provenance map	First-pass assessment only
283	Polynomial-time unitary synthesis from a Boolean oracle	Oracle construction and unitary-synthesis cost audit	031
284	The optimal quartic separation between randomized and quantum queries	Quantum-query complexity comparison ledger	031

Number theory

Family	Result and source directory	Latest application assessment	Dossier IDs
001	Milne’s rationality conjecture and algebraic specialization	Arithmetic cycle specialization provenance	First-pass assessment only
002	The full BSD formula from low Selmer corank	Elliptic-curve BSD evidence ledger	First-pass assessment only
003	The quasi-Riemann hypothesis	L-function assumption and algorithm dependency audit	016
004	Hilbert’s tenth problem over ℚ	Rational constraint solver completeness contract	First-pass assessment only
005	Irrationality of Catalan’s constant	Exact constant provenance teaching module	First-pass assessment only
006	Goldfeld’s conjecture: densities and mean analytic rank	Quadratic twist rank distribution reference	First-pass assessment only
007	Ordinary two-point correlations and the corrected Elliott conjecture	Multiplicative sequence correlation benchmark	First-pass assessment only
008	The Deligne–Drinfeld conjecture	Grothendieck–Teichmüller graded algebra reference	First-pass assessment only
009	Function-field reconstruction from Milnor K-theory and Galois data	Function-field reconstruction formal adapter	First-pass assessment only
010	Unrestricted pro-modularity at the prime two	Dyadic modularity assumption ledger	First-pass assessment only
011	Prime-factor statistics of $`p-1`$	Prime-minus-one factor statistics reference	First-pass assessment only
012	Independent largest prime factors of consecutive integers	Consecutive integer factor statistics benchmark	First-pass assessment only
013	Ostmann’s inverse Goldbach conjecture	Additive prime decomposition limitation reference	First-pass assessment only
014	Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan	Langlands characteristic and assumption coverage graph	First-pass assessment only
015	Torus-packet equidistribution in prime, quartic, and sextic degrees	Arithmetic packet equidistribution research benchmark	First-pass assessment only
016	Zilber–Pink in abelian varieties and the Siegel threefold	Atypical subvariety provenance library	First-pass assessment only
017	The irrationality exponent of <i>π</i> is 2	Constant approximation and Flint–Hills reference	First-pass assessment only
018	The Margulis–Platonov conjecture over global fields	Arithmetic group normal-subgroup classification adapter	First-pass assessment only
019	The local <i>p</i>-adic section conjecture and global consequences	Rational point/Galois section correspondence ledger	First-pass assessment only
020	Squarefree quartics and power-free polynomial values	Polynomial value density reference	First-pass assessment only
021	A quadratic bound for Jacobsthal’s function	Coprime search interval budget research	First-pass assessment only
022	The weak inhomogeneous Duffin–Schaeffer conjecture	Inhomogeneous approximation assumption reference	First-pass assessment only
023	Patterson's first moment for cubic Gauss sums	Cubic Gauss sum exact reference library	First-pass assessment only
024	An asymptotic formula for the number of totients	Totient counting asymptotic reference	First-pass assessment only
025	Short Egyptian fractions	Egyptian fraction construction research library	First-pass assessment only
026	Positive lower density of large prime gaps	Prime-gap distribution reference	First-pass assessment only
027	Potential integral density on curve character varieties	Character variety integral-point provenance	First-pass assessment only
028	Uniformly bounded components of Gaussian-prime graphs	Gaussian prime graph component research explorer	First-pass assessment only
029	Primitive roots for every admissible integer base	Primitive-root prime generation assumption audit	016
030	Modularity of elliptic curves over imaginary quadratic fields	Imaginary-quadratic elliptic curve modularity adapter	First-pass assessment only
031	Uchida’s conjecture for open homomorphisms of Galois groups	Galois homomorphism reconstruction reference	First-pass assessment only

Operator algebras

Family	Result and source directory	Latest application assessment	Dossier IDs
285	Counterexamples to Baum–Connes and Kadison–Kaplansky	Assembly and projection counterexample ledger	First-pass assessment only
286	Rigidity and arithmetic of lattice von Neumann algebras	Rigid lattice bimodule classification reference	First-pass assessment only
287	Isomorphism of the free group factors	Free-group-factor equivalence evidence	First-pass assessment only
288	Kadison's similarity conjecture	C-star similarity hypothesis contract	First-pass assessment only
289	Strong Kadison–Kastler stability and its spatial boundaries	Operator-algebra perturbation scope audit	First-pass assessment only
290	Relative bicentralizers and modular spectral recovery	Bicentralizer inclusion evidence ledger	First-pass assessment only
291	Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability	Nuclear C-star regularity equivalence adapter	First-pass assessment only
292	Kirchberg's $`\mathcal O_2`$ norm-ultrapower embedding problem	Nuclear ultrapower embedding counterexample	First-pass assessment only
293	Invariant projections, hyperinvariant subspaces, and transitive algebras	Infinite-operator invariant-subspace fixture	First-pass assessment only
294	Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness	Quasitrace and tensor-finiteness counterexample	First-pass assessment only
295	The Kadison–Ringrose cohomology conjecture	Bounded cohomology primitive contract	First-pass assessment only
296	The generator problem for finite factors	Single-generator representation research adapter	First-pass assessment only
297	A ZFC counterexample to Naimark's problem	Nonseparable representation contract	First-pass assessment only
298	Two notions of free entropy differ even when both are finite	Free entropy notion separation reference	First-pass assessment only
299	The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability	Central-sequence regularity contract	First-pass assessment only
300	Approximation and quadratic strong-operator paving	Compressed strong paving scope audit	First-pass assessment only
301	Trace cones and Razak–Jacelon stabilization	Stabilized trace-cone classification reference	First-pass assessment only
302	Radius of comparison equals half the mean dimension	Mean-dimension comparison evidence	First-pass assessment only
303	Weak pure infiniteness and Cuntz-algebra absorption	Pure infiniteness and absorption contract	First-pass assessment only

Partial differential equations

Family	Result and source directory	Latest application assessment	Dossier IDs
362	Global smoothness for relativistic Vlasov–Maxwell	Relativistic plasma model assumption and solver benchmark library	029
363	Nonuniqueness with local conservation for the hard-sphere Boltzmann equation	Kinetic solution selection and conservation audit	029
364	Kinetic limits and fluctuations over the Boltzmann lifespan	Molecular-to-kinetic simulation validity planner	First-pass assessment only
365	Joint metric and connection recovery from one boundary patch	Inverse boundary measurement evidence workbench	026
366	The planar Mumford–Shah regularity conjecture and local weak-<i>L</i><sup>4</sup> gradient bounds	Variational segmentation structure diagnostic	028
367	The critical dimension for the one-phase Bernoulli problem	Free boundary critical-dimension research benchmark	028
368	The three-dimensional Ball–Evans approximation problem	Diffeomorphic deformation feasibility research toolkit	First-pass assessment only
369	The hot spots conjecture for simply connected planar domains	Neumann eigenmode extrema validation	First-pass assessment only
370	The Lane–Emden and Hénon–Lane–Emden conjectures	Nonlinear elliptic model nonexistence assumption map	First-pass assessment only
371	Stable blowup for the defocusing Schrödinger equation	Scientific Fourier discretization audit	029
372	Global uniqueness in smooth isotropic elasticity	Elastic material identifiability experiment workbench	027
373	Nonattainment of the three-marginal Coulomb Monge problem	Multi-marginal transport representation audit	First-pass assessment only
374	Sharp one-third stability of Brenier maps	Transport sharpness benchmark and source-domain audit	015, 020
375	De Giorgi's conjecture in dimension eight	Allen–Cahn classification and phase-field research reference	First-pass assessment only
376	Universal computation in forced Navier–Stokes flows	Scientific prediction claim and computational-limit audit	002, 029
377	Interior $`C^{1,\alpha}`$ regularity for infinity-harmonic functions	Infinity-harmonic interpolation regularity reference	029

Probability and statistical mechanics

Family	Result and source directory	Latest application assessment	Dossier IDs
211	The geometric phase diagram, diffusion, and spectra of random planar maps	Random-map geometry and diffusion research benchmark	First-pass assessment only
212	Planar first-passage geometry and the absence of bigeodesics	First-passage routing model asymptotic audit	First-pass assessment only
213	Critical percolation on every quasi-transitive graph	Percolation criticality research simulator	First-pass assessment only
214	The Benjamini–Schramm nonuniqueness conjecture	Nonamenable network phase reference	First-pass assessment only
215	Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics	Spin-model continuum and mass reference calculator	First-pass assessment only
216	Critical and near-critical XY scaling and BKT universality	BKT model scaling analysis workbench	First-pass assessment only
217	The low-temperature Sherrington–Kirkpatrick fluctuation law	Low-temperature SK fluctuation normalization benchmark	First-pass assessment only
218	Conformal universality for weakly interacting and random-bond Ising models	Ising perturbation universality benchmark	First-pass assessment only
219	GOE bulk universality for regular graphs with weak Anderson disorder	Random regular graph spectral calibration	First-pass assessment only
220	Directional zero–one laws beyond iid environments and iid ballisticity	Random-environment transport assumption audit	First-pass assessment only
221	The Mézard–Parisi formula for diluted spin glasses	Diluted spin-glass variational model library	First-pass assessment only
222	Perceptron free energies and microscopic jamming exponents	Perceptron capacity and jamming regime planner	First-pass assessment only
223	Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling	Random-cluster interface reference library	First-pass assessment only
224	Critical and quenched near-critical universality for Poisson–Voronoi percolation	Voronoi critical crossing calibration	First-pass assessment only
225	Gaussian free field limits throughout the balanced six-vertex regime	Six-vertex height variance reference	First-pass assessment only
226	The double-dimer loop ensemble converges to CLE<sub>4</sub>	Double-dimer loop scaling research suite	First-pass assessment only
227	Critical SK autocorrelation processes and dynamics across the temperature transition	Spin sampler clock and regime audit	First-pass assessment only
228	Continuum phase transitions for radial pair potentials	Continuum phase-transition model reference	First-pass assessment only
229	Exact three- and four-state reconstruction thresholds and four-state tree capacity	Graph reconstruction feasibility audit	032
230	Exact Hausdorff gauges for SLE	SLE fractal-measure normalization reference	First-pass assessment only
231	The free uniform spanning forest is a factor of IID	Root-free randomized graph construction research	First-pass assessment only
232	Gaussian fields and interfaces for triangular-lattice Lipschitz heights	Lipschitz random-height reference suite	First-pass assessment only
233	The joint critical Ashkin–Teller current limit	Ashkin–Teller joint structure benchmark	First-pass assessment only
234	All-temperature pressure of orthogonally invariant Ising spin glasses	Invariant spin-glass pressure reference	First-pass assessment only
235	Limiting random SAT thresholds, sharp variance and computability	Random SAT benchmark regime calibration	033
236	The exact factor-of-IID threshold for free Ising spins on trees	Tree Ising distributed-sampling feasibility map	First-pass assessment only
237	The three-quarter exponent for honeycomb self-avoiding walk	Self-avoiding polymer scaling reference	First-pass assessment only
238	Optimal logarithmic mixing of the Thorp shuffle	Queryable permutation framework and randomness bridge	040
239	Sharp singularity rates for symmetric random sign matrices	Random symmetric matrix singularity reference	First-pass assessment only

Real and complex analysis

Family	Result and source directory	Latest application assessment	Dossier IDs
071	Koebe’s circle-domain conjecture	Conformal domain representation research toolkit	First-pass assessment only
072	Brennan's conjecture and the integral-means spectrum	Conformal derivative and boundary spectrum audit	First-pass assessment only
073	The Falconer distance conjecture	Fractal distance-set research explorer	First-pass assessment only
074	Kakeya in three and four dimensions	Directional tube analysis reference	First-pass assessment only
075	The $`L\log L`$ Fourier-convergence conjecture	Fourier convergence assumption reference	First-pass assessment only
076	Real ultraflat Littlewood polynomials and unbounded binary merit factors	Binary waveform spectrum and correlation contract audit	035
077	Fourier restriction for positively curved surfaces	Curved-surface Fourier extension reference	First-pass assessment only
078	The three-dimensional Bochner–Riesz conjecture	Spectral filter boundedness reference	First-pass assessment only
079	Local smoothing in three dimensions	Wave propagation regularity assumption ledger	First-pass assessment only
080	The exact Sobolev endpoint for Schrödinger convergence	Schrödinger initial-data convergence reference	First-pass assessment only
081	Riesz transforms and rectifiability in higher codimension	Geometric measure regularity evidence map	First-pass assessment only
082	Annular variation and dyadic absolute bounds for the triangular Hilbert transform	Multilinear singular-integral reference suite	First-pass assessment only
083	Hilbert transforms along Lipschitz directions	Directional filtering scale contract	First-pass assessment only
084	The geometric case of the Erdős similarity conjecture	Geometric-pattern avoidance research construction	First-pass assessment only
085	Endpoint Sobolev regularity of centered disk averages	Scale-space maximal filter regularity audit	First-pass assessment only
086	An <i>L</i><sup>3</sup> bound for the trilinear Hilbert transform	Trilinear Hilbert transform research benchmark	First-pass assessment only

Theoretical computer science

Family	Result and source directory	Latest application assessment	Dossier IDs
102	The Unique Games Conjecture and optimal approximation thresholds	Approximation hardness review service	First-pass assessment only
103	Exact derandomization of logarithmic space: $`\mathsf L=\mathsf{RL}=\mathsf{BPL}`$	Small-space decision and compiler readiness audit	020
104	Quasipolynomial algorithms for mean-payoff, stochastic and parity games	Mean payoff strategy synthesis plugin	003, 018
105	Perfect completeness for 2-to-1 games	Perfect-completeness reduction benchmark library	First-pass assessment only
106	Hardness of coloring three-colorable graphs	Graph-coloring solver stress suite	First-pass assessment only
107	Matrix multiplication with exponent at most 9/4	Matrix tensor construction research toolkit	003, 020, 022
108	A cubic permanent–determinant lower bound	Symbolic representation complexity audit	First-pass assessment only
109	Integer multiplication below $`n\log n`$	Big integer asymptotic feasibility worksheet	020
110	Optimal-order randomized <i>k</i>-server on arbitrary metrics	Metric movement and activation audit	010, 020
111	One-sample matroid prophet inequalities against an almighty adversary	Matroid allocation research simulator	020, 021
112	Beyond the square-root exponent for depth-three circuits	Depth-three circuit lower-bound teaching and benchmark module	First-pass assessment only
113	Approximate counting and entropy of perfect matchings	Matching counts: literal schedule exclusion and bounded fragility reference	001, 006
114	Approximate counting of common integer polymatroid bases	Constraint allocation space counter	First-pass assessment only
115	Sampling and counting contingency tables with arbitrary margins	Fixed-margin tables: law distinction and literal schedule exclusion	001, 004, 041
116	Uniform black-box noncommutative identity testing across characteristics	Noncommutative identity testing library	012
117	Uniform sparsest cut: hardness and semidefinite gaps	Cut relaxation calibration suite	First-pass assessment only
118	Bin packing and unbounded configuration-LP gaps	Bin packing approximation claim auditor	First-pass assessment only
119	The Courtade–Kumar and Hellinger conjectures	Noisy binary summary information benchmark	First-pass assessment only
120	Almost-linear-time exact matching and prescribed-degree factors in general graphs	General matching: literal backend prerequisite prevents a practical speedup claim	007
121	Almost-linear approximation of edit distance	Approximate long-document edit comparison	008
122	Quantitative trace-reconstruction bounds with a uniform decoder	Deletion-channel reconstruction validation suite	First-pass assessment only
124	Polynomial-time scheduling on three identical machines	Three-machine scheduling: source model and practical boundary	001, 009, 020
125	The metric <i>k</i>-median approximation threshold and recovery	Metric facility placement planner	011
126	Exponential semidefinite complexity of perfect matching	Matching formulation complexity audit	First-pass assessment only
127	Average sensitivity of polynomial threshold functions	Polynomial threshold sensitivity benchmark	First-pass assessment only
128	A factor-two approximation for shortest common superstring	Superstring construction, byte objectives and practical packaging	023
129	Exponential state costs for two-way automata	Automata transformation resource estimator	First-pass assessment only
130	Exact Fourier transforms below $`n\log n`$	Exact Fourier algorithm feasibility worksheet	020
131	Rapid mixing of graph switches for every degree sequence	Graph switching: host, event-time and exactness boundaries	003, 005
132	A superquadratic separation of sensitivity and block sensitivity	Sensitivity counterexample generator	First-pass assessment only
133	The computational complexity of Weisfeiler–Leman refinement	Graph refinement cost and capability evaluator	First-pass assessment only
134	Generalized star height at most three	Generalized regular expression research compiler	First-pass assessment only
135	Homogeneous depth-five lower bounds for iterated matrix multiplication	Arithmetic circuit restriction benchmark	First-pass assessment only
136	A quasilinear PCP theorem for PPAD	Robust local verification research toolkit	First-pass assessment only
137	One-tape time simulation in two-fifths-power space	Memory-constrained simulation explorer	First-pass assessment only
138	Subset Sum in $`O(2^{0.49n})`$ time	Subset-sum decisions: resource, identity and assurance boundary	017
139	Subpolynomial query complexity for log-concave sampling	Oracle sampling cost audit	020
140	Memory–sample lower bounds for noiseless Gaussian regression	Streaming inference precision budget model	First-pass assessment only
141	Existential–universal real sentences in the counting hierarchy	Real constraint complexity scope explorer	First-pass assessment only
142	Deterministic polynomial factorization over prime fields	Deterministic finite-field factorization backend	016

Topology

Family	Result and source directory	Latest application assessment	Dossier IDs
304	The Hilbert–Smith conjecture in every dimension	Continuous symmetry classification contract	First-pass assessment only
305	Four-dimensional disk embedding and Wall's conjecture	Four-dimensional embedding obstruction fixture	First-pass assessment only
306	The purely cosmetic surgery conjecture	Knot surgery distinctness research contract	First-pass assessment only
307	Failure of rational injectivity for maximal coarse assembly	Coarse assembly counterexample ledger	First-pass assessment only
308	Finite Smith–Toda complexes at every height	Chromatic finite-complex existence catalog	First-pass assessment only
309	The Kervaire invariant problem at the prime three	Odd-primary Kervaire witness registry	First-pass assessment only
310	Quillen's conjecture in rational homology	Finite-group poset homology contract	First-pass assessment only
311	The Hovey–Strickland and Chai conjectures	Chromatic tensor-ideal classification ledger	First-pass assessment only
312	The Grothendieck homotopy hypothesis	Infinity-groupoid model correspondence evidence	First-pass assessment only
313	Finite generation for the $`K(n)`$-local sphere	Localized homotopy finite-generation contract	First-pass assessment only
314	Cyclic length and chromatic fixed-point loss	Cyclic-chain length is computable for bounded finite examples	First-pass assessment only
315	The four-dimensional Singer conjecture	Four-dimensional L2 homology contract	First-pass assessment only
316	Curtis’s conjecture	Stable Hurewicz image reference	First-pass assessment only
317	Thomason model structures in all strict higher dimensions	Categorical homotopy model interoperability	First-pass assessment only
318	Chromatic splitting: filtrations and counterexamples	Chromatic splitting regime audit	First-pass assessment only
319	Counterexamples to finite generation at chromatic height two	Finite-building assumption counterexample	First-pass assessment only
320	Nonhomeomorphic closed aspherical four-manifolds	Homotopy-to-homeomorphism inference fixture	First-pass assessment only
321	A counterexample to Wall's finite D(2) problem	Cell-dimension simplification contract	First-pass assessment only