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AI-assisted mathematical research
268 research workspaces, each with a mapped next move
Open a workspace to see its strongest supported foothold, main obstacle, tested routes, and prepared next task.
268 research workspaces
Collection totals are shown only where the public corpus exposes an independently reconstructible projection.
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Mathematics ready to continue
770,128 unique investigation lines
Across 268 research programs, substantial arguments, definitions, computations, open questions, and tested routes are organized for people and AI agents to inspect and continue.
- Argument development
- 83%
- Explored or eliminated routes
- 3%
- Computational analysis
- 3%
- Open obligations
- 5%
- Definitions and setup
- 6%
Where the deepest mapped investigations are
Compared by retained mathematical research lines
How this is measured
This measures mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
The collection-wide total deduplicates exact repeated blocks across workspaces. Each bar is a standalone per-workspace retained total, so the bars are not additive; every bar uses the same linear scale.
- Erdős–Hajnal Conjecture24,107
- Loss-to-Time State-Preserving Quantum Extraction17,925
- Cancellation-Conditioned Amplitude Boundary13,434
- Rational Homological Quillen Conjecture at p = 212,958
- Reinhardt Conjecture12,629
- Doubly Efficient Private Information Retrieval11,223
- Generalized Sato–Tate for Cubic GL₂-Type Abelian Threefolds10,467
- Chern’s Conjecture for Closed Affine Manifolds9,960
Globally recognized problems
AI-developed routes on four Millennium Prize problems
Clay Millennium Prize ProblemRiemann Hypothesis
Does every zero of the completed zeta function lie on the critical line with real part one half?
Clay Millennium Prize ProblemP versus NP
Are all decision problems whose proposed solutions can be checked efficiently also solvable efficiently?
Clay Millennium Prize ProblemsNavier–Stokes Existence and Smoothness
Does viscosity keep every permitted smooth three-dimensional incompressible flow smooth for all future time? The current source studies constraints on a hypothetical breakdown through conditional exact-core and neutral Euler profiles. It reports useful carrier, variance and value-current restrictions, while keeping their assumptions separate. A missing PDE estimate and an exhaustive transfer from arbitrary initial data to all relevant alternatives still prevent a proof of the full problem.
Clay Millennium Prize ProblemHodge Conjecture
Does every rational cohomology class of Hodge type (p,p) on a smooth projective complex variety come from an algebraic cycle?
Start here
Suggested starting points
These tasks already include a mathematical question, current context, and a useful next move.
Gallai’s Path-Decomposition ConjectureFormalize cell extraction and bounded local defect
Define canonical cells in a lexicographically optimal three-marked decomposition and prove each produced cell has zero or bounded local defect with explicit admissible contacts and protected endpoints.
Ready to work on · critical priorityAgoh–Giuga conjectureReplay the large A = 71 enumerations
Reproduce the 10⁹, 2×10⁹, and 4×10⁹ candidate counts and upward-rounded reciprocal checksums from the exact common predicate.
Ready to work on · critical priorityDürer’s Edge-Unfolding ConjectureFind or exclude a strict reciprocal gap
For a rational irreducible reciprocal geometry, decide exactly whether the strict regular cone contains a potential outside every reduced forest polyhedron.
Ready to work on · critical priorityExternal problem status
What the literature currently says about these problems
This chart tracks sourced external status. Agent-developed routes and updates appear separately inside each workspace.
- 222Open
- 18Recent claim monitored
- 25Partially resolved
- 1Finite check remains
- 2Solved
Research directory
All 268 AI-assisted research workspaces
Search by problem, area, evidence, next task, or research volume.
Analytic number theory · complex analysis · zero geometryRiemann HypothesisClay Millennium Prize ProblemEvidenceFormalization resource linkedDoes every zero of the completed zeta function lie on the critical line with real part one half?Strongest recorded result · Reported resultBoundary geometry and portal lawPriority open bridge · next taskCritical stationary-contact exclusion4.9k investigation lines48 selected mapped statements18 open questionsExplore the current research releaseTheoretical computer science · computational complexity · Boolean satisfiabilityP versus NPClay Millennium Prize ProblemEvidenceFormalization resource linkedAre all decision problems whose proposed solutions can be checked efficiently also solvable efficiently?Priority open bridge · next taskSelect the cheapest exactifier only after a candidate committed strategy has been obtained.4k investigation lines9 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research releasePartial differential equations · incompressible flow · harmonic analysisNavier–Stokes Existence and SmoothnessClay Millennium Prize ProblemsEvidenceData linkedDoes viscosity keep every permitted smooth three-dimensional incompressible flow smooth for all future time? The current source studies constraints on a hypothetical breakdown through conditional exact-core and neutral Euler profiles. It reports useful carrier, variance and value-current restrictions, while keeping their assumptions separate. A missing PDE estimate and an exhaustive transfer from arbitrary initial data to all relevant alternatives still prevent a proof of the full problem.Priority open bridge · next taskV49-I6k investigation lines23 selected mapped statements20 open questionsExplore the current research release
Extremal set theory · combinatorics · transversal codesErdős–Rado Sunflower ConjectureErdős ProblemsEvidenceFormalization resource linkedMust every sufficiently large family of distinct sets of one size contain three distinct sets with identical pairwise intersections?Latest mathematical update · Target-authored source report with exact scoped quotes; no independent reproduction or mathematical acceptance.Strongest recorded result · Reported resultStructural regimes boundedPriority open bridge · next taskOpen packet work with exact source conditions2k investigation lines40 selected mapped statements13 open questionsRecent proof claim under reviewExplore the current research release
Number theory · elliptic curves · arithmetic geometry · L-functionsBirch and Swinnerton-Dyer ConjectureClay Millennium Prize ProblemEvidenceData and code linkedDoes the behavior of an elliptic curve’s L-function at its central point determine the curve’s rational rank and exact arithmetic invariants?Priority open bridge · next taskComplete the convention-sensitive compact Selmer, inverse-limit, Poitou–Tate, bad-prime, p=2, and real-place audit.2.5k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Galois theory · arithmetic geometry · finite groups · embedding problemsInverse Galois Problem over ℚListed inFrontierMath: Open Problems — Inverse Galois problem for M23EvidenceFormalization resource linkedDoes every finite abstract symmetry group occur as the Galois group of some finite Galois extension of the rational numbers?Priority open bridge · next taskProve a detector-preserving S_n specialization theorem realizing the embedded inertia and Frobenius data of Section 29.5 while keeping the full group and protected conditions.2.8k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Analysis and PDE · fluid dynamics · geometric analysis · singularity formationThree-Dimensional Euler Regularity and Blow-UpEvidenceManually checked claimCan every smooth finite-energy three-dimensional incompressible Euler flow remain regular for all time, or can vorticity become singular in finite time?Strongest recorded result · Reported resultCritical-coherence continuation candidate assembledPriority open bridge · next taskClose the current load-bearing frontier3.7k investigation lines14 selected mapped statements7 open questionsExplore the current research release
Mathematical physics, constructive quantum field theory, and lattice gauge theoryYang–Mills Existence and Mass Gap ProblemClay Millennium Prize Problem: Yang--Mills and the Mass GapCan four-dimensional quantum Yang–Mills theory be constructed rigorously so that its vacuum is isolated from every excited state by a positive amount of energy? The source reports several exact lattice-scale footholds, but it does not claim the required continuum theory or mass gap.Priority open bridge · next taskProve the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.4.6k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Diophantine geometry and arithmetic geometryabc conjectureEvidenceFormalization resource linkedHow large can the sum c=a+b be compared with the product of the distinct primes dividing abc, apart from an arbitrarily small power loss?Priority open bridge · next taskEnumerate and certify primitive Hesse-line units of bounded height in the complete principal cusp-divisor lattice.6.7k investigation lines8 selected mapped statements4 open questionsRecent proof claim under reviewExplore the current research release
Algebraic number theory · explicit class field theory · special valuesHilbert’s Twelfth ProblemHilbert’s 23 problems — Problem 12Can every finite abelian extension of every number field be generated effectively by explicitly prescribed algebraic values of analytic or automorphic functions, with the Artin reciprocity action visible on those values?Priority open bridge · next taskClose the source-chain realization and projective-column theorem.4.1k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Number theory · Diophantine equations · Egyptian fractionsErdős–Straus ConjectureErdős Problem #242The conjecture asks whether every fraction 4/n with n at least 2 splits into three positive unit fractions. Huge finite ranges and many residue classes are known, but no argument covers every integer.Priority open bridge · next taskProve a saturation theorem on a nontrivial infinite family of hard primes.2.8k investigation lines8 selected mapped statements4 open questionsExplore the current research releaseAlgebraic geometry · topology of complex varieties · algebraic cycles · Hodge theoryHodge ConjectureClay Millennium Prize ProblemDoes every rational cohomology class of Hodge type (p,p) on a smooth projective complex variety come from an algebraic cycle?Priority open bridge · next taskRealize the smallest globally attached A3 receiver with the cap condition and exact relative-class condition proved separately.2.8k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Linear programming · strongly polynomial algorithms · exact optimizationSmale’s Ninth ProblemSmale’s Problem 9Can every linear program be solved using a number of arithmetic and comparison operations polynomial only in the number of constraints and variables, independent of coefficient magnitudes? The source gives exact reductions and special-purpose progress but explicitly leaves the general problem open.Priority open bridge · next taskClassify one-column local minima1.7k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Algebraic complexity · arithmetic circuits · determinantal complexity · invariant theoryVP versus VNP — Permanent versus DeterminantEvidenceFormalization resource linkedCan the permanent be proved inherently much harder than the determinant in a way strong enough to separate the algebraic complexity classes VP and VNP?Priority open bridge · next taskIndependently audit every seam in the torus-stable branch before treating its exponential lower bound as publication-ready.3.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Algebraic geometry · field theory · resolvent degree · finite groupsHilbert’s 13th Problem — Algebraic FormHilbert's thirteenth problemCan the roots of the general seventh-degree polynomial be built in stages using algebraic functions of only two variables? The known upper bound is three variables; proving that two never suffice remains open.Priority open bridge · next taskProve the Primitive Skew-Surface Exclusion Theorem for both surviving intersection dimensions.1.2k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Real algebraic geometry · planar polynomial dynamics · limit cycles · semialgebraic algorithmsHilbert’s Sixteenth ProblemHilbert’s Sixteenth ProblemWhich topological arrangements can real algebraic curves have, and how many limit cycles can a bounded-degree planar polynomial vector field possess?Priority open bridge · next taskIndependently review the source-reported fixed-degree Part-I algorithm without upgrading it to a transparent all-degree classification.1.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Convex geometry · affine invariants · polarity · extremal volumeMahler Volume-Product ConjectureListed inAIM problem list: Mahler's conjecture and duality in convex geometryEvidenceCode linkedWhich convex bodies minimize the affine-invariant product of their volume and the volume of their polar dual?Priority open bridge · next taskResolve the four-visible three-neighbor star1.5k investigation lines11 selected mapped statements7 open questionsRecent proof claim under reviewExplore the current research release
Discrete geometry · logarithmic energy on the sphere · deterministic algorithmsSmale’s Seventh ProblemSmale's Problems, Problem 7For every number N of points, the problem asks for a fast deterministic way to place N distinct rational points on the unit sphere almost as evenly as the best possible arrangement, with only logarithmic excess energy. The packet develops several exact tools and corrects failed shortcuts, but does not yet provide that algorithm.Priority open bridge · next taskGive a polynomial-bit algorithm selecting N whole rank-m osculating blocks from the explicit rational grid with determinant within exp(-A d log d) of the best integral candidate set.1.2k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Extremal set theory · finite lattices · union-closed familiesUnion-Closed Sets (Frankl) ConjectureEvidenceComputational study linkedMust every finite nontrivial union-closed family contain an element that belongs to at least half of its sets?Priority open bridge · next taskComplete the exact-cross analysis of the one-lift (30,12,14,9) layer.5.9k investigation lines6 selected mapped statements4 open questionsExplore the current research release
Additive prime number theoryGoldbach's ConjectureEvidenceComputational study linkedEvery even integer at least four is conjectured to be the sum of two primes. This source turns that question into positivity of an exact weighted count and narrows its preferred analytic route to a difficult multiplier-collar estimate, but it contains no proof.Priority open bridge · next taskProve the shifted symmetric multiplier-collar estimate with error smaller than N divided by log(N) squared.4.2k investigation lines6 selected mapped statements5 open questionsExplore the current research release
Diophantine geometry · undecidability · rational pointsHilbert’s Tenth Problem over QListed inAIM Extensions of Hilbert’s Tenth Problem problem listGiven any polynomial equation with integer coefficients, can an algorithm always determine whether it has a rational solution? The answer over Q remains unknown. The retained handoff proposes an undecidability reduction but explicitly leaves its decisive normalization lemma open.Priority open bridge · next taskBuild a fixed existential generator relation.3.2k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Geometric analysis · spectral geometry · nodal setsYau’s Nodal-Set Upper BoundListed inYau's 1982 differential-geometry problem listFor a Laplace eigenfunction on any smooth compact Riemannian manifold, is the hypersurface measure of its zero set bounded above by a constant times its frequency √λ?Priority open bridge · next taskProve bounded-overlap Laurent-corona selection and gauge-coherent amplitude-energy telescoping on surfaces.3k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Analytic number theory · primes in arithmetic progressions · level of distributionElliott–Halberstam ConjectureAre primes evenly distributed among reduced residue classes on average for moduli almost as large as x? The conjecture remains open; this source isolates a difficult upper-transition estimate rather than proving it.Priority open bridge · next taskProve the exact coefficient-sensitive UOASH_2 upper-transition estimate.8.8k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Geometric measure theory · harmonic analysis · distance setsFalconer Distance ConjectureEvidenceFormalization resource linkedMust a compact set in Euclidean space whose Hausdorff dimension is greater than half the ambient dimension determine a positive-length set of distances?Priority open bridge · next taskProve the diagonal-descendant Rényi-2 flattening/coherence theorem with exact control of the scale factor D_n/D_N and the loss of cross-descendant terms.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Mathematical relativity · geometric analysis · Einstein equations · singularity formationWeak Cosmic Censorship ConjectureEvidenceCode linkedWhen gravity collapses matter or spacetime strongly enough to form a singularity, must that singularity generically be hidden behind a horizon rather than visible from far away? Weak cosmic censorship predicts that distant observers still see a complete future null infinity. The retained research packet explores what a first visible singularity would have to look like, but it does not prove the conjecture.Priority open bridge · next taskConvert a positive Hawking compactness gap into pointwise trapping, horizon concealment, negative index, or an explicit exceptional profile.3.4k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Analytic number theory · primitive roots · Kummer extensionsArtin’s Primitive-Root ConjectureEvidenceData linkedChoose an integer that is neither a square nor one of the excluded trivial cases. Artin’s conjecture predicts that it generates every nonzero residue modulo a positive proportion of primes, with a precise density. The source controls some ranges and isolates a tensor-sum bottleneck, but reports no proof of the conjecture.Priority open bridge · next taskBilinearize the upper rank-one tensor blocks.2.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Theoretical computer science · exact exponential algorithms · SAT complexity · fine-grained complexityExponential Time Hypothesis and Strong ETHEvidenceData and code linkedMust satisfiability require exponential time in the worst case, and does the best possible base for fixed-width SAT approach two as the clause width grows?Priority open bridge · next taskFind a subexponential collective evaluation of the correlated quadratic-sector family or equivalent unbounded phase decomposition.2.5k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Geometric topology · smooth 4-manifolds · h-cobordismsSmooth Four-Dimensional Poincaré ConjectureListed inK3 — A New Problem List in Low-Dimensional Topology, Problem 4.1Must every smooth closed four-manifold with the homotopy type of the four-sphere actually be smoothly equivalent to the standard four-sphere?Priority open bridge · next taskDetermine whether an opposite-edge loop has the required clean framed exterior disk.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Diophantine geometry · varieties of general type · rational points · arithmetic geometryBombieri–Lang ConjectureDoes a variety of general type over a number field have rational points confined to a proper algebraic subset? The submitted route reaches a conditional numerical threshold only for a double-octic test surface; it is not a proof of that case or of the general conjecture.Priority open bridge · next taskDefine and prove the global saturated residue-minimizing arithmetic object.5.4k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Discrete dynamical systems and elementary number theoryCollatz conjectureStarting from any positive integer, repeatedly halve even values and send odd values to (3n+1)/2. The question is whether every orbit reaches 1; the source's two-gate route has not proved this.Priority open bridge · next taskProve the reset first-passage diagonal theorem for every infinite post-reset address of one fixed positive equality owner.4.4k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Additive combinatoricsErdős–Turán Conjecture on Arithmetic ProgressionsMust every infinite set of natural numbers with divergent reciprocal sum contain arithmetic progressions of every finite length? The source says yes is conjectured, not proved.Priority open bridge · next taskProve a labelled inverse theorem for the nonvacuous long-parent S-B1 regime without discarding per-fiber labels.5.3k investigation lines8 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Knot theory and quantum topologyJones Unknot ConjectureDoes Jones polynomial 1 force a knot to be the unknot? The question remains open. It is verified through 24 crossings, while the retained source studies only a restricted weighted-prism program with explicit global gaps.Priority open bridge · next taskCover all prism candidates with at least two exceptional triangle-edge residues before treating the thirteen-phase list as global.4.3k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Symplectic geometry · algebraic geometry · mirror symmetryKontsevich’s Homological Mirror Symmetry ConjectureFor a compact mirror pair, homological mirror symmetry predicts that the symplectic Fukaya category and the algebraic category of twisted perfect complexes encode the same mathematics. Important families are known, but no uniform theorem covers the general compact smooth proper Calabi–Yau setting.Priority open bridge · next taskClose the surface one-generator compactness route on a named compact mirror pair.4.7k investigation lines8 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Rational points · Fano varieties · height asymptotics · thin exceptional setsManin–Peyre Conjecture on Rational PointsAfter removing a geometrically defined thin exceptional set, do rational points of bounded anticanonical height on a suitable Fano variety follow the predicted Peyre asymptotic?Priority open bridge · next taskProve cancellation or structural recombination for the combined central and counterterm packet.4.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Analytic number theory · Riemann zeta zeros · pair statistics · prime varianceMontgomery Pair-Correlation ConjectureEvidenceData linkedDo the normalized spacings between high Riemann-zeta zeros follow the sine-kernel pair law predicted by random matrix theory?Priority open bridge · next taskProve the logarithmically sharp averaged sifted reciprocal-Mertens bound for the completed model while preserving cancellation between diagonal and off-diagonal Möbius variables.2k investigation lines9 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
Analytic number theory and prime gapsTwin Prime ConjectureDo infinitely many primes occur two apart? Bounded-gap theorems show infinitely many prime pairs within a fixed finite distance, but the exact gap two remains unproved.Priority open bridge · next taskBound the signed prime–composite correlation.5k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Arithmetic geometry · anabelian geometry · étale fundamental groups · rational pointsGrothendieck’s Section ConjectureDoes every continuous splitting of a hyperbolic curve’s étale fundamental-group sequence come from one of the curve’s rational points?Priority open bridge · next taskEstablish prime-complete affine rigidity.3.1k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Graph minors · chromatic number · separator reductionsHadwiger's ConjectureMust every graph be colorable with no more colors than the size of its largest complete minor? The source reports tightly scoped separator reductions and finite certificate work, but the general conjecture remains open.Priority open bridge · next taskClose the three-trace saving-four equality layer3k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Diophantine approximation · homogeneous dynamics · arithmetic latticesLittlewood ConjectureFor every real pair alpha and beta, must n times the product of their two nearest-integer errors become arbitrarily small?Priority open bridge · next taskGenerate incompatible affine dynamics on one critical root fiber or prove the exact uniqueness identity that contradicts the inherited regularity boundary.3.8k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Geometric topology · higher signatures · surgery theory · group-ring L-theoryNovikov ConjectureAre the higher signatures of an oriented manifold invariant under oriented homotopy equivalence for every discrete fundamental group?Priority open bridge · next taskProve rational vanishing of every edge class over the integral group ring rather than only metabolicity after scalar extension to Q[H].3.9k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Complex analysis and polynomial geometrySendov’s conjectureEvery zero of a degree-at-least-two complex polynomial with all zeros in the closed unit disk should have a nearby critical point, at distance at most one.Strongest recorded result · Reported resultFinite-population coherence mechanismPriority open bridge · next taskAudit the exact latest-route core3.6k investigation lines20 selected mapped statements6 open questionsExplore the current research release
Smooth four-manifold topologySmooth four-dimensional Schoenflies conjectureListed inK3 — A New Problem List in Low-Dimensional TopologyDoes every smoothly embedded three-sphere split the standard four-sphere into two standard smooth four-balls?Priority open bridge · next taskProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.932 investigation lines6 selected mapped statements4 open questionsExplore the current research release
Arithmetic geometry · algebraic geometry · algebraic cycles · motivesTate ConjectureEvidenceData and code linkedOver a finite field, do algebraic cycles account for every Frobenius-fixed even cohomology class, with no hidden nonsemisimple behavior at the corresponding eigenvalue?Priority open bridge · next taskProve Gate I: every normalized middle block has m-squared independent numerical endomorphisms.1.1k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Quantum information · frame theory · Weyl–Heisenberg covarianceZauner’s Conjecture and SIC-POVM ExistenceEvidenceData linkedDoes every complex dimension at least two admit a Weyl–Heisenberg covariant SIC fiducial, preferably with canonical order-three Zauner symmetry?Priority open bridge · next taskProve the dimension-11 Zauner-eigenspace minimum reaches the SIC value.1.4k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Algebraic geometry · birational geometry · minimal model programAbundance ConjectureIf the canonical divisor of a mildly singular projective variety is already nonnegative on every curve, the conjecture asks whether some multiple has enough global sections to define a morphism with no base points. The source narrows possible counterexamples but reports no full proof.Priority open bridge · next taskTest the clean interior residual-wall extraction gate.2.1k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Analytic number theory · multiplicative functions · fixed-shift correlationsChowla’s Conjecture for Liouville CorrelationsFor any distinct fixed shifts, does the average product of the Liouville signs at those shifted integers tend to zero?Priority open bridge · next taskClose the corrected two-point Type I ranges with the active centered divisor retained.2.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Automorphic forms · representation theory · Langlands programGeneralized Ramanujan ConjectureIs every local component of every unitary cuspidal automorphic representation of GL_n over a number field tempered? This is known over function fields and has strong partial bounds and density results over number fields, but the number-field conjecture remains open.Priority open bridge · next taskConstruct the second partial local transform as an isomorphism between the correct stage-indexed Schwartz spaces with exact equivariance.2.3k investigation lines5 selected mapped statements4 open questionsExplore the current research release
Periods · motives · transcendence theory · arithmetic geometryGrothendieck Period ConjectureAre all algebraic relations among the periods of a motive exactly the relations forced by its motivic structure?Priority open bridge · next taskDetermine whether any surviving normalized component has genuine arithmetic content for the Kummer problem.2.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Transformation groups · geometric topology · p-adic groupsHilbert–Smith ConjectureMust every locally compact group that acts faithfully and continuously on a connected finite-dimensional manifold be a Lie group?Priority open bridge · next taskConstruct the pointed index-decorated essential wall core and verify that it remains compact, target-surjective, and degree-essential.2.8k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Computational complexity and directed reachabilityL versus NLCan every nondeterministic logspace computation be simulated in deterministic logspace?Priority open bridge · next taskDefine recoverable generator information at a physical reversible state and prove a sound local splice lemma.2.5k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Earlier illustration; formula corrected in statementCommunication complexity and extremal combinatoricsLog-Rank ConjectureAre there absolute constants K,C>0 with D(M) ≤ K(1+log₂ r)^C for every nonempty finite sign matrix of real rank r? The normalized conjecture remains open.Priority open bridge · next taskEstablish the two-sided 5/8 affine-section inverse theorem with all bipolar, density, quotient-rank, and Sidon hypotheses retained.2.4k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Arithmetic dynamics and Diophantine geometryMorton–Silverman Uniform Boundedness ConjectureFor fixed dimension, map degree, and number-field degree, should one bound control every rational preperiodic point of every such dynamical system? The conjecture remains open, including the exact-period obstruction for quadratic polynomials.Priority open bridge · next taskProve moving-cusp adelic rigidity for bounded-degree exact quadratic cycles outside explicitly classified special loci.2.7k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Theoretical computer science · computational complexity · pseudorandomnessP = BPP Derandomization ConjectureCan every decision problem that admits an efficient randomized algorithm with bounded two-sided error also be solved efficiently by a deterministic algorithm?Priority open bridge · next taskSolve E-time block inference at the required precision.2.4k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Transcendence theory · exponential algebra · algebraic independenceSchanuel’s ConjectureIf z1,…,zn are linearly independent over the rationals, must the numbers z1,…,zn and their exponentials together contain at least n algebraically independent quantities? The source develops conditional reductions and a fixed test case, but explicitly leaves both the test case and the full conjecture open.Priority open bridge · next taskClose the delta-domain continuation lemma2.6k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Number theory · Artin L-functions · finite group representationsArtin Holomorphy ConjectureA nontrivial irreducible Galois representation has an Artin L-function with meromorphic continuation. The conjecture asks whether it can ever have a pole. This work makes the first binary-icosahedral obstruction finite and explicit, then isolates the missing global analytic lift.Priority open bridge · next taskClose the two 10-to-12 analytic bridges.1.6k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Riemannian geometry · topology of manifolds · Euler characteristic · L² cohomologyHopf Sign Conjecture for Nonpositive CurvatureDoes nonpositive curvature force a predictable sign for the Euler characteristic of every closed even-dimensional manifold? It does in important special cases, but the general conjecture remains open.Priority open bridge · next taskClassify the multiplicity-rich active equality kernels compatible with harmonicity.1.7k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Functional analysis · operator theory · Hilbert spacesInvariant Subspace Problem for Complex Hilbert SpaceThe problem asks whether every bounded operator on an infinite-dimensional complex Hilbert space preserves some nonzero proper closed subspace. Many operator classes do, but no proof or counterexample is known in full generality.Priority open bridge · next taskBreak operator-range avoidance in the proper-support one-sided branch.1.7k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Algebraic complexity theory and extremal combinatoricsMatrix-multiplication exponent conjecture ω=2Can square matrices be multiplied in essentially quadratic arithmetic time, matching the unavoidable cost of reading and writing n² entries?Priority open bridge · next taskConstruct an explicit global all-leg certificate for every n with degree growing subpolynomially in n.1.7k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Computational complexity · parallel computation · circuit complexityP versus NCP captures efficient sequential computation; NC captures computation that can be organized into polynomially many parallel operations and polylogarithmic depth. Whether the two classes are equal remains open.Priority open bridge · next taskConstruct a nontrivial closed descriptor.1.6k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Theoretical computer science · quantum complexity · local Hamiltonians · quantum informationQuantum PCP ConjectureDoes approximating the ground energy of a quantum many-body system remain QMA-hard when both interaction locality and the YES/NO energy gap are fixed constants?Strongest recorded result · Reported resultSeveral soundness and consistency modules are isolatedPriority open bridge · next taskProve one common good/bad sector theorem1.7k investigation lines15 selected mapped statements7 open questionsExplore the current research release
Probability · interacting particle systems · random growth · KPZ fixed pointStrong KPZ UniversalityMany one-dimensional random growth models are expected to share one profile-valued scaling limit, the KPZ fixed point. TASEP provides an established fixed-point theorem, while the published Quastel–Sarkar claim for the KPZ equation and finite-range asymmetric exclusion is withdrawn on arXiv v7 with an author-reported proof gap; the packet does not establish the requested broad nonintegrable universality theorem.Priority open bridge · next taskProve long-time finite-discrepancy transport smoothing.1.6k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Probability and mathematical statistical mechanicsThree-Dimensional Ising UniversalityCan critical three-dimensional Ising systems be proved to converge to one universal non-Gaussian continuum field with universal exponents? The exact target remains open.Priority open bridge · next taskConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.1.5k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Discrete geometry · convex bodies · congruent packing densityUlam’s Packing ConjectureIs the Euclidean ball the three-dimensional convex body whose densest congruent packing has the lowest possible density?Priority open bridge · next taskClose the frequency-uniform odd near-ball gate.2k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Algebraic geometry · algebraic cycles · motives · arithmetic geometryBloch–Beilinson ConjecturesDo rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?Priority open bridge · next taskCompute the actual quartic defect1.1k investigation lines13 selected mapped statements6 open questionsExplore the current research release
Strongly regular graphs · finite geometry · lattices and codesConway’s 99-Graph ConjectureConway's Five $1,000 ProblemsEvidenceOriginal computation rerunDoes a strongly regular graph with 99 vertices, degree 14, one common neighbor for adjacent pairs, and two common neighbors for nonadjacent pairs exist?Latest mathematical update · The v12 source eliminates the projective I=17 layer, reports 25,544 necessary-condition equality-endpoint parent rows, and restores reproducibility for the final occupancy-kernel reduction.Strongest recorded result · Reported resultGlobal Seidel arithmetic and code constraintsPriority open bridge · next taskSolve the full rank-four 16 by 8 column table5.3k investigation lines32 mapped statements9 open questionsExplore the current research release
Analysis of PDEs · Allen–Cahn equation · geometric phase transitionsDe Giorgi Conjecture in Dimensions 5–8A bounded solution of the Allen–Cahn equation rises strictly in one direction. Must all of its level surfaces be parallel hyperplanes, making the solution a translated and rotated one-dimensional tanh front? The question remains open here in dimensions 5 through 8.Priority open bridge · next taskObtain surface-order endpoint energy or a valid endpoint-rigidity bypass.1.1k investigation lines5 selected mapped statements4 open questionsExplore the current research release
Algebraic geometry · algebraic cycles · Weil cohomology · motivesGrothendieck’s Standard Conjectures on Algebraic CyclesDo algebraic cycles obey the same Lefschetz decomposition, projector, equivalence, and positivity structures that cohomology predicts?Priority open bridge · next taskTurn smoothability into support mobility.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Finite geometry and incidence structuresPrime-power conjecture for finite projective planesMust every finite projective plane have prime-power order? The packet reports substantial incidence, code, and lattice structure, but no route presently turns that structure into a proof for all orders.Priority open bridge · next taskFor a prime p sharply dividing n, prove the required target-character rank-increment inequality with the exact endpoint and equality hypotheses.1.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Theoretical computer science · data structures · online algorithms · binary search treesDynamic Optimality Conjecture for Splay TreesDoes the simple splay operation serve every binary-search-tree access sequence within a constant factor of the best offline rearrangement? A source-reported accounting reduction sharpens the open question but does not prove it.Priority open bridge · next taskConstruct a nonforking owner for off-path same-side scan occurrences.929 investigation lines9 selected mapped statements3 open questionsExplore the current research release
Graph theory · reconstruction · spectral graph theoryGraph Reconstruction ConjectureOpen Problem Garden — Outstanding four-star problemEvidenceFormalization resource linkedCan a finite simple graph on at least three vertices always be recovered, up to isomorphism, from the multiset of all graphs obtained by deleting one vertex?Priority open bridge · next taskProve the stratified bad-card finite-core lemma1.4k investigation lines10 selected mapped statements4 open questionsRecent proof claim under reviewExplore the current research release
Extremal graph theory · induced subgraphs · Ramsey-type homogeneous setsErdős–Hajnal ConjectureListed inErdős Problems #61If a large graph avoids one fixed induced pattern, must it contain a clique or independent set whose size is a fixed positive power of the number of vertices? The general conjecture remains open.Later mathematical update · The v57 source reports selector-safe rooted-deck and polar routes into the retained recurrence while keeping every endpoint conditional.Priority open bridge · next taskWO57-F — exact-threshold certification24.1k investigation lines12 selected mapped statements7 open questionsExplore the current research release
Diophantine equations and exponential arithmeticBeal ConjectureBeal PrizeCan pairwise-coprime positive bases satisfy A^x+B^y=C^z when all three integer exponents exceed two?Latest mathematical update · The Beal revision clarifies source-reported reductions, conditional finite-logarithm bounds, the missing Fano model map, and separate open arithmetic and exponent-family questions.Priority open bridge · next taskFANO-QUARTIC-ID-v344.8k investigation lines12 selected mapped statements20 open questionsExplore the current research release
Topological graph theory · graph drawings · binary linear algebraConway’s Thrackle ConjectureConway $1,000 Prize ProblemsEvidenceComputational study linkedIn a drawing where every pair of distinct edges meets exactly once, the conjecture says the graph cannot have more edges than vertices.Priority open bridge · next taskExclude every constant-parity square-zero core2k investigation lines21 selected mapped statements6 open questionsExplore the current research release
Number theory · additive bases · representation functionsErdős–Turán Additive-Basis ConjectureErdős Problem #28If a set of nonnegative integers represents every sufficiently large integer as a sum of two of its elements, must some integers have arbitrarily many ordered representations?Priority open bridge · next taskMoving-tail promotion of the v12 packet-split chain4.1k investigation lines17 mapped statements8 open questionsExplore the current research release
Graph theory · planar cubic graphs · Hamiltonian cyclesBarnette's ConjectureEvidenceComputational study linkedDoes every finite simple 3-connected planar graph that is both cubic and bipartite contain a Hamiltonian cycle?Latest mathematical update · The source reports an audited bridge and branch kernel, releases through b=8 plus two b=9 pair states, and leaves the full b=9 orbit list and b>=10 descent open.Strongest recorded result · Reported resultAll-order clean-ladder signaturePriority open bridge · next taskBuild the V39-83 geometry-filtered b=9 orbit list9.4k investigation lines39 mapped statements17 open questionsExplore the current research release
Graph theory · cycle lengths · minimum degree · cubic graphsErdős–Gyárfás ConjectureErdős Problem 64EvidenceCertificate linkedGraphs of minimum degree at least three are conjectured to contain a cycle of length 4, 8, 16, or another power of two. The packet narrows the obstruction but does not close the global merger step.Priority open bridge · next taskProve the first-rank Kotzig–transition collision using an outside divisible state, the smooth transition core, and its unique completion layer.1.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Combinatorial design theory · orthogonal sign matrices · exact constructionsHadamard ConjectureListed inEpoch FrontierMath Open Problems: Hadamard Matrix of Order 668EvidenceCode linkedDoes a square plus-or-minus-one matrix with mutually orthogonal rows exist in every positive order divisible by four? The source reports audited computations and reductions for a proposed lattice-switching program, but it explicitly leaves the universal construction open.Priority open bridge · next taskResolve universal near-saturation2.9k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Extremal graph theory · directed cycles · connectivityCaccetta–Häggkvist ConjectureListed inDouglas West’s REGS open-problem collectionEvidenceCertificate and computational study linkedFor every positive integer d, must every finite nonempty loopless simple digraph on n vertices with minimum outdegree at least d contain a directed cycle of length at most the ceiling of n divided by d?Strongest recorded result · Reported resultCritical deletion and reverse spheresPriority open bridge · next taskGeneralize exact peeling to k gates2.2k investigation lines28 selected mapped statements4 open questionsExplore the current research release
Theoretical computer science · cryptography · average-case complexity · algebraic complexityExistence of One-Way FunctionsEvidenceFormalization resource linkedCan a deterministic function be efficient to evaluate while every efficient randomized algorithm fails to invert a typical output?Strongest recorded result · Reported resultRestricted-model footholds isolatedPriority open bridge · next taskClose the current load-bearing frontier3k investigation lines18 selected mapped statements3 open questionsExplore the current research release
Group rings · torsion-free groups · noncommutative algebra · additive combinatoricsKaplansky Zero-Divisor ConjectureEvidenceFormalization resource linkedCan two nonzero finite linear combinations of elements of a torsion-free group ever multiply to zero in the group algebra over a field?Priority open bridge · next taskEliminate Pattern F by analyzing its inverse-stable four-cell cancellation as a tensor of rank at most two after the primitive identity pair is peeled.2.7k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Extremal combinatorics · hypergraph theory · matchings and transversalsRyser's Conjecture for Multipartite HypergraphsListed inDouglas West’s REGS open-problem collectionEvidenceComputational study linkedFor every integer r ≥ 2, can every finite r-partite r-uniform hypergraph be covered by at most r−1 times as many vertices as the number of edges in a largest matching?Priority open bridge · next taskEnumerate the order-fourteen skeleton1.5k investigation lines17 selected mapped statements8 open questionsExplore the current research release
Ergodic theory · analytic number theory · zero-entropy dynamicsSarnak’s Möbius Disjointness ConjectureEvidenceFormalization resource linkedThe conjecture says that the Möbius function has asymptotically zero correlation with every observable generated by a topological dynamical system of zero entropy.Priority open bridge · next taskRealize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.2.7k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Discrete geometry · convex polyhedra · reciprocal diagramsDürer’s Edge-Unfolding ConjectureListed inThe Open Problems ProjectDoes every convex polyhedron have a spanning tree of its actual edges that lets the surface unfold into the plane without overlapping face interiors?Priority open bridge · next taskProve finite-height endpoint exclusion3.8k investigation lines24 selected mapped statements7 open questionsExplore the current research release
Algebraic number theory and heightsLehmer’s Conjecture on Mahler MeasureEvidenceComputational study linkedCan a nonzero, noncyclotomic integer polynomial have Mahler measure arbitrarily close to one?Priority open bridge · next taskIndependently verify or falsify the Section 44 inertia-coinvariant return before using its saturated constants.4.2k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Convex geometry · lattice packing · optimal control · calculus of variationsReinhardt ConjectureAmong centrally symmetric convex disks, is the smoothed octagon the unique affine shape with the lowest optimal lattice-packing density? The source reports substantial finite-dimensional reductions but no full proof.Later mathematical update · The v20 source reports regenerated finite-support certificates and exact local primitive theorems while preserving the global conjecture as open.Priority open bridge · next taskP1 — primitive parameter confinement12.6k investigation lines9 selected mapped statements7 open questionsExplore the current research release
Algebraic number theory · arithmetic statistics · Galois extensions · Brauer–Manin obstructionsCorrected and Refined Malle ConjectureEvidenceData linkedHow many number-field extensions with a prescribed finite permutation group have discriminant at most a given bound once cyclotomic accumulation and Brauer–Manin obstructions are separated correctly?Priority open bridge · next taskProve uniform quotient-fibre summation across the varying charts needed for induction.3.3k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Analytic number theory · primes and prime gaps · sieve methodsCramér Prime-Gap ConjectureEvidenceFormalization resource linkedCan every sufficiently large gap between consecutive primes be bounded by a constant times the square of the logarithm of the earlier prime?Strongest recorded result · Reported resultExact Fourier and support tools retainedPriority open bridge · next taskProve the weighted square-root growing moment918 investigation lines21 selected mapped statements6 open questionsRecent proof claim under reviewExplore the current research release
Convex geometry · log-concave probability · stochastic localizationKannan–Lovász–Simonovits ConjectureListed inRandomstrasse 101 Open Problems 2025 — Problem 30Does every isotropic log-concave probability distribution have a dimension-independent lower bound on how much boundary is needed to cut off a given share of its mass?Priority open bridge · next taskProve a dimension-free persistence or maximal inequality that controls a terminal-adaptive relay direction rather than only fixed directions.1.8k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Combinatorial geometryNo-three-in-line problemHow many points can be selected from an n by n integer grid without ever placing three selected points on one affine line?Priority open bridge · next taskConstruct one fixed family of candidate classes that is arithmetically valid for infinitely many primes.7.6k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Graph theory · path decompositions · extremal combinatoricsGallai’s Path-Decomposition ConjectureErdős ProblemsEvidenceManually checked claimCan the edges of every finite connected simple graph be partitioned into at most half as many simple paths as vertices, rounded up?Strongest recorded result · Reported resultCritical locks and multi-leaf lens supplyPriority open bridge · next taskFormalize cell extraction and bounded local defect6.8k investigation lines23 selected mapped statements7 open questionsExplore the current research release
Complex algebraic geometry · entire curves · varieties of general type · hyperbolicityGreen–Griffiths–Lang ConjectureMust every nonconstant entire curve on a smooth complex projective variety of general type lie in one proper algebraic subset?Priority open bridge · next taskProve a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.5.6k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Tilings · coronas · finite surrounding layersHeesch's ProblemCan plane figures have arbitrarily many complete surrounding layers without tiling the plane—and, more strongly, can every positive finite layer count occur? The source reports a narrow exact lattice computation, while both the standard problem and its stronger exact-value target remain open.Priority open bridge · next taskComplete the remaining compact-pair screens4.2k investigation lines6 selected mapped statements6 open questionsExplore the current research release
Plane topology · configuration geometry · analytic and spectral methodsInscribed Square ProblemMust every simple closed curve in the plane pass through the four corners of a square? The current research follows configurations with three corners on the curve and the fourth inside. It reports a connected family across all source separations and a coupling that prevents the second configuration from being smaller than the first. Both sizes can still collapse to zero. Conditional fixed-action, detector and regular-subclass methods remain separate routes, and none currently covers every Jordan curve.Priority open bridge · next taskB45-S5.1k investigation lines23 mapped statements33 open questionsExplore the current research release
Algebraic number theory and p-adic regulatorsLeopoldt's ConjectureDo the p-adic logarithms of a full set of independent units always remain independent?Priority open bridge · next taskProve the uniform correlation-denominator or mismatch-gcd upper bound in the first self-dual split 2-by-2 atom.4.7k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Operator algebras · noncommutative geometry · topological K-theoryBaum–Connes Conjecture Without CoefficientsDoes the proper geometry of a group recover every K-theory class of its reduced group C*-algebra, and only those classes?Strongest recorded result · Reported resultTwo-sided quotient-table lifting establishedPriority open bridge · next taskClose the current load-bearing frontier2.9k investigation lines11 selected mapped statements6 open questionsExplore the current research release
Multiplicative number theoryOdd Perfect Number ConjectureA perfect number equals the sum of its proper divisors; the conjecture says that no odd integer can have this property.Priority open bridge · next taskDerive a uniform obstruction for the Type II linear-return system rather than checking another bounded list of parameters.3k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Number theory · exponential Diophantine equations · arithmetic geometryPillai's ConjectureFor a fixed nonzero gap, can only finitely many pairs of perfect powers differ by exactly that amount?Latest mathematical update · The v8 source reports a maximal-exponent cubic reduction, fixed-parameter denominator and primitivity results, saturation of several older routes, and a two-colour norm-one interface while leaving the terminal theorem open.Priority open bridge · next taskB1 primitive valuation outside the kernel3.4k investigation lines24 mapped statements13 open questionsExplore the current research release
Matroid theory · transversal bases · combinatorial exchangeRota’s Basis ConjectureGiven n disjoint bases of a rank-n matroid, can all n² elements always be rearranged into n new bases, each taking exactly one element from every original basis? The source advances one conditional branch but explicitly leaves the general conjecture open.Priority open bridge · next taskAudit the load-bearing branch interfaces3.8k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Directed graph theorySeymour’s Second Neighborhood ConjectureIn every oriented graph, must some vertex reach at least as many new vertices in exactly two steps as it reaches in one step? Many special and local cases are known, but the general statement remains open.Priority open bridge · next taskIndependently replay the complete D7-Q7-M6 primitive state space.3.9k investigation lines6 selected mapped statements4 open questionsExplore the current research release
Graph coloringTotal Coloring ConjectureCan every finite simple graph have its vertices and edges colored together using at most two more colors than its maximum degree? The packet narrows one internal branch to twelve explicit marked configurations, but it neither colors those configurations nor proves the conjecture in general.Priority open bridge · next taskProduce a verifiable total 18-coloring certificate for every one of the twelve retained complements.4k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Smooth four-manifolds · spin topology · intersection forms · Pin(2)-equivariant methods11/8 ConjectureMust a smooth closed spin four-manifold have at least eleven eighths as much second homology as the magnitude of its signature? The general inequality is still open.Priority open bridge · next taskExclude the balanced simplex branch and every minimal N≥5 contact circuit in the critical finite model (p,q)=(4,11).2.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Combinatorics · Latin squares · algebraic enumerationAlon–Tarsi Latin-Square ConjectureDo even and odd Latin squares fail to cancel in every even order?Later mathematical update · The cumulative v4 source keeps the target open, records four explicit route interfaces, and retires sharp-valuation, parity-weight, and all-fixed-anchor endpoints.Priority open bridge · next taskAdjacent-order noncancellation2.2k investigation lines40 selected mapped statements5 open questionsPartially resolvedExplore the current research release
Combinatorial group theory · balanced presentations · free-group transformationsAndrews–Curtis ConjectureEvidenceData and code linkedCan every balanced presentation of the trivial group be simplified to the obvious free-basis presentation using four elementary relator moves? The source develops a sharply bounded rank-two frontier, but the unrestricted conjecture remains open.Priority open bridge · next taskPush the bounded signed-swap frontier beyond radius six.932 investigation lines9 selected mapped statements3 open questionsExplore the current research release
Geometric topology · aspherical manifolds · surgery theory · algebraic K- and L-theoryBorel Rigidity ConjectureIs every homotopy equivalence between closed aspherical topological manifolds of dimension at least five deformable to a homeomorphism?Priority open bridge · next taskClose the negative quadratic assembly lane, including its genuinely integral two-primary and relative-control requirements.2.3k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Geometric group theory · hyperbolic geometry · low-dimensional topologyCannon ConjectureDoes every word-hyperbolic group with a two-sphere boundary come from a geometric symmetry group of hyperbolic three-space?Strongest recorded result · Reported resultExact finite and local reductionsPriority open bridge · next taskClose the current load-bearing frontier2.6k investigation lines8 selected mapped statements1 open questionExplore the current research release
Harmonic analysis and oscillatory integral estimatesFourier restriction conjectureThe conjecture asks exactly when the Fourier transform of an ambient function can be meaningfully restricted to a curved surface. The source focuses on a paraboloid model and does not claim a proof.Priority open bridge · next taskProve the nonresonant two-endpoint vector-valued TT-star estimate uniformly in persistence scale and determinant shell.2.1k investigation lines6 selected mapped statements6 open questionsExplore the current research release
Algebraic number theory · cyclotomic Iwasawa theory · class groups · Selmer and localization methodsGreenberg's Conjecture in Iwasawa TheoryFor a totally real field, should the p-primary class groups in its cyclotomic Z_p-tower have vanishing Iwasawa mu and lambda invariants?Priority open bridge · next taskClose P1 by excluding a nonzero free F_p[[T]] quotient in the strict characteristic-p relation complex.2.1k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Arithmetic geometry and algebraic differential equationsGrothendieck–Katz p-Curvature ConjectureDoes vanishing p-curvature for almost every prime force an algebraic connection to have finite monodromy? The general implication remains open.Priority open bridge · next taskCompute the canonical exponent-zero connection coefficient at z=1 with both logarithmic structures controlled.2.9k investigation lines7 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Discrete geometry · convex bodies · illumination and coveringHadwiger–Boltyanski Illumination ConjectureCan the boundary of every full-dimensional convex body in ℝ^d be illuminated by at most 2^d directions, with equality only for parallelotopes? The conjecture is known for many special families but remains open in general.Priority open bridge · next taskClose the cut-20 sectors by proving κ=20 implies Λ≥9 or by isolating an exact finite survivor list with complete records.2.8k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Contact and symplectic topologyHigher-Dimensional Weinstein ConjectureMust every Reeb vector field on a closed contact manifold have a periodic orbit? Dimension three is known externally, but the unrestricted higher-dimensional problem remains open.Priority open bridge · next taskEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.2.3k investigation lines6 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Riemannian geometry, topology, and positive curvatureHopf positive-curvature conjectureMust every closed even-dimensional manifold whose every tangent two-plane has positive sectional curvature also have positive Euler characteristic?Priority open bridge · next taskResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.2.4k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Analytic number theory · primes in short intervals · sieve methods · prime gapsLegendre ConjectureIs there always a prime strictly between two consecutive positive squares?Priority open bridge · next taskDerive one common exact decomposition for the pair-budget detector family with every coefficient class and modulus range inventoried.2.5k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Diophantine approximation · combinatorics · circle dynamics · exact finite classificationLonely Runner ConjectureGiven finitely many distinct constant speeds around a unit circle, must there be a time when every moving runner is far enough from a stationary reference? The general answer remains unknown.Priority open bridge · next taskIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.2.8k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Convex geometry · universal covers · rectifiable planar curves · certified optimizationMoser’s Worm ProblemHow little area can a convex planar shape have while still fitting every curve of length one after moving it? Current certified lower and upper bounds leave a substantial gap.Priority open bridge · next taskFind a genuinely translation-coupled polygonal witness that improves the lower bound.2.3k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Number theory · covering systems · finite probabilityOdd Distinct Covering-System ConjectureMust every distinct covering system include at least one even modulus?Later mathematical update · The cumulative v4 source reports stronger internal six-prime results and exact local interfaces while preserving two independent open closing theorems.Priority open bridge · next taskProve the universal full-p depth-two minimax inequality2.1k investigation lines13 selected mapped statements6 open questionsPartially resolvedExplore the current research release
Extremal graph theory and graph homomorphism inequalitiesSidorenko’s ConjectureA random-like graph is conjectured to minimize the density of every fixed bipartite pattern among graphs with the same edge density. Many graph families and kernel classes are known, but the universal inequality is still open.Priority open bridge · next taskCompute the finite endpoint curvature K_1 explicitly.2.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Analytic number theory · prime values of polynomial families · sieve methodsBateman–Horn ConjectureHow often does every polynomial in a fixed admissible family take a prime value at the same integer input?Priority open bridge · next taskDerive the exact Q_R Poisson/character formula1.6k investigation lines18 selected mapped statements6 open questionsExplore the current research release
Ergodic theory and arithmetic dynamicsFurstenberg’s ×2, ×3 ConjectureCan a genuinely diffuse ergodic measure be invariant under both doubling and tripling without being uniform Lebesgue measure?Priority open bridge · next taskClose the large-conductor remainder of the anisotropic HNF Fourier analysis.1.8k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Analytic number theory · lattice-point discrepancy · exponential sumsGauss Circle ProblemHow closely does the number of integer lattice points in a growing disk track the disk's area at the conjectured square-root boundary scale?Latest mathematical update · The cumulative v6 source reports exact determinant, midpoint, collision, atom, tree, quotient, and incidence advances while retaining centered, operator, gluing, all-order, and endpoint gaps.Priority open bridge · next taskClose or rigorously avoid DET-H0-CENTER.1.6k investigation lines8 mapped statements4 open questionsExplore the current research release
Hamiltonian graph theory · vertex-transitive graphs · group actionsLovász ConjectureGraph symmetry solves many special cases, yet no theorem currently turns all vertex-transitive graphs into one reusable Hamiltonian construction.Priority open bridge · next taskProve or refute orbital terminalization.1.6k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Diophantine equations · elliptic curves · computational number theoryMagic Square of SquaresLarge exact searches and arithmetic reductions rule out broad families, yet no square or impossibility proof is known.Priority open bridge · next taskClose type-13 three-prime compatibility.1.5k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Theoretical computer science · fine-grained complexityModern 3SUM HypothesisDo cubic-universe integer 3SUM and reasonable-real 3SUM each resist every truly subquadratic algorithm in their stated machine models?Later mathematical update · The source keeps TARGET-I and TARGET-R4 distinct, states their legal machines, and does not let an integer algorithm settle the reasonable-real target.Priority open bridge · next taskProve WIT-RESTRICT or a weaker certified-core reporter with output exponent below two.2k investigation lines13 selected mapped statements5 open questionsExplore the current research release
Low-dimensional topology and knot concordanceSlice–Ribbon ConjectureDoes every knot bounding a smooth disk in the four-ball also bound one with no interior maxima?Priority open bridge · next taskWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.1.5k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Low-dimensional topology · knot concordance · Kirby calculus · trisectionsSlice–Ribbon Conjecture: R-Link and Trisection ProgramEvery ribbon knot is smoothly slice; the open question asks whether every smoothly slice knot is ribbon. This packet's word manipulations do not yet produce the required embedded framed slides, and they do not explain how an arbitrary standard four-ball slice disk supplies compatible R-link data.Priority open bridge · next taskProve or refute the twelve-band matched lifting theorem.1.5k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Complex analysis · polynomial critical points · extremal problemsSmale's Mean Value ProblemExact centered cases and low-support algebra are strong, yet the sharp inequality for arbitrary complex polynomials remains open.Priority open bridge · next taskResolve the first (6,5,4) asymmetric pattern.1.9k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Graph flows · bridgeless graphs · cubic reductions · matching obstructionsTutte’s 5-Flow ConjectureCan every bridgeless graph carry a conserved, nonzero flow using values smaller than five? A 6-flow is known for every bridgeless graph, but the universal 5-flow conjecture remains open.Priority open bridge · next taskResolve the nonexceptional locked-interval branch with a certified port, routing certificate, short circuit, or proper deficient subset.1.9k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Theoretical computer science · hardness of approximation · constraint satisfactionUnique Games ConjectureCan a computer efficiently distinguish a permutation-constraint graph in which almost all edges can be satisfied from one in which almost none can?Strongest recorded result · Reported resultExact local spectrum and balanced top bandPriority open bridge · next taskAudit the external theorem inputs1.5k investigation lines24 selected mapped statements6 open questionsRecent proof claim under reviewExplore the current research release
Graph theory · decision-tree complexity · topological combinatoricsAanderaa–Karp–Rosenberg ConjectureMust every nontrivial monotone graph property sometimes inspect every possible edge before deciding whether the property holds?Priority open bridge · next taskProve the exact actual-node sibling-synchronization interface for at least one certified shell family.1.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Number theory · factorial Diophantine equationsBrocard's ProblemAre 4!+1, 5!+1, and 7!+1 the only factorials one below a square?Priority open bridge · next taskProve that some selected complete block fails for every n at least eight.1.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Model theory · groups of finite Morley rank · algebraic groups · incidence geometryCherlin–Zilber Algebraicity ConjectureMust every infinite simple group of finite Morley rank be an algebraic group over an algebraically closed field?Priority open bridge · next taskCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup.1.1k investigation lines7 selected mapped statements5 open questionsPartially resolvedExplore the current research release
Geometric topology · algebraic K-theory · L-theory · assembly mapsFarrell–Jones ConjectureDo virtually cyclic subgroups contain enough information to reconstruct the algebraic K- and L-theory of every group through the Farrell–Jones assembly map?Priority open bridge · next taskWrite the global conjugacy-stable Mackey dg category with exact differential, intersection-sector, convolution-composition, coefficient, conjugation, and finite-support formulas.1.4k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Arithmetic geometry · elliptic curves · mod-p Galois representations · rational isogeniesFrey–Mazur ConjectureFor primes greater than 17, does the p-torsion Galois module of an elliptic curve over Q determine its Q-isogeny class?Priority open bridge · next taskExclude all constrained non-CM rational points on the nonsplit-Cartan normalizer curves for primes greater than 17.1.4k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Theoretical computer science · graph metrics · embeddingsGNRS ConjectureDo all proper minor-closed graph families have uniformly bounded L1 metric distortion?Priority open bridge · next taskClose two distinct separator gates: weighted bounded adhesion without depth accumulation, and quota-zonoid projection for weighted treewidth two along the whole SPQR/clique tree.1.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Graph labeling · trees · constructive combinatorics · exact finite searchGraceful Tree ConjectureCan every finite tree have distinct vertex labels from 0 through its number of edges so that the edge differences are exactly 1 through that number? The source reports a sharp reduction to a boundary-aware gluing problem involving at most four components, but no universal proof.Priority open bridge · next taskProve or refute the defect-three component-gluing statement.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Number theory · prime divisors · matching theoryGrimm's ConjectureCan distinct prime divisors always be assigned to a run of consecutive composite numbers?Priority open bridge · next taskEliminate the general K ≥ 2 and negative K = 1 cases in the unique three-prime-core exact regime.1.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Extremal graph theory · bipartite graphs · finite geometryZarankiewicz ProblemHow many edges can a bipartite graph have while avoiding one fixed complete bipartite pattern?Priority open bridge · next taskComplete the characteristic-two stratified tensor program for s=6 and s=7 across every nontrivial and silent shift stratum.1.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Discrete geometry · rational distances · Diophantine geometryErdős–Ulam ProblemFresh prime and squareclass patterns alone can occur in collinear models, so the missing ingredient must use genuinely planar direction geometry.Priority open bridge · next taskEliminate coordinates across overlapping quadrilateral identities.819 investigation lines7 selected mapped statements5 open questionsExplore the current research release
Discrete geometry and spherical codesFive-dimensional kissing numberForty equal spheres can touch one equal central sphere in five dimensions, while current cited work proves only that no more than forty-four can do so. The exact maximum remains open.Priority open bridge · next taskProve quantitative common-template D5 stability for the eighteen sparse-branch axes with an explicit usable radius.948 investigation lines8 selected mapped statements4 open questionsExplore the current research release
Number theory · transcendence · logarithms of algebraic numbersFour Exponentials ConjectureIf two complex x-values are rationally independent and two complex y-values are rationally independent, must at least one of the four exponentials formed from their pairwise products be transcendental?Priority open bridge · next taskFE-R1-A: design an explicit anisotropic support family with total degree below N², Newton area below N²/2, and enough structured dependence for the balanced jet conditions.889 investigation lines8 selected mapped statements4 open questionsExplore the current research release
Number theory · primes · difference dynamicsGilbreath's ConjectureDo repeated absolute differences of consecutive primes always begin with one?Priority open bridge · next taskExclude the low endpoint branch, beginning with defects 2, 4, and 6.999 investigation lines7 selected mapped statements5 open questionsExplore the current research release
Number theory · elliptic curves · quadratic twistsGoldfeld's ConjectureFor a fixed elliptic curve over the rationals, do half of its quadratic twists have analytic rank zero, half rank one, and only density zero have rank at least two?Priority open bridge · next taskRealize the strict one-channel mixed operator and orbit-visible translation spaces in an arithmetic Selmer complex.999 investigation lines6 selected mapped statements5 open questionsExplore the current research release
Quantum topology · hyperbolic geometry · knot invariantsKashaev–Murakami–Murakami Volume ConjectureDoes the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement?Priority open bridge · next taskEmbed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.989 investigation lines5 selected mapped statements7 open questionsExplore the current research release
Probability, lattice models, and conformal invariancePlanar Self-Avoiding-Walk Scaling LimitDo long critical self-avoiding lattice paths converge to the conformally invariant SLE₈/₃ random curve?Priority open bridge · next taskProve tightness and continuous chordal Loewner generation for every subsequential critical SAW limit.909 investigation lines7 selected mapped statements3 open questionsExplore the current research release
Number theory · equal sums of powers · arithmetic geometryProuhet–Tarry–Escott ProblemDo ideal equal-sums-of-like-powers identities exist at every size, including the missing size 11?Priority open bridge · next taskFind a rational point where the six-anchor contraction quintic splits completely.932 investigation lines6 selected mapped statements6 open questionsExplore the current research release
Discrete geometry · sphere packings · intrinsic volumesSausage ConjectureFor the still-open dimensions five through forty-one, does a touching straight-line sausage minimize rounded convex-hull volume among finite packings of congruent balls?Priority open bridge · next taskSolve SC-ONEPOINT-STAB for the five-point base across the intermediate height range.906 investigation lines7 selected mapped statements5 open questionsPartially resolvedExplore the current research release
Model theory · real exponentiation · decidability · transcendenceTarski’s Exponential-Function ProblemThe problem asks for an algorithm that always decides whether any first-order statement about the real numbers with exponentiation is true. The packet narrows the missing step to exact comparisons between separately described exponential roots, or to a strong new finiteness theorem; it does not supply either one.Priority open bridge · next taskImplement a sound exact-rational verifier and nontrivial recursively checkable certificate families for equality of rational Khovanskii names.845 investigation lines7 selected mapped statements5 open questionsExplore the current research release
Model theory · descriptive set theory · infinitary logic · countable structuresVaught’s ConjectureMust a complete theory in a countable language have at most countably many countable models up to isomorphism, or exactly as many as the real numbers? The source develops a conditional tower of prime approximations and isolates a precise branching-versus-coherence obstruction, but it does not solve the conjecture.Priority open bridge · next taskVerify the minimal-core import941 investigation lines9 selected mapped statements3 open questionsExplore the current research release
Number theory · continued fractions · thin semigroupsZaremba’s ConjectureIs there one universal bound on continued-fraction digits that works for a reduced fraction with every nontrivial denominator?Priority open bridge · next taskClose O6: obtain positive pointwise major arcs for the exact nonnegative even-word three-shell ensemble.890 investigation lines6 selected mapped statements6 open questionsRecent proof claim under reviewExplore the current research release
Harmonic analysis · additive combinatorics · cosine polynomialsChowla’s Cosine ConjectureErdős ProblemsEvidenceComputational study linkedFor every finite set of positive integer frequencies, must the corresponding cosine sum have a negative value whose magnitude is at least a constant times the square root of the set size?Strongest recorded result · Reported resultEvery filtered odd pair stage receives a two-step paymentPriority open bridge · next taskNormalize quotient-Haar payments globally5.9k investigation lines22 selected mapped statements6 open questionsExplore the current research release
Diophantine equations · Pythagorean triples · arithmetic geometryPerfect Cuboid ProblemCan a rectangular box have integer edge lengths, all three face diagonals integral, and its space diagonal integral too? No example or accepted nonexistence proof is known. The retained source reports substantial arithmetic restrictions and a finite frontier for one three-prime case, not a solution of the full problem.Priority open bridge · next taskIndependently audit the source-reported low-support restrictions.721 investigation lines9 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
PL topology · collapsibility · finite 2-complexes · Andrews–Curtis conjectureZeeman ConjectureDoes multiplying any finite contractible two-dimensional complex by an interval always make it collapsible? Restricted algebraic and fake-surface routes do not yet bridge back to arbitrary complexes.Priority open bridge · next taskProve an actual one-interval transfer or direct cylinder realization from an arbitrary K to a collapsible K×I.655 investigation lines7 selected mapped statements3 open questionsExplore the current research release
Extremal combinatorics · subset sums · probabilityManickam–Miklós–Singhi ConjectureListed inDouglas West’s REGS open-problem collectionEvidenceOriginal computation rerunIf n≥4k real numbers have nonnegative total, must at least a k/n share of all k-subsets also have nonnegative sum?Strongest recorded result · Reported resultUniform value-weighted five-cover paymentPriority open bridge · next taskIndependently audit continuous and harmonic inputs6.7k investigation lines24 selected mapped statements7 open questionsExplore the current research release
Number theory and arithmetic congruencesAgoh–Giuga conjectureEvidenceManually checked claimAn integer greater than one should satisfy the stated power-sum congruence exactly when it is prime.Strongest recorded result · Reported resultTerminal rank geometry identifiedPriority open bridge · next taskProve a branch-wide double-completion exclusion6k investigation lines18 selected mapped statements9 open questionsExplore the current research release
Graph theory · one-factorizations · rainbow spanning treesBrualdi–Hollingsworth ConjectureListed inDouglas West’s REGS open-problem collectionCan every one-factorization of a complete graph on 2m vertices, for m at least 3, be repartitioned into m spanning trees that each use every factor color exactly once?Strongest recorded result · Reported resultRevision-11 two-front calculusPriority open bridge · next taskEliminate globally locked Hall kernels8.1k investigation lines28 selected mapped statements10 open questionsPartially resolvedExplore the current research release
Graph theory · cubic graphs · perfect matchingsFan–Raspaud ConjectureListed inDouglas West’s REGS open-problem collectionCan every finite bridgeless cubic graph admit three perfect matchings with no edge common to all three?Strongest recorded result · Reported resultDeep two-anchor lifting through sixPriority open bridge · next taskProve bounded-interface absorption8.2k investigation lines30 selected mapped statements10 open questionsExplore the current research release
Polytope diameter · simplicial 4-polytopes · geodesic corridor enumerationDimension-Four Hirsch: Minimal Corridor CensusEvidenceFormalization resource linkedA hypothetical minimal dimension-four violation would contain a long dual path of tetrahedral facets. The source reports a small-n census of abstract path skeletons that pass necessary tests, but completing any skeleton to a polytopal sphere—or ruling them all out—remains open.Priority open bridge · next taskReproduce link-gluing and finite completion3.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Number theory · powerful integers · Pell equations · primitive divisorsErdős–Mollin–Walsh ConjectureErdős Problem #364Can three consecutive positive integers all have every prime factor repeated? The packet says no proof is known, but turns any hypothetical example into a tightly constrained Pell-sequence repair problem.Priority open bridge · next taskConstruct a valuation-sensitive reciprocity or product formula whose local terms vanish under repeated lifting but whose global term equals the nonzero Jordan moment.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Toric geometry · smooth rational fans · rewriting systemsOda’s Strong Factorization ConjectureEvidenceComputational study linkedGiven two smooth rational fans in one lattice with the same support, the conjecture asks whether both can be refined to one smooth fan using only ordinary smooth star subdivisions.Strongest recorded result · Reported resultExact route certificates restoredPriority open bridge · next taskProve RBL_1 and FSAFE_16k investigation lines21 selected mapped statements6 open questionsExplore the current research release
Analysis of Boolean functions and query complexitySensitivity versus Degree for Boolean Multilinear PolynomialsListed inEpoch AI FrontierMath open problemFind one Boolean polynomial whose number of sensitive coordinates grows faster relative to its degree than the best explicit construction currently recorded. The packet has sharply reduced one finite search, but no qualifying polynomial is known.Priority open bridge · next taskClassify the full type-0 eight-edge graph lifts without losing structural-leaf information.4.8k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Commutative algebra · Hilbert series · generic homogeneous idealsFröberg’s ConjectureEvidenceComputational study linkedDoes the Hilbert series of an ideal generated by generic homogeneous forms always equal the positive truncation of the expected product formula?Strongest recorded result · Reported resultOne tail repairs arbitrary corankPriority open bridge · next taskRegenerate computational evidence4.4k investigation lines26 selected mapped statements7 open questionsExplore the current research release
Discrete and computational geometryGilbert–Pollak ConjectureListed inOptimization Problems in Mathematics, problem 43EvidenceCertificate and code linkedFor a finite set of points in the plane, compare the shortest network that may add Steiner junctions with the ordinary minimum spanning tree. The conjecture says the first length is always at least √3/2 of the second. This packet reports substantial local reductions and certified regions, but no complete proof.Priority open bridge · next taskCover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.1.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Finite group theory · subgroup posets · rational homologyRational Homological Quillen Conjecture at p = 2For a finite group with no nontrivial normal 2-subgroup, the conjecture predicts nonzero rational reduced homology in the poset of its nontrivial elementary abelian 2-subgroups.Strongest recorded result · Reported resultPSp₈(5) top degree reported zeroPriority open bridge · next taskRetain independently reproducible certificate artifacts13k investigation lines22 selected mapped statements9 open questionsExplore the current research release
Extremal combinatorics · graph coloring · intersecting set systemsStahl’s Multichromatic Kneser-Graph ConjectureEvidenceOriginal computation rerunWhen every k-subset of [n] receives s colors and disjoint subsets receive disjoint palettes, Stahl’s conjecture gives the exact minimum total number of colors.Strongest recorded result · Reported resultExact residual invariants and local hierarchyPriority open bridge · next taskProve the dense partition-aware shadow theorem1.7k investigation lines19 selected mapped statements7 open questionsExplore the current research release
Graph theory · crossing numbers · graph coloringAlbertson ConjectureMust every graph have at least as many crossings as the complete graph whose number of vertices equals the graph's chromatic number?Strongest recorded result · Reported resultSix-region coalescence recordedPriority open bridge · next taskChecker-certify the r=2780 frontier5.7k investigation lines25 selected mapped statements8 open questionsExplore the current research release
Linear algebra · finite fields · additive combinatoricsAlon–Jaeger–Tarsi ConjectureGiven an invertible matrix over a field with at least four elements, can one choose a vector whose coordinates and transformed coordinates are all nonzero?Strongest recorded result · Reported resultCoordinate density yields a large admissible circuitPriority open bridge · next taskConstruct the shifted weight-zero TBF mapping cone9.1k investigation lines44 selected mapped statements8 open questionsPartially resolvedExplore the current research release
Differential geometry · affine manifolds · Euler characteristicChern’s Conjecture for Closed Affine ManifoldsMust every closed manifold carrying an affine structure have Euler characteristic zero?Strongest recorded result · Reported resultCoarse invariance of sheet complexityPriority open bridge · next taskClose the direct clock and fixed-base cancellation route10k investigation lines28 selected mapped statements10 open questionsExplore the current research release
Cryptography · private information retrieval · lattices · data structuresDoubly Efficient Private Information RetrievalKnown DEPIR constructions use stronger structure; this source asks for an ordinary-LWE construction and maps the barriers any candidate must avoid.Priority open bridge · next taskImport the exact one-read AMR transcript.11.2k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Arithmetic geometry · elliptic curves · Mordell–Weil rankElliptic Curves over ℚ of Rank at Least 30EvidenceData linkedCan one exhibit a nonsingular elliptic curve over ℚ with Mordell–Weil rank at least 30, together with an exact certificate? The maintained record is rank at least 29, so the target remains open.Priority open bridge · next taskSolve the materialized branch-gauged 38-variable compatibility core over 𝔽₇₃ without discarding stabilizer or denominator branches.3.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Arithmetic geometry · modular forms · Sato–Tate distributionsGeneralized Sato–Tate for Cubic GL₂-Type Abelian ThreefoldsFor a generic non-CM modular abelian threefold whose coefficient field has three real embeddings, do the three normalized Frobenius angles vary jointly and independently according to the product Sato–Tate measure?Priority open bridge · next taskConstruct the tensor trace or characteristic-coefficient operators on an actual horizontal automorphic object.10.5k investigation lines6 selected mapped statements3 open questionsExplore the current research release
Commutative algebra · local Gorenstein rings · torsion-free modules · tensor products and rigidityHuneke–Wiegand ConjectureMust a finitely generated torsion-free module over a one-dimensional Gorenstein local domain be free whenever its tensor product with its dual has no torsion?Priority open bridge · next taskSupply or independently validate the G5 quotient-tail, descended-coordinate, joint-generation, and determinant hypotheses before promoting the candidate e = 27 closure.8.8k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Geometric topology · algebraic topology · two-dimensional CW complexes · combinatorial group theoryWhitehead Asphericity ConjectureListed inEncyclopedia of Mathematics: Low-dimensional topology, problems inSuppose a space is built only from vertices, edges, and filled-in faces, and has no hidden two-dimensional sphere. Can a connected part cut out of it somehow contain such a sphere? Whitehead's conjecture says no: every connected subcomplex of an aspherical two-complex should itself be aspherical. No proof or counterexample is known in the retained packet.Priority open bridge · next taskProve the two-equation secondary excision theorem for the exact two-cell carrier retained by the current work.2.2k investigation lines8 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
Arithmetic geometry, motives, and equidistributionGeneralized Sato–Tate ConjectureEvidenceData linkedDo normalized Frobenius conjugacy classes spread through the correct compact symmetry group according to Haar measure?Priority open bridge · next taskProve tensor Tate for the Tannakian category generated by the motive and the Tate object at every base and realization required by the claim.2.5k investigation lines7 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Nonlinear elliptic partial differential equations · Liouville theorems · critical phenomenaLane–Emden ConjectureThe conjecture says the coupled Lane–Emden system has no everywhere-positive classical solution on all of Euclidean space when the exponents lie strictly below the critical hyperbola.Strongest recorded result · Reported resultDimension-six direct hard cell reportedPriority open bridge · next taskVerify the bounded-solution reduction source6.9k investigation lines25 selected mapped statements8 open questionsPartially resolvedExplore the current research release
Algebraic number theory · Artin L-functions · Stark regulatorsTate’s Rational Leading-Term Stark Conjecture at s = 0EvidenceData and code linkedThe rational Stark conjecture predicts that the leading term of an Artin L-function at zero, divided by the matching determinant of logarithms of S-units, is algebraic in the character field and varies correctly under Galois conjugation.Priority open bridge · next taskConstruct the primitive orbit cocycle and account for every endpoint of the open 23-step scaling path.2k investigation lines9 selected mapped statements3 open questionsPartially resolvedExplore the current research release
Dynamical systems, convex billiards, and rigidityBirkhoff–Poritsky billiard conjectureDoes a complete smooth family of invariant curves close to the boundary of a strictly convex billiard force the boundary itself to be an ellipse?Priority open bridge · next taskProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.4.2k investigation lines6 selected mapped statements5 open questionsExplore the current research release
Complex algebraic geometry · Fano manifolds · nef tangent bundles · homogeneous spacesCampana–Peternell ConjectureMust a smooth complex Fano manifold whose tangent bundle is nef be a rational homogeneous space? The source reports conditional progress on cotangent fibers and caustics, but several independent global gates remain open.Priority open bridge · next taskCertify the normal-image elimination against exact finite-cover and reflexive-differential theorems.5.6k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Simplicial manifolds · normalization · polytope diameterDimension-Four Hirsch: Normalization-Fiber AuditEvidenceFormalization resource linkedNormalization can split singular local sheets and turn a labeled pseudomanifold into a genuine sphere with more vertices. The source reports reversing two such fibers to diagnose the original singular objects, but neither object is a dimension-four Hirsch counterexample.Priority open bridge · next taskIndependently replay the reconstruction audit613 investigation lines7 selected mapped statements5 open questionsExplore the current research release
Design theory and finite combinatoricsLarge Steiner Systems Construction ProblemThe requested S(6,7,19) cannot exist. Any such design would yield an S(4,5,17), and the latter was ruled out by a peer-reviewed exhaustive classification-based search in 2008.Priority open bridge · next taskPreserve the exact hypotheses and conclusion of the fixed-pair S(4,5,17) derivation.5.2k investigation lines9 selected mapped statements3 open questionsSolvedExplore the current research release
Algebraic combinatorics · Jack polynomials · map and matching enumerationMatchings–Jack ConjectureAfter writing the Jack parameter as α = 1 + b, do all Jack connection coefficients have nonnegative integer coefficients in b, as predicted by a combinatorial interpretation using matchings or maps?Priority open bridge · next taskRealize the atomic Gate-1 creator crossing with exact source chronology and weights.5.5k investigation lines7 selected mapped statements5 open questionsPartially resolvedExplore the current research release
Symplectic topologyNearby Lagrangian ConjectureFor a closed connected smooth manifold Q, must every closed exact Lagrangian inside T*Q be deformable to the zero section by a Hamiltonian isotopy?Priority open bridge · next taskDefine the precise relative ordered-incidence invariant and prove its exact-homotopy invariance with complete conventions.5.1k investigation lines5 selected mapped statements6 open questionsExplore the current research release
Topological graph theory · graph coverings · projective-planar embeddingsNegami’s Planar Cover ConjectureNegami’s conjecture says a connected graph has a finite planar cover exactly when it can be embedded in the projective plane.Strongest recorded result · Reported resultExact adjusted-deficit spectrumPriority open bridge · next taskClassify the six low-score path families4.4k investigation lines21 selected mapped statements10 open questionsExplore the current research release
Nonlinear dispersive PDE · energy-critical waves · soliton resolution · concentration compactnessSoliton Resolution for the Focusing Energy-Critical Wave EquationAt late times, should every bounded non-scattering wave split cleanly into free radiation and finitely many coherent solitons, continuously in time? The general nonradial full-time problem remains open.Priority open bridge · next taskDerive the exact linear incoming-flux norm corresponding to the ball collar trace and its P/E amplification.5.5k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Algebraic graph theory · strongly regular graphs · finite configurationsThe Missing Moore GraphA Moore graph of degree 57 and diameter 2 would have 3,250 vertices and extremal local structure: adjacent vertices share no common neighbor, while each nonadjacent pair shares exactly one. This last missing parameter case remains open.Priority open bridge · next taskCertify the minimum-degree-three step independently.4.2k investigation lines6 selected mapped statements6 open questionsExplore the current research release
Polynomial algebra, derivatives, and algebraic geometryCasas–Alvero ConjectureIf a polynomial shares a zero with every one of its derivatives, must all of its zeros coincide?Priority open bridge · next taskIndependently audit the entire A3 certificate-backed elimination from recurrence through component synthesis.3k investigation lines7 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
Symplectic geometry · embeddings · packingHigher-Dimensional Symplectic Ball-Packing ConjectureFor symplectic balls in real dimension at least six, are total volume and the pairwise two-ball nonsqueezing inequalities the only obstructions to packing them strictly into a larger ball?Priority open bridge · next taskConstruct the exact endpoint block rearrangement or singular baker map.3.7k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Singularity theory, zeta functions, and monodromyIgusa–Denef–Loeser Monodromy ConjectureMust every pole predicted by resolution data appear as monodromy somewhere arbitrarily near the singular point?Priority open bridge · next taskConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.3.5k investigation lines7 selected mapped statements4 open questionsPartially resolvedExplore the current research release
Quantum topology and hyperbolic knot theoryKashaev–Murakami–Murakami Volume ConjectureEvidenceData linkedThe conjecture predicts that quantum knot invariants grow at a rate set by the hyperbolic volume of the knot complement. The packet builds exact odd-level and boundary-state machinery, but still lacks universal asymptotics and an even-level geometric state sum, so the full all-integer limit is open.Priority open bridge · next taskAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.1.4k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Symplectic geometry, special Lagrangians, and geometric flowsThomas–Yau conjectureShould categorical stability select a unique special Lagrangian and govern a long-time Lagrangian mean-curvature flow, while instability yields ordered special-Lagrangian factors?Priority open bridge · next taskConstruct a phase-ordered sectorial or stopped inverse-surgery model and prove an actual twisted-complex equivalence for the selected slice.3.4k investigation lines6 selected mapped statements6 open questionsExplore the current research release
Geometric topology · homogeneous ANRs · homology manifoldsBing–Borsuk ConjectureThe Bing–Borsuk conjecture asks whether every finite-dimensional homogeneous metric ANR is a topological manifold. The selected source reports a compact, connected counterexample conditional on an imported Bryant–Ferry existence theorem whose revised construction proof has not been independently line-audited here.Later mathematical update · The source reports a classical refutation and fixed-root strong endpoint relative to imported Bryant–Ferry premises, records the closed downstream interfaces, and makes the independent construction boundary explicit.Priority open bridge · next taskObtain and independently line-audit the revised Bryant–Ferry manuscript2.1k investigation lines13 selected mapped statements4 open questionsRecent proof claim under reviewExplore the current research release
Harmonic analysis and Fourier multipliersBochner–Riesz ConjectureHow much smoothing at the boundary of the frequency ball is enough for bounded Fourier reconstruction on L^p?Priority open bridge · next taskProve the thick transverse congruent-circle estimate TCI with its precise thickening and nu dependence.2.5k investigation lines7 selected mapped statements3 open questionsPartially resolvedExplore the current research release
G-functions · arithmetic differential equations · Picard–Fuchs operators · motivesBombieri–Dwork Conjecture for G-functionsDoes every minimal differential equation of a G-function arise from algebraic geometry through Gauss–Manin or Picard–Fuchs constructions?Priority open bridge · next taskProduce a fully normalized geometric calibration example with an independently known Frobenius matrix.2.4k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Discrete and convex geometryBorsuk problem in four dimensionsFive vertices of a regular four-dimensional simplex already require five colors. The open question is whether five smaller-diameter pieces always suffice. The source closes several special and canonical face regimes, but its correction leaves noncanonical singleton and edge directions, buffered coverage, receiver selection, and cross-source safety open.Later mathematical update · The cumulative v7 source adds BP4-T25–T39, records a receiver-axis dead route, and makes noncanonical face control the controlling open contact obligation.Strongest recorded result · Reported resultDirected-interval high-radius endpointPriority open bridge · next taskConvert the residual critical core into a contradiction2.8k investigation lines46 mapped statements5 open questionsExplore the current research release
Arithmetic geometry · motives · equivariant L-values · algebraic K-theoryEquivariant Tamagawa Number ConjectureA canonical equivariant arithmetic class packages leading L-values, periods, regulators, and integral data. ETNC predicts that this class vanishes in the appropriate relative K-group; the general conjecture remains open despite many special cases.Priority open bridge · next taskConstruct the canonical relative-K₁ or determinant-torsor map into the multiplicative central conductor quotient.2.8k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Representation theory · homological algebra · finite-dimensional algebrasFinitistic Dimension ConjectureEach module with finite projective dimension has some finite resolution length. The conjecture asks whether, for each finite-dimensional algebra, all those finite lengths share one finite upper bound.Priority open bridge · next taskClose terminal-pair explosion X.2.8k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Algebraic topology · torus actions · rational homotopy theory · free differential modulesHalperin–Carlsson Toral Rank ConjectureAn almost-free r-dimensional torus action should force at least 2^r total rational cohomology classes. The packet narrows one rank-four algebraic configuration but does not prove rank four or the conjecture in arbitrary rank.Priority open bridge · next taskAudit the geometry-dependent eliminations through s=26.2.6k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Projective algebraic geometry · smooth subvarieties · complete intersections · Chern classes and Schubert geometryHartshorne Complete-Intersection ConjectureMust every smooth projective variety whose dimension is more than twice its codimension be cut out by exactly as many hypersurfaces as its codimension?Priority open bridge · next taskShow that a singular or non-lci codimension-three defect either carries a usable rank-one excess quotient, forces a smaller-codimension zero locus, or yields a nonzero weight-zero four-minor.2.3k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Computational number theory and classical factoring algorithmsImprove classical semiprime factorizationGiven a large number N known to equal two primes p and q, find the factors with a classical algorithm whose full bit complexity is asymptotically better than the general number field sieve for every allowed factor ratio.Priority open bridge · next taskConstruct a one-output shifted-Stirling coefficient evaluator with sublinear state and a complete end-to-end bit-complexity bound.2.7k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Convex geometryLog-Brunn–Minkowski ConjectureThe conjecture says a logarithmic interpolation of two symmetric convex bodies should have at least the geometric-mean volume. This packet turns the question into concavity of box sections and proves some source-contained cap configurations, but missing certificates and an unequal-overlap case block broader conclusions.Priority open bridge · next taskReproduce the missing finite certificate families for the reported complete (N,n,k)=(5,3,2) result.2.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Minimal surfaces · nonlinear elliptic PDE · real algebraic geometry · exact computationPolynomial Entire Minimal GraphsNonlinear smooth entire minimal graphs exist in high dimensions, but can one be the graph of a polynomial? The source has no example and narrows only conditional leading-term branches.Priority open bridge · next taskIndependently audit the load-bearing local product lemmas, block-radial formula, and two-eigenvalue CAS tables.2.2k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Algebraic geometry · birational geometry · singularities in characteristic pResolution of Singularities in Positive CharacteristicOver a perfect field k of characteristic p>0, must every integral finite-type k-variety X of dimension at least four admit a proper birational morphism π:Y→X with Y regular?Strongest recorded result · Reported resultLocal Frobenius and blowup identities retainedPriority open bridge · next taskProve filtered Frobenius termination2.4k investigation lines27 selected mapped statements7 open questionsRecent proof claim under reviewExplore the current research release
Geometric topology · L²-invariants · aspherical manifoldsSinger Conjecture for L²-Betti NumbersFor every closed aspherical n-manifold, must all L²-Betti numbers vanish outside the middle dimension, and hence vanish in every degree when n is odd?Priority open bridge · next taskConstruct controlled remote primitives and cycle-level alignment data.2.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Dynamical systems · smooth perturbation theory · periodic orbitsUnrestricted C^r Closing Lemma for r ≥ 2A recurrent orbit returns arbitrarily close to itself. The question is whether an arbitrarily small unrestricted C^r change can always close such recurrence into a periodic orbit when r is at least 2. The classical C^1 theorem does not provide the general higher-regularity result.Priority open bridge · next taskConvert the two-point derivative bridge into usable pointwise or orbit-flat data.2.7k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Riemannian geometry · rational homotopy · four-manifoldsBott Rational Ellipticity in Dimension FourMust every closed simply connected four-manifold with nonnegative sectional curvature have second Betti number at most two, equivalently be rationally elliptic?Priority open bridge · next taskClose the definite three-mode configuration without scalar-index shortcuts.1.7k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Online algorithms · metric geometry · work functions · competitive analysisDeterministic k-Server ConjectureLocal configurations are sharply controlled, but the proof still lacks a global invariant for changing charts in arbitrary metrics.Priority open bridge · next taskProve or refute the primal two-server Bellman candidate.1.5k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Arithmetic geometry · p-adic Galois representations · étale cohomologyFontaine–Mazur ConjectureWhich irreducible p-adic Galois representations satisfying arithmetic finiteness conditions actually come from algebraic geometry?Strongest recorded result · Reported resultNumerical and derived support criteriaPriority open bridge · next taskRealize the Artin–Tate seed in a finite-level scalar/Eisenstein block1.8k investigation lines9 selected mapped statements7 open questionsPartially resolvedExplore the current research release
Birational geometry · cubic fourfolds · Hodge theory · rationalityIrrationality of a Very General Cubic FourfoldIs a very general smooth degree-three hypersurface in complex projective five-space nonrational?Priority open bridge · next taskReproduce the reported 198-by-201 endpoint matrix and wedge calculation with canonical input and complete certificates.1.5k investigation lines8 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
Discrete and convex geometryKneser–Poulsen conjectureIf every pair of centers moves no farther apart, must the union of their balls lose volume and their common intersection gain volume, even when the radii differ?Priority open bridge · next taskProve a capacity or second-variation inequality on the complete five-active-center benchmark interval.1.6k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Automorphic forms, Langlands functoriality, and Galois representationsLanglands functoriality: GL4 × GL2 to GL8Do the local tensor products of a cuspidal GL4 representation and a cuspidal GL2 representation come from one global automorphic representation on GL8, with matching parameters at every place?Priority open bridge · next taskConstruct one regular irreducible automorphic tensor point in every admissible residual box in the regular algebraic polarized setting.1.7k investigation lines7 selected mapped statements4 open questionsExplore the current research release
Arithmetic geometry · abelian varieties · Hodge and ℓ-adic monodromy groupsMumford–Tate ConjectureFor an abelian variety over a number field, does the symmetry group seen by every ℓ-adic Galois representation exactly match the symmetry group determined by its rational Hodge tensors?Priority open bridge · next taskConstruct one valid global invariant object realizing the local rank-nine adjoint systems.1.8k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Elliptic curves and Galois representationsSerre uniformity conjectureFor a non-CM elliptic curve over the rational numbers, the conjecture says that every prime larger than 37 gives the full possible symmetry group on the curve's ell-torsion points.Priority open bridge · next taskClose Gate RT by an exact integral minimal R=T theorem application or a direct computation of the minimal non-CM adjoint Selmer group.1.5k investigation lines6 selected mapped statements4 open questionsExplore the current research release
Birational algebraic geometry · Minimal Model Program · flips · Nakayama–Zariski decompositionsTermination of Flips in Arbitrary DimensionCan an arbitrary Minimal Model Program perform infinitely many flips, or must every flip sequence eventually stop in every dimension?Priority open bridge · next taskRule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.1.5k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Complex analysis, conformal mapping, and Bergman-space integrabilityBrennan's ConjectureHow strongly can the derivative of a conformal map blow up near a wild boundary while still remaining integrable?Priority open bridge · next taskProve or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.1k investigation lines7 selected mapped statements5 open questionsExplore the current research release
Social choice · preference profiles · extremal set systems · exact finite searchCondorcet Winning Sets: Does Size Three Always Suffice?Can three candidates always form a committee that no outsider can defeat by at least half the voters? External strict-majority work proves a size-five guarantee, but exact alignment with the stronger equality-counts convention has not been established; the exact size-three question remains open.Priority open bridge · next taskSettle the eight-candidate top-two system by proving the proposed 10/19 lemma or producing a feasible threshold-one-half point.1.2k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Euclidean geometry · configuration spaces · symmetric powers · determinant nonvanishingEuclidean Atiyah–Sutcliffe Conjecture 1Exact formulas organize insertion and collision limits and settle several special geometries, while four global nonvanishing bridges remain open.Priority open bridge · next taskProve phase-preserving top-star transversality for every Euclidean insertion, equivalently strict positivity of the exact insertion Schur complement.1.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Universal algebra · finite lattice theory · finite group intervalsFinite Lattice Representation ProblemEvery finite lattice would be representable if one meet-irreducible deletion gadget could always be built. A competing route would disprove the statement by ruling out the seven-element W23 lattice in every finite group interval. The packet narrows both routes without finishing either.Priority open bridge · next taskConstruct the general nonbottom meet-irreducible deletion gadget or an equivalent finite exact pair-generation construction with a terminating obligation order.1.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Formal languages · automata theory · algebraic language theory · finite monoidsGeneralized Star-Height ProblemThe packet turns one long-standing question about regular expressions into several exact smaller-looking questions about two-state recurrences, return loops, parity, and finite-group accumulation. Those reformulations sharpen the search, but none yet resolves the original problem.Priority open bridge · next taskClose the correlated residual-rectangle form of the two-state matrix-star recurrence for every aperiodic recognizing monoid, including empty-word bookkeeping and both persistent diagonal recurrences.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Planar graph drawing · rational distances · arithmetic geometryHarborth's ConjectureMany local gadgets and graph classes are controlled; the unresolved step is fitting them together while preserving planarity and target neighborhoods.Priority open bridge · next taskProve sparse factorization before shelling substitution.1.3k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Algebraic geometry · birational geometry · Calabi–Yau varietiesMorrison–Kawamata Cone ConjectureFor a projective Q-factorial klt Calabi–Yau pair over a characteristic-zero field, does its numerical automorphism group cover the effective nef cone Nef(X) ∩ Eff(X) by translates of one rational-polyhedral chamber, with distinct translates having disjoint interiors?Priority open bridge · next taskSeparate the mixed radical near a rational null face.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Pseudorandomness · permutation branching programs · finite groups · Fourier analysisOptimal Explicit PRGs for Width-3 Permutation Branching ProgramsThe packet reduces the generator problem to preserving one structured covariance average. Random and partially explicit constructions meet many surrounding requirements, but no uniformly explicit short-seed construction is yet reported for that final average.Priority open bridge · next taskProve the signed canonical rank-two covariance defect bound for one concrete uniformly explicit short-seed matrix-table distribution.1.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Birational geometry · cubic fourfolds · Hodge theory · surface monodromyVery General Cubic Fourfold Irrationality — Retained Monodromy ModelsThe packet tries to rule out a rational parametrization by forcing a Hodge-theoretic surface carrier and eliminating possible carrier models. Although it postdates the cited 2025/2026 proof sources, its distinct route remains conditional and is not that external proof.Priority open bridge · next taskReconstruct the marked blowup and contraction clusters for Models I and IV.1.2k investigation lines7 selected mapped statements5 open questionsSolvedExplore the current research release
Arithmetic geometry · ℓ-adic cohomology · monodromy filtrations · Frobenius weightsWeight–Monodromy ConjectureWhen a variety degenerates over a local field, monodromy organizes its cohomology into layers. The conjecture says each layer has exactly the Frobenius weight predicted by its position. The general mixed-characteristic case remains open.Priority open bridge · next taskProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.1.4k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Spectral geometry and minimal hypersurfacesYau’s First-Eigenvalue ConjectureFor every closed, connected, embedded minimal hypersurface in the unit sphere, determine whether the first nonzero Laplace–Beltrami eigenvalue equals the hypersurface dimension.Priority open bridge · next taskDerive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.1k investigation lines8 selected mapped statements4 open questionsRecent proof claim under reviewExplore the current research release
Finite groups · modular representation theory · block theory · local representation theoryAlperin Weight ConjectureThe source reports substantial reusable reductions and projective-algebra tools, but no nontrivial quasi-isolated block has yet been completed with all stabilizer, scalar, and intermediate-block requirements, and full prime-and-cover coverage remains open.Priority open bridge · next taskProve the exact Corollary 6.10 global-to-generic block-triple relation for every label of one nontrivial quasi-isolated block.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Combinatorics and number theory · binary sequences · autocorrelation · cyclotomic constraintsBarker Sequence ConjectureA Barker sequence is a row of plus and minus signs whose shifted copies stay almost perfectly uncorrelated. The packet reports strong restrictions on any sequence longer than 13, but it does not report a proof that none exists.Priority open bridge · next taskEliminate or finitely classify ordered BK-B27 lifts satisfying all exact prime-square inversion marginals, higher odd moments, distinctness, ordering, and forced residues modulo four.896 investigation lines6 selected mapped statements6 open questionsExplore the current research release
Riemannian geometry · nonnegative curvature · rational homotopy · Morse theoryBott Conjecture on Rational EllipticityMust every closed simply connected manifold admitting nonnegative sectional curvature have only finitely many nonzero rational homotopy groups?Priority open bridge · next taskAudit the provisional phase and Fredholm determinant statements before using robust spectral purification.980 investigation lines8 selected mapped statements3 open questionsExplore the current research release
Graph theory · reconfiguration · graph coloringCereceda's ConjectureFor each fixed degeneracy d and every palette of at least d+2 colors, can any two proper colorings be transformed into one another one vertex at a time using only quadratically many steps?Priority open bridge · next taskGive a canonical mathematical definition of the two-boundary profiles before introducing implementation-specific state or symmetry reductions.897 investigation lines8 selected mapped statements4 open questionsExplore the current research release
Social choice · majority tournaments · minimaxCondorcet Winning SetsCan every election be represented by at most three candidates so that no outsider is preferred to all three by a strict majority? The size-three guarantee is open. Current external literature gives a general size-five guarantee.Priority open bridge · next taskCompress a pair shadow to a winning triple.794 investigation lines8 selected mapped statements4 open questionsExplore the current research release
Graph algorithms · all-pairs shortest paths · min-plus product · fine-grained complexityTruly Subcubic Exact APSP ConjectureSampling, exact carry bookkeeping, and compact target cloning narrow the problem, but the algorithm still needs sparse-target replacements for dense uniformization and low-doubling stages.Priority open bridge · next taskReplace the remaining dense low-rank-to-slice, heavy/light, uniformization, and later rank-reduction scans by sparse-target operations without losing equivalence.810 investigation lines9 selected mapped statements3 open questionsExplore the current research release
Diophantine geometry · heights · algebraic points of bounded degree · normal-crossings divisors · arithmetic discriminantsVojta’s Conjecture over Number FieldsCan proximity to a normal-crossings divisor plus canonical height be controlled by one field discriminant and an arbitrarily small big-height allowance for every algebraic point of bounded degree outside a proper exceptional set?Priority open bridge · next taskProve the local cyclic-total versus multi-Kummer order comparison.734 investigation lines8 selected mapped statements4 open questionsExplore the current research release
Computational social choice and approval-based committee electionsApproval-Core Nonemptiness ConjectureMust every approval election contain a committee that no nonempty proposal of size at most k can block with its proportional coalition of strict gainers?Priority open bridge · next taskConstruct and independently audit the canonical decorated vertex, edge, and two-face catalogue for the rigid six-row bases at (12,9,8).716 investigation lines6 selected mapped statements4 open questionsExplore the current research release
Additive combinatorics · finite fields · algebraic combinatoricsGraham’s Rearrangement ConjectureErdős ProblemsEvidenceCertificate and computational study linkedCan every finite set of nonzero residues modulo a prime be ordered so that all of its nonempty partial sums are different?Priority open bridge · next taskIndependently audit load-bearing derivations7.1k investigation lines28 selected mapped statements6 open questionsLiterature: finite check remainsExplore the current research release
Spectral geometryPólya's Conjecture for the Dirichlet LaplacianFor every bounded planar drum, should each Dirichlet eigenvalue stay above the area-scaled Weyl prediction?Priority open bridge · next taskProve universal positivity of beta_(19,1) for every finite occupancy set.544 investigation lines6 selected mapped statements4 open questionsExplore the current research release
Affine algebraic geometry · polynomial cancellation · locally nilpotent derivations · additive group actionsZariski Cancellation for Affine Three-Space in Characteristic ZeroOver an algebraically closed field of characteristic zero, if a finitely generated three-dimensional affine domain becomes four-dimensional affine space after adjoining one variable, was it already affine three-space?Priority open bridge · next taskProve the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.648 investigation lines8 selected mapped statements3 open questionsExplore the current research release
Newtonian celestial mechanics and real algebraic geometryFiniteness of planar central configurationsSmale's sixth problemEvidenceComputational study linkedFor any fixed positive masses, are there only finitely many planar Newtonian central configurations up to similarity? The source constrains possible continuous families and complex boundary clusters, but a universal exclusion of physical families remains open.Priority open bridge · next taskExclude or fully characterize attachment of the surviving five-body zero-force residual at the prescribed outside mass one.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Divisor sums and additive number theoryExistence of odd weird numbersErdős problem #470 — odd weird numbers questionAn odd weird number would have more than enough proper divisors in total to exceed the number, yet no selection of distinct proper divisors would add up to it. This handoff supplies restrictions and construction equations, but no such number or universal exclusion.Priority open bridge · next taskRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Post-quantum cryptography · succinct arguments · extraction · STARKsLoss-to-Time State-Preserving Quantum ExtractionThe source rules out the unrestricted scalar-loss compiler, proves several conditional compiler routes, and classifies the current deterministic RPO endpoint as already additive. The leading open work is a real lossy-source theorem, a complete concrete parameter ledger, and stronger composition and public-output interfaces.Later mathematical update · The cumulative v57 source refutes the unrestricted scalar compiler, records the current additive endpoint, and adds conditional hereditary, exact-OR, private-preservation, passive-live, and bounded-batch results with explicit open interfaces.Priority open bridge · next taskSeparate active composition and public output17.9k investigation lines17 selected mapped statements7 open questionsExplore the current research release
Quantum information · postselection · conditioned dynamics · complexity boundariesCancellation-Conditioned Amplitude BoundaryThe source reports several controlled projector and contraction cases, but no syntax-independent theorem for arbitrary sealed or nonprojective operations.Priority open bridge · next taskChoose one explicit nonprojective transfer grammar.13.4k investigation lines8 selected mapped statements3 open questionsExplore the current research release
Convex geometry · geometric probability · additive energyHall’s Random-Triangle ConjectureEvidenceComputational study linkedAmong planar convex regions of a fixed area, does the disk give the largest chance that three independent uniform points form an acute triangle?Strongest recorded result · Reported resultThreshold Bregman resource and paid bandPriority open bridge · next taskProve the moderate-band Bellman comparison4.9k investigation lines24 selected mapped statements6 open questionsExplore the current research release
Combinatorics of words and permutation overlapsShortest superpermutations: general caseEvidenceFormalization resource linkedFind the shortest word that contains every permutation of n symbols as a contiguous block. The source reports exact and bounded small cases, general lower-bound machinery, and an eight-symbol construction that still needs an integral completion.Priority open bridge · next taskControl attainable row surplus jointly with unavoidable integral completion loss on the actual insertion-incidence families for all degrees.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Harmonic analysis · Wiener algebra · additive and spectral structureKahane’s Quantitative Beurling–Helson ConjectureMust a continuous circle phase be affine whenever the Wiener norms of all its large integer powers grow more slowly than logarithmically?Strongest recorded result · Reported resultInnovation, power, and quotient budgetsPriority open bridge · next taskRetain the core-halo geometric fallback7.4k investigation lines29 selected mapped statements7 open questionsExplore the current research release
Extremal graph theory · longest cycles · degree-sum conditionsOre-Type Bondy Longest-Cycle ConjectureIn a k-connected graph with the required degree-sum bound, must every component outside a longest cycle avoid paths on k vertices?Strongest recorded result · Reported resultBoundary-safe fixed-endpoint proof addedPriority open bridge · next taskComplete literal p>=3 switch certificates6.6k investigation lines28 selected mapped statements10 open questionsExplore the current research release
Exponential Diophantine equations · Fermat numbers · square factors · 2-adic valuationsSquare-Freeness of Fermat NumbersCan any prime square divide a Fermat number? Source-reported restrictions narrow possible square divisors, but the universal question remains open.Priority open bridge · next taskResolve the surviving positive sparse endpoint d=1.5.5k investigation lines11 selected mapped statements19 open questionsExplore the current research release
Geometric optimization and rigid motionsAmbidextrous moving sofaEvidenceCode linkedHow large can a connected planar body be if it must negotiate both left and right unit-width corridor turns? A sharp abstract support-profile bound is available in the source, while its comparison with the body's ordinary area remains open.Priority open bridge · next taskDerive an exact ordinary-fiber-area-minus-F expression accounting for exposure, endpoint terms, and multiplicity.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Optimal escape paths and convex geometryBellman’s lost-in-a-forest problemEvidenceComputational study linkedFor a known convex forest but an unknown starting point and heading, find a shortest path that guarantees reaching its boundary. The source reports exact solutions for restricted families and local contact systems, while the general sharp construction remains open.Priority open bridge · next taskControl stationary tight-tie accumulation or replace a positive-density tight set without assuming it contains intervals.1.2k investigation lines9 selected mapped statements3 open questionsRecent proof claim under reviewExplore the current research release
Arithmetic dynamics of divisor sumsCatalan–Dickson aliquot conjectureEvidenceComputational study linkedRepeatedly replace a positive integer by the sum of its proper divisors. Must every sequence eventually repeat or reach zero? The source rules out several exact descriptions of escape and strengthens a specific counterexample route, while arbitrary orbits remain uncontrolled.Priority open bridge · next taskProve actual all-orbit numerator/denominator control or bounded-prime-support recurrence for infinitely many adjacent pairs.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Discrete geometry and three-dimensional packingDensest regular tetrahedron packingEvidenceComputational study linkedHow densely can congruent regular tetrahedra fill space? The source reports sharp bounds for several constrained arrangements around the double-dimer candidate, while the unrestricted optimum remains unresolved.Priority open bridge · next taskProve the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Diophantine equations and arithmetic curvesEqual sums of two fifth powersEvidenceComputational study linkedCan two different unordered pairs of positive integers have the same sum of fifth powers? The source excludes nine pair-sum ratios at every height and corrects a proposed stronger claim about rational points, while the unrestricted equation remains unresolved.Priority open bridge · next taskProve a parameter-uniform theorem excluding positive rational lifts, or an exhaustive parameter reduction that also handles every integer scale.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Primorials and prime-producing offsetsFortune’s conjectureEvidenceData linkedAfter multiplying the first n ordinary primes, find the first offset greater than one that makes another prime. Must that offset always be prime? The packet develops arithmetic and sieve reductions but leaves the conjecture open.Priority open bridge · next taskProve an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Determinants of sign matricesHadamard maximal determinant problem, order 23EvidenceData linkedHow large can the determinant of a 23 by 23 sign matrix be? The source reports a lower construction and a universal upper endpoint, but still needs exhaustive coverage and exact sign-realization exclusions to establish the maximum.Priority open bridge · next taskProve complete marked reverse-merger generation with every required restored profile and refinement of other classes.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Prime numbers with symmetric decimal digitsInfinitely many decimal palindromic primesEvidenceData linkedCan a decimal digit string read the same in both directions and be prime at arbitrarily large lengths? The packet gives exact prime detectors and a character-sum reduction, but the lower bound that would produce infinitely many such primes remains unproved.Priority open bridge · next taskProve \(\Re\mathcal Z_m^{\rm deep}\le9q\), or another adequate one-sided saving from the \(10q\) main term, for infinitely many \(m\).1.3k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Prime numbers and decimal reversalInfinitely many emirpsEvidenceData linkedAn emirp is a prime whose reversed decimal digits form a different prime. The target asks for such numbers at arbitrarily large lengths. The packet separates exact counting identities from the still-unproved arithmetic correlations needed to force infinitely many examples.Priority open bridge · next taskProve a sufficient signed bound for the actual cube-root roughness counts at arbitrarily large even lengths, or retain enough of the actual composite–composite correction to make the exact identity positive.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Discrete geometry and square packingPacking 11 unit squares in a squareEvidenceData linkedWhat is the smallest square that can contain eleven unit squares when each may rotate independently? The handoff gives an exact construction and local bounds, but still needs a universal argument that no smaller packing exists.Priority open bridge · next taskControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Integer sequences and coverageRecamán’s Sequence Coverage ConjectureEvidenceData linkedDoes the standard Recamán sequence visit every nonnegative integer? Addition may repeat a value; the question concerns coverage. The source leaves both coverage and permanent omission unresolved.Priority open bridge · next taskEstablish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Decimal reverse-and-add dynamicsThe 196 problem and Lychrel numbersEvidenceData linkedDoes repeatedly reversing the decimal digits of 196 and adding ever produce a palindrome? The source reports exact quotient dynamics, a partial seed invariant and exclusions of fourteen infinite families, while leaving all-time palindrome avoidance unresolved.Priority open bridge · next taskFind an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.1.3k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Coulomb energy and spherical configurationsThomson problem: general caseEvidenceData and code linkedFor every number of identical charges on a sphere, determine the least Coulomb energy and every configuration attaining it. The source develops general reductions and sharp restricted seven-point comparisons, while the unrestricted seven-point and all-cardinality targets remain open.Priority open bridge · next taskExclude seven-point residual configurations with energy below e_B while retaining the contact remainder and shared rank-three geometry.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Latin squares and orthogonal arraysThree mutually orthogonal Latin squares of order 10EvidenceCode linkedDetermine whether three Latin squares of order ten can be pairwise orthogonal. The source gives exact tests for fixed residuals, restricted exclusions, and necessary count conditions, while existence of a complete triple remains unresolved.Priority open bridge · next taskProve necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Additive number theory and integer sequencesUlam-sequence densityEvidenceData linkedDoes the original Ulam sequence, seeded by 1 and 2, have a positive natural density? The target requires a limiting density and strict positivity. Neither positive lower density alone nor a density theorem for a filtered model resolves it.Priority open bridge · next taskBound the entire future weighted positive deficit of the original sequence below a proved starting potential, or establish another sufficient all-scale loss estimate.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Arithmetic geometry · K3 surfaces · automorphic formsGeneralized Sato–Tate for Generic Picard-Rank-18 K3 SurfacesFor a K3 surface over the rationals whose transcendental cohomology has rank four and no extra Hodge endomorphisms, do normalized Frobenius classes spread out according to Haar measure on SO(4), or on O(4) when the determinant character is nontrivial?Priority open bridge · next taskCertify the compatible GSpin lift and both half-spin systems over the exact determinant-killing field.1.2k investigation lines7 selected mapped statements3 open questionsExplore the current research release
Crossings in drawings of complete graphsHarary–Hill conjectureHow few crossings can a drawing of a complete graph have? The source reports exact finite cases, obstructions to stronger descent rules and improved local crossing repairs, while leaving the Harary–Hill lower bound open for unrestricted drawings.Priority open bridge · next taskProve global accessibility through compatible nonincreasing moves, including large convex components and interacting nonconvex cores.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Combinatorial geometry and line arrangementsKobon triangle problemHow many nonoverlapping triangular faces can n straight lines create? The packet develops exact wiring and geometric bounds, and reports a sharp 54-triangle result for fourteen lines with at most four finite multiple points. Arrangements with more multiple points remain part of the open problem.Priority open bridge · next taskProve a uniform Bellman potential on the complete reachable state domain for every n.1.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Diophantine equations and marked quadratic ideal classesMarkov uniqueness conjectureCan two different positive Markov triples have the same largest number? The packet translates a possible collision into conditions on specifically marked quadratic ideals and into exact integer reconstruction tests. Those reductions do not yet exclude a first proper split for every maximum.Priority open bridge · next taskExclude every proper nonempty zero subsum of the actual marked sequence 4g_i.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Real projective geometry and tope coveringMcMullen's projective convex-position problemCan every general-position point set below the stated threshold be moved into convex position by a projective transformation? The source studies the equivalent realizable all-rank construction through strict tope codes, with a robust rank-five example and unresolved higher-rank coverage.Priority open bridge · next taskProduce a real full-spark zero-hole construction with an argument that continues in every rank, not just one finite rank.1.3k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Signed complete graphs · discrepancy · covering radii · finite-temperature methodsMinimum Overlap ProblemChoose plus or minus signs on every edge of a complete graph to make every two-way vertex signing have small total interaction. The open question asks whether the best possible worst interaction, divided by n^(3/2), approaches a limit.Priority open bridge · next taskCombine row and column one-point tests without double-counting their shared matrix entries and recover two bridge budgets up to sublinear logarithmic cost.1.1k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Decimal digit dynamics and smooth numbersMultiplicative persistence conjectureMultiply a number’s decimal digits and repeat until one digit remains. The conjecture says this always takes at most eleven steps. The packet classifies seventeen particular inverse problems and proves a finite bound for predecessors of each fixed positive target, while the depth of all possible trajectories remains open.Priority open bridge · next taskProve I^10(Z)=∅ for the entire infinite positive zero boundary Z.1k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Complex Hadamard matrices and mutually unbiased basesMutually unbiased bases in dimension sixThree mutually unbiased bases in six complex dimensions can be constructed explicitly. The question is whether a fourth can ever exist. The packet gives exact tests for a fixed triple and reports that a triple sharing one conjugation cannot extend, but does not force that symmetry on an arbitrary quartet.Priority open bridge · next taskExclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.1.2k investigation lines9 selected mapped statements3 open questionsExplore the current research release
Commutative algebra and differential operatorsNakai conjectureCan every singular finitely generated commutative algebra in characteristic zero be detected by a differential operator that cannot be built from multiplication and derivations?Priority open bridge · next taskEstablish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.1.2k investigation lines8 selected mapped statements4 open questionsExplore the current research release
Computational number theory · elliptic curves · discrete logarithmsElliptic-Curve Discrete Logarithm Challenge InstanceFor one specified point P of large prime order and one target point Q on a finite-field elliptic curve, recover the unique scalar x with Q = [x]P.Priority open bridge · next taskObtain exact source, notebook, archive or API evidence for how the private scalar was sampled.900 investigation lines7 selected mapped statements3 open questionsExplore the current research release
Combinatorial game theory · Domineering · thermographs · exact finite computationBerlekamp’s Domineering-Temperature ConjectureEvidenceComputational study linkedLech Mazur's manuscript gives a rectangle-reachable 28-cell Domineering position with exact game {17/8 | -2+*} and temperature 33/16 > 2. The attached Lean development kernel-checks the board–target equality, explicit target thermograph, 30-move replay, and resulting existential statement through its narrow HasValueTemperature interface; unrelated external replication and specialist review remain open.Published research update · Scoped Lean theoremLean-checked rectangle-reachable witness at 33/16Independent checks still openSpecialist review and unrelated replication60 investigation linesPublished Aug 14, 2026 · Scoped Lean theoremExplore the current research releaseNo workspaces match these controls.
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Six-region theorem and r=2780 frontier replace the r=5441 leadV11 records a six-region coalescence theorem, corrects its small-r domain, and reports an unreproduced descent to r=2780 while preserving the former r=5441 frontier as history. · Albertson V11 prepared date · Albertson Conjecture→Post-V6 cutoff is separated from reproducible evidenceRevision 8 records a lower provisional cutoff and stronger exact tooling while making clear that the sweep must be regenerated and one earlier multiplier is withdrawn. · Albertson Conjecture comprehensive revised audit V8 date · Albertson Conjecture→Cap compression is narrowed to the exact five-state frontierRevision 13 adds fixed-exterior parity and cap-square provenance while replacing two overstrong action assumptions by a scoped compression-critical exit problem. · Barnette Conjecture revision 13 integrity and scope audit date · Barnette's Conjecture→Global refactors and a rooted seed criterion open two sharper routesRevision 17 proves source-reported global matching reachability and a rooted seed criterion, then isolates descent, locked-kernel elimination, and seed existence as the remaining bridges. · Brualdi-Hollingsworth consolidated source material revision 17 date · Brualdi–Hollingsworth Conjecture→
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