Mo Putera comments on Mo Putera’s Shortform

[Mo Putera]

I asked GPT-5.6 Sol to list unsolved math problems it would rate ≥60 on VibeMathed significance.

Sol’s list of ≥60 significance-rated unsolved math problems

A decades-old named problem known throughout one specialty is normally 30–40. Problems with monographs, textbooks, or a subfield organised around them occupy roughly 50–60. A score of 60 therefore requires something extra: cross-field mathematical dependence, a premier cross-disciplinary problem list, or exceptional fame. When torn, I took the lower score.

The fixed comparison points are Jacobian 65, cycle double cover 55, Connes rigidity 45, and sofic groups 60. VibeMathed’s three current 60+ entries are the 67.25% zeta-zero record at 68, the general Jacobian conjecture at 65, and the sofic-groups question at 60. They are used as anchors below.

Several rows overlap logically. GRH contains RH-like cases; Hardy–Littlewood contains twin-prime-type cases; Bateman–Horn contains Bunyakovsky; full BSD contains Tate–Shafarevich finiteness; and broad Langlands statements contain Artin, Ramanujan, Sato–Tate, and other cases. They remain separate because mathematics treated each as an independently named target with its own literature and attack history. Counting them is appropriate for a watchlist of problems, but summing the row count does not measure “how many independent mysteries” exist.

The list excludes broad aspirations without a stable proposition, such as “understand turbulence,” “complete the Langlands programme,” or “solve the three-body problem.” It also excludes independence results such as the continuum hypothesis, where the original yes-or-no statement is no longer an ordinary open conjecture in ZFC.

Finally, this ranks significance only. It says nothing about which problems are most likely to fall to AI soon. Formalizability, availability of intermediate lemmas, counterexample-search space, proof length, and feedback quality are separate axes, and they would reorder this list radically.

Score	Problem	Area	One-sentence significance justification
100	Riemann hypothesis	Number theory	The fixed reference point: a Millennium problem with an enormous web of conditional theorems across number theory and beyond.
97	Generalized and grand Riemann hypotheses	Number theory	The broader L-function versions support a comparably vast conditional superstructure, while lacking the original RH’s singular cultural primacy.
97	P versus NP	Complexity theory	A Millennium problem central to algorithms, cryptography, optimization, logic, and the limits of efficient reasoning, with fame far beyond mathematics.
90	Goldbach’s conjecture	Number theory	A Landau problem and one of the canonical household-name conjectures, attacked continuously since the eighteenth century.
90	Navier–Stokes existence and smoothness	PDE and fluid mechanics	A Millennium problem governing the mathematical coherence of three-dimensional viscous flow and a standard reference point across PDE and physics.
89	Twin prime conjecture	Number theory	A Landau problem, a household-name question about primes, and the endpoint of the modern bounded-gaps programme.
89	Birch and Swinnerton-Dyer conjecture	Arithmetic geometry	A Millennium problem that organizes the arithmetic of elliptic curves and links ranks, rational points, L-functions, and major machinery across number theory.
88	Hodge conjecture	Algebraic geometry	A Millennium problem at the centre of algebraic cycles, topology, complex geometry, and the theory of motives.
87	Yang–Mills existence and mass gap	Mathematical physics	A Millennium problem asking for rigorous foundations for a central quantum field theory and its physically observed mass gap.
86	Hardy–Littlewood prime-tuples conjecture	Number theory	The master quantitative conjecture for simultaneous prime patterns subsumes twin-prime-type questions and guides much of sieve theory and additive prime number theory.
85	Langlands functoriality	Number theory and representation theory	Described by Arthur as one of the central questions of contemporary mathematics, it is the main transfer principle in the Langlands programme and connects arithmetic, harmonic analysis, and representation theory.
85	Global Langlands reciprocity	Number theory and representation theory	Langlands’s proposed ultimate reciprocity law generalizes the modularity paradigm to broad classes of Galois representations and automorphic forms and is a distinct central pillar of the programme alongside functoriality.
84	Hilbert’s sixteenth problem	Dynamical systems and real algebraic geometry	One of the most visibly unresolved Hilbert problems, it generated the modern qualitative theory of polynomial differential equations and real plane curves.
83	Smooth four-dimensional Poincaré conjecture	Four-manifold topology	The unique unresolved smooth-dimensional case of the Poincaré story sits at the heart of exotic smoothness and four-dimensional topology and appears on DARPA’s challenge list.
82	Cosmic censorship	General relativity and PDE	Penrose called it possibly the most important unsolved problem in classical general relativity, and its strong and weak forms govern determinism and visibility of singularities.
81	Three-dimensional Euler regularity and blow-up	PDE and fluid mechanics	The inviscid counterpart to Navier–Stokes is a central singularity problem in mathematical fluid mechanics with major numerical and partial-result programmes; recent boundary blow-up results do not settle the classical full-space or periodic problem.
80	Collatz conjecture	Number theory and dynamics	The prompt’s own approximately-80 calibration: extraordinary fame and decades of attack despite comparatively isolated mathematical consequences.
80	Tate conjecture	Arithmetic geometry	The companion to Hodge over nonclosed fields is one of the two dominant conjectures on algebraic cycles and underlies major parts of arithmetic geometry and motives.
79	Generalized Hodge conjecture	Algebraic geometry and motives	Grothendieck’s strengthened companion to Hodge organizes coniveau, Hodge structures, algebraic cycles, and motives and now has a dedicated survey in the Journal of Open Mathematical Problems.
79	abc conjecture	Number theory	A celebrated “one inequality explains many Diophantine phenomena” conjecture with a huge consequence tree; Mochizuki’s published claim remains unaccepted by the mainstream community.
79	General Boolean circuit lower bounds, especially NP not in P/​poly	Complexity theory	Superpolynomial lower bounds for explicit NP functions are a central route to P≠NP, and an entire theory of barriers and restricted-circuit lower bounds is organized around them.
78	Rigorous construction of a nontrivial four-dimensional quantum field theory or QCD	Mathematical physics	Simon’s foundational mathematical-physics list and decades of constructive QFT treat four-dimensional interacting theories as a defining unsolved foundations problem beyond the narrower Yang–Mills mass-gap formulation.
77	NP versus coNP and superpolynomial proof lower bounds	Complexity and proof theory	This is the central question of whether short proofs exist for all efficiently refutable statements, tying complexity separation directly to propositional proof systems.
77	Kakeya conjecture in dimensions at least four	Harmonic analysis and geometric measure theory	A towering crossroad for restriction theory, incidence geometry, PDE, and additive combinatorics; the landmark three-dimensional result leaves the general conjecture open.
77	Grothendieck’s standard conjectures on algebraic cycles	Algebraic geometry	They were designed to make motives and Weil cohomology behave as expected and would supply foundational positivity and equivalence results throughout algebraic geometry.
76	Bloch–Kato and Tamagawa-number conjectures	Arithmetic geometry	This far-reaching special-value framework unifies class-number formulas, BSD-type statements, Galois cohomology, and motives across modern number theory.
76	Bloch–Beilinson filtration and conjectures on Chow groups	Algebraic geometry and arithmetic geometry	The conjectural filtration is a principal organizing framework for algebraic cycles and mixed motives and links Chow groups to regulators and the orders and leading terms of L-functions.
76	Schanuel’s conjecture	Transcendence theory	The master conjecture of transcendence theory implies most classical algebraic-independence conjectures about exponentials, logarithms, e, and π.
76	Hilbert’s twelfth problem	Algebraic number theory	The major unresolved explicit-class-field-theory problem from Hilbert’s list seeks the nonabelian or general-number-field analogue of complex multiplication.
75	Invariant subspace problem	Functional analysis	One of analysis’s best-known open problems asks whether bounded operators on a separable Hilbert space must preserve a nontrivial closed subspace, after shaping operator theory for decades.
75	Existence of one-way functions	Cryptography and complexity	This is the foundational yes-or-no question beneath essentially all modern computational cryptography and connects average-case hardness, pseudorandomness, and complexity separations.
75	Hilbert’s tenth problem over the rationals	Number theory and logic	The surviving rational version of Hilbert’s decision problem sits at the intersection of Diophantine geometry, computability, and undecidability.
74	Bombieri–Lang conjecture	Arithmetic geometry	Often described as one of arithmetic geometry’s biggest open problems, it extends Faltings’s finiteness picture from curves to varieties of general type and has broad uniformity consequences.
74	Resolution of singularities in positive characteristic	Algebraic geometry	Resolution is basic infrastructure throughout geometry and arithmetic, while characteristic-positive dimension at least four has resisted every general replacement for Hironaka’s theorem.
74	Inverse Galois problem over the rationals	Algebra and number theory	The problem asks whether every finite group occurs arithmetically over Q and has driven major work linking group theory, algebraic geometry, and number theory.
73	Vojta’s conjecture	Diophantine geometry	A master height inequality unifying Nevanlinna theory with Diophantine approximation and implying abc, Mordell-type, and many finiteness statements.
73	Beilinson conjectures on special values of L-functions	Arithmetic geometry	These conjectures organize regulators, motivic cohomology, and special L-values and encompass much of the expected arithmetic of motives.
73	Generalized Ramanujan conjecture	Automorphic forms	A central Langlands prediction controlling local components and Fourier coefficients, with consequences across number theory, representation theory, expander graphs, and spectral theory.
72	Existence of odd perfect numbers	Number theory	One of antiquity’s most famous surviving questions has accumulated a vast theory of necessary conditions and remains visible well outside number theory.
72	Artin holomorphy conjecture for L-functions	Number theory	A foundational reciprocity prediction for nontrivial irreducible Galois representations whose cases and consequences are woven through the Langlands programme.
72	BPP equals P and full derandomization	Complexity theory	Whether randomness adds polynomial-time power is a central complexity question linked by hardness-versus-randomness theorems to major circuit lower bounds.
72	P versus PSPACE	Complexity theory	The foundational time-versus-space separation asks whether every polynomial-space computation can be performed in polynomial time and has resisted even indirect progress for half a century.
70	VP versus VNP	Algebraic complexity	Valiant’s algebraic analogue of P versus NP anchors arithmetic-circuit complexity and the geometric-complexity-theory programme.
70	Bateman–Horn conjecture	Number theory	The quantitative master law for prime values of polynomial systems unifies Bunyakovsky, prime tuples, and many prime-producing heuristics.
70	Lindelöf hypothesis	Analytic number theory	The canonical growth conjecture on the critical line is implied by RH and controls zeta moments, subconvexity, and error terms throughout analytic number theory.
70	Homological mirror symmetry	Geometry and mathematical physics	Kontsevich’s programme predicts a categorical equivalence linking symplectic and algebraic geometry and has reorganized both fields together with string-theoretic mathematics.
69	BQP versus BPP	Quantum complexity	Whether quantum computation has superpolynomial power over probabilistic classical computation is the foundational asymptotic question behind quantum advantage and a large complexity, algorithms, and physics programme.
68	Novikov conjecture	Topology, geometry, and operator algebras	Homotopy invariance of higher signatures is a central organizing problem with deep links to assembly maps, group actions, K-theory, and manifold classification.
68	Finiteness of Tate–Shafarevich groups	Arithmetic geometry	The missing finiteness statement is indispensable to the full BSD formula and to the arithmetic classification of rational points on abelian varieties.
68	Matrix-multiplication exponent ω equals 2	Algorithms and algebraic complexity	The exponent is a universal algorithmic constant affecting linear algebra, graph algorithms, optimization, and complexity, with a half-century record race.
68	Classical polynomial-time integer factorization	Algorithms, number theory, and cryptography	One of computation’s most famous concrete open problems underlies deployed public-key cryptography and is the emblematic exponential quantum speedup supplied by Shor’s algorithm.
68	Manin’s conjecture on rational points of bounded height	Arithmetic geometry and analytic number theory	A monograph-scale programme predicts the quantitative distribution of rational points on Fano varieties and joins Diophantine geometry, harmonic analysis, and birational geometry.
67	Abundance conjecture	Algebraic geometry	A principal remaining pillar of the minimal model programme, it would turn nef canonical divisors into morphisms and complete the expected birational classification in all dimensions.
66	Baum–Connes conjecture	Topology and operator algebras	A central assembly-map conjecture connecting group geometry, K-theory, index theory, and C*-algebras, with Novikov-type consequences and decades of machinery built around it.
66	Montgomery pair-correlation conjecture	Analytic number theory and random matrices	The conjecture launched the zeta-zero and random-matrix correspondence and remains a standard structural target beyond merely locating the zeros.
66	Farrell–Jones conjecture	Topology, geometric group theory, and algebraic K-theory	A major assembly-conjecture programme with broad Borel- and Novikov-type consequences and applications across topology, group theory, algebra, and operator algebras.
66	Nonexistence of Landau–Siegel zeros	Analytic number theory	Eliminating exceptional real zeros would make major estimates effective and directly controls class numbers, primes in arithmetic progressions, and central nonvanishing across the theory of L-functions.
66	The polynomial hierarchy does not collapse	Complexity theory	The hierarchy is a basic structural model of alternating quantifiers and oracle computation, and proving it infinite would separate P from NP while organizing a large body of conditional collapse theorems.
65	Littlewood conjecture	Diophantine approximation and dynamics	A flagship problem connecting simultaneous approximation, homogeneous dynamics, measure rigidity, and exceptional-set theory.
65	Fontaine–Mazur conjecture	Arithmetic geometry	A major organizing principle says which p-adic Galois representations come from geometry or automorphic forms, bridging deformation theory and Langlands reciprocity.
65	Rigorous three-dimensional Ising critical behaviour	Probability and statistical mechanics	The archetypal model of phase transitions remains without a rigorous derivation of its three-dimensional critical exponents, despite shaping renormalization and universality across physics and mathematics.
65	Strongly polynomial algorithm for linear programming	Optimization and algorithms	Smale placed it on his next-century list, and it asks whether the foundational tractable optimization problem admits an algorithm independent of numerical bit length.
65	Simplicity of all nontrivial zeta zeros	Analytic number theory	Simplicity is a standard independent target in the RH ecosystem, tied to zeta derivatives, moments, and random-matrix predictions; it is distinct from putting every zero on the line.
65	Deterministic polynomial-time polynomial identity testing	Algebra and complexity	Full derandomization of PIT is a central algorithmic question whose hardness-versus-randomness equivalences would force major arithmetic or Boolean circuit lower bounds.
64	Artin’s primitive-root conjecture	Number theory	A canonical prime-distribution conjecture with a famous conditional solution under GRH and deep links to Galois and sieve methods.
64	Grothendieck period conjecture	Algebraic geometry and transcendence	It predicts that all algebraic relations among periods arise geometrically, joining motives, algebraic cycles, and transcendence in one master statement.
64	Full nonlinear stability of the subextremal Kerr family	General relativity and PDE	The complete rotating-black-hole stability problem is a central test of Einstein evolution and black-hole physics beyond the major slowly rotating and restricted results.
64	General Sato–Tate conjecture	Arithmetic geometry	The original elliptic-curve case is settled, while the general automorphic equidistribution programme remains a major Langlands target governing Frobenius statistics.
64	Infinitely many primes of the form n²+1	Number theory	One of Landau’s four “unattackable” prime problems, and the emblematic unsolved case of prime values of a nonlinear polynomial.
64	Mumford–Tate conjecture	Arithmetic geometry	This central bridge among Hodge, Tate, and Galois theory predicts that the arithmetic monodromy of an abelian variety or motive is exactly the group visible from its Hodge structure.
64	Zilber–Pink conjecture	Diophantine geometry and model theory	The master unlikely-intersections conjecture unifies the Manin–Mumford, Mordell–Lang, and André–Oort worlds and has built a broad programme using heights, o-minimality, Hodge theory, and Shimura varieties.
64	Schinzel’s hypothesis H	Number theory and Diophantine geometry	This famous qualitative master conjecture for simultaneous prime values contains Bunyakovsky and prime-tuples cases and is widely used conditionally in arithmetic geometry.
63	Cramér’s conjecture on prime gaps	Number theory	The standard quantitative model for extreme prime gaps connects probability heuristics, zeta estimates, and one of the public-facing frontiers of prime distribution.
63	Critical exponents and scaling limits of self-avoiding walk in dimensions two and three	Probability, combinatorics, and statistical mechanics	A Simon-list problem with books and a cross-disciplinary renormalization programme, it is the canonical mathematical model of long polymer chains.
63	Absence of an infinite cluster at criticality in three-dimensional percolation	Probability and statistical mechanics	The most basic unresolved phase-transition question for the canonical percolation model sits inside the larger programme on critical exponents, universality, and renormalization in three dimensions.
63	Classical polynomial-time discrete logarithms in finite fields and elliptic-curve groups	Algorithms, number theory, and cryptography	The assumed classical hardness of discrete logarithms in the standard cryptographic groups supports Diffie–Hellman and elliptic-curve cryptography, while quantum algorithms solve these cases in polynomial time.
63	Permanent versus determinant and unbounded determinantal complexity	Algebraic complexity and geometry	Valiant’s flagship concrete VP-versus-VNP problem anchors geometric complexity theory and imports algebraic geometry and representation theory into lower bounds.
63	Asymptotic sphere-packing density in high dimensions	Discrete geometry, coding, and information theory	The exponential gap between the best general upper and lower bounds is a foundational geometry-of-numbers and coding problem whose special dimensions have produced landmark mathematics.
63	Strominger–Yau–Zaslow conjecture	Geometry and mathematical physics	The geometric mechanism proposed to explain mirror symmetry has driven major work on special Lagrangian fibrations, Calabi–Yau geometry, and string-inspired mathematics.
62	Hadwiger’s conjecture	Graph theory	Often treated as the deepest generalization of the four-colour theorem, it organizes graph-minor structure and colouring across modern graph theory.
62	Quantum PCP conjecture	Quantum information and complexity	A defining open problem of Hamiltonian complexity asks whether robust approximation of ground-state energy remains QMA-hard and connects computation to many-body entanglement.
62	Global regularity for three-dimensional magnetohydrodynamics	PDE and mathematical physics	The coupled fluid–magnetic analogue of Navier–Stokes is a central nonlinear PDE problem with direct physical importance and a large regularity programme.
62	Hadwiger–Nelson problem	Discrete geometry and graph theory	The chromatic number of the plane combines a universally intelligible statement and exceptional public fame with deep unit-distance graph theory; the 2018 breakthrough left 5, 6, or 7 open.
62	Asymptotic growth of diagonal Ramsey numbers	Combinatorics and theoretical CS	Determining the exponential scale and constants of R(k,k) is one of probabilistic combinatorics’ oldest central targets, with deep links to pseudorandomness and information theory.
62	Grothendieck section conjecture	Arithmetic geometry	Grothendieck’s central anabelian prediction seeks to recover rational points from étale fundamental groups and has generated a distinctive crossroad of geometry and Galois theory.
62	Unique Games conjecture	Complexity and approximation algorithms	A central hardness conjecture would determine optimal approximation thresholds for a broad range of problems and has generated extensive algorithmic and analytic machinery.
62	Shub–Smale τ-conjecture	Algebraic complexity and number theory	Smale’s fourth problem bounds integer roots by straight-line complexity and was designed as an arithmetic route to an algebraic analogue of P≠NP.
62	11⁄8 conjecture for smooth spin four-manifolds	Four-manifold topology	Princeton’s account calls it the last uncharted territory in the fundamental geography problem of which simply connected topological four-manifolds admit smooth structures.
61	Kaplansky zero-divisor conjecture	Algebra and topology	The question for torsion-free group rings links group theory, topology, operator algebras, and the cluster of conjectures around soficity and assembly maps.
61	Density of hyperbolicity for complex polynomials	Complex dynamics	Smale’s eleventh problem remains open even for the quadratic family and is a central rigidity and parameter-space target intertwined with the Mandelbrot-set programme.
60	Stark conjectures	Algebraic number theory	This broad special-value programme generalizes the analytic class-number formula and guides explicit class field theory, units, and equivariant L-value conjectures.
60	Leopoldt’s conjecture	Algebraic number theory	The nonvanishing of the p-adic regulator is basic expected infrastructure for Iwasawa theory and the arithmetic of number fields.
60	Global smooth solutions of the Boltzmann equation for general large data	Kinetic theory and PDE	A foundational equation of statistical mechanics still lacks a general global smooth theory away from perturbative regimes, linking PDE, probability, and thermodynamics.
60	Selberg’s 1⁄4 conjecture	Spectral and automorphic theory	A central spectral-gap prediction for congruence surfaces, equivalent to a key generalized-Ramanujan case and influential in expanders and homogeneous dynamics.
60	Chowla conjecture on correlations of the Möbius or Liouville function	Number theory and dynamics	A master randomness conjecture for multiplicative functions underlies much modern work on Möbius orthogonality, prime patterns, and ergodic methods.
60	Smooth four-dimensional Schoenflies conjecture	Four-manifold topology	A close structural cousin of smooth four-dimensional Poincaré asks whether every smooth 3-sphere in the 4-sphere is standard, squarely inside the exceptional dimension-four problem complex.
60	Hilbert–Smith conjecture	Topology and transformation groups	The last major topological extension of Hilbert’s fifth problem asks whether manifolds admit faithful actions only by Lie-like locally compact groups.
60	Weinstein conjecture in higher dimensions	Contact geometry and dynamics	One of the most famous periodic-orbit questions in symplectic and contact topology asks whether every Reeb flow on a closed contact manifold has a closed orbit, with the general higher-dimensional case still open.
60	Volume conjecture for knots	Quantum topology and geometry	This flagship quantum-topology conjecture predicts that colored Jones asymptotics recover hyperbolic volume, linking knot invariants, geometry, and quantum field theory.
60	Legendre’s conjecture	Number theory	A Landau problem with an elementary statement and century-long prime-gap significance, though it has fewer downstream dependencies than n²+1 or twin primes.
60	Bunyakovsky conjecture	Number theory	The qualitative one-polynomial predecessor to Schinzel and Bateman–Horn includes Landau’s n²+1 problem and is a standard statement of what prime-producing polynomials should do.
60	Infinitely many Mersenne primes and even perfect numbers	Number theory	A centuries-old problem with exceptional public visibility, large distributed searches, and the complete Euclid–Euler classification tying it to even perfect numbers.
60	Graph isomorphism in polynomial time	Algorithms and complexity	Babai’s quasipolynomial breakthrough left one of algorithms’ most famous intermediate-status problems, connecting groups, combinatorics, logic, and complexity classification.
60	Shannon zero-error capacity of graphs	Information theory and graph theory	Introduced by Shannon as foundational communication theory, the capacity remains unknown even for simple graph families and supports a substantial bridge between information theory, combinatorics, and quantum methods.
60	Grothendieck–Katz p-curvature conjecture	Arithmetic geometry and differential equations	A deep local-to-global conjecture characterizing algebraic differential equations through reductions modulo primes and linking monodromy, Galois theory, and arithmetic geometry.
60	Atiyah conjecture on L²-Betti numbers	Topology, groups, and operator algebras	Originating in Atiyah’s L²-index theory, it connects topology of coverings, group rings, von Neumann algebras, and zero-divisor-type conjectures.
60	Mandelbrot set local connectivity conjecture	Complex dynamics	The MLC conjecture combines exceptional fame beyond its field with a decades-long programme controlling parameter-space rigidity, density of hyperbolicity, and the topology of quadratic dynamics.
60	Anderson delocalization in three dimensions	Mathematical physics and spectral theory	Simon’s problem list singled out the existence and structure of extended states in the 3D Anderson model, a foundational unresolved phase of localization theory.
60	Crystallization conjecture	Mathematical physics and discrete geometry	The emergence of periodic crystalline ground states from broad particle interactions is a famous long-running bridge problem in physics, analysis, and geometry and appears on Simon’s list.
60	Finiteness of central configurations in the Newtonian n-body problem	Dynamical systems and celestial mechanics	Smale’s sixth problem asks for finiteness of relative equilibria for arbitrary positive masses and links a classical physics question to algebraic geometry and global n-body dynamics.
60	Brauer–Manin sufficiency for rationally connected varieties	Arithmetic geometry	The conjecture that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation for rationally connected varieties organizes the modern local-to-global study of rational points.
60	Global classical regularity for the three-dimensional relativistic Vlasov–Maxwell system	Kinetic PDE and plasma physics	This longstanding large-data problem governs a foundational collisionless-plasma model and has generated a major continuation-criteria and weak-solution programme across kinetic theory and hyperbolic PDE.

Sol’s list of famous unsolved problems that just missed the 60-rating cut

Provisional score	Problem	Why it misses 60
58	Kannan–Lovász–Simonovits conjecture	The prompt itself places KLS in the 50–60 textbook/​subfield band; it is central and consequential but does not clearly beat the sofic anchor.
58	Cannon conjecture	A major geometric-group-theory programme with topology at its boundary, but mostly invisible outside that research complex.
58	Telescope conjecture	One of stable homotopy theory’s great organizing problems, yet its audience and dependencies remain concentrated within a specialty.
58	Borel conjecture	A major topological-rigidity statement, but much of its modern significance is mediated through the broader Farrell–Jones programme.
58	Hopf Euler-characteristic conjecture	A classic global-geometry conjecture with a substantial literature, but largely concentrated in one mature specialty.
58	Gauss’s real-quadratic class-number-one problem	A famous classical arithmetic question, but its present-day dependency footprint is narrower than the problems admitted at 60.
58	Restriction conjecture	It organizes a major part of harmonic analysis and reaches PDE and geometry, but its recognition still fits the top of the textbook/​subfield band.
58	Elliott–Halberstam conjecture	It is powerful sieve-theoretic infrastructure and gained visibility through bounded prime gaps, but remains less independently prominent than the 60 anchor.
58	Falconer distance conjecture	A central geometric-measure problem in the Kakeya and restriction constellation, but primarily a one-research-complex flagship.
58	Graph reconstruction conjecture	Universally stated and field-wide in graph theory, yet its consequence network and cross-field reach fall short of soficity.
58	Sarnak’s Möbius disjointness conjecture	A major bridge between dynamics and multiplicative number theory, but younger and less institutionally entrenched than the 60 anchor.
57	Andrews–Curtis conjecture	Famous and longstanding across combinatorial group theory and low-dimensional topology, but less structurally depended upon than soficity.
57	Mahler volume conjecture	A central convex-geometry extremal problem with broad techniques, but it fits the upper end of the textbook/​subfield band.
57	KLS-adjacent slicing or hyperplane conjecture	The convex-geometry programme is large, though its wider fame and conditional superstructure do not clear the threshold confidently.
57	Hadamard matrix conjecture	It connects designs, coding, and statistics, but is still best calibrated as a famous combinatorics flagship rather than a cross-mathematical dependency hub.
57	Lehmer’s conjecture on Mahler measure	It has genuine links to dynamics, knots, and Diophantine geometry, yet most of its sustained attention comes from a specialist complex.
57	Fermat–Catalan conjecture	Its connection to Fermat and abc gives it unusual visibility, but comparatively little mathematics is organized around this exact statement.
56	Frankl union-closed sets conjecture	Extremely well known in combinatorics and easy to state, but it remains structurally isolated and below cycle-double-cover-to-sofic territory.
56	Rota basis conjecture	A celebrated matroid and linear-algebra problem with cross-links, but still primarily a field-famous workhorse.
56	Four exponentials conjecture	A major intermediate statement in transcendence theory, but it is largely known and used inside one specialty and does not beat the 60 anchor.
55	Cycle double cover conjecture	Fixed VibeMathed anchor.
55	Caccetta–Häggkvist conjecture	A central digraph extremal problem with decades of work, but a classic example of the 50s rather than something beyond a subfield.
54	Lonely runner conjecture	Broadly known and attractive, yet neither a major dependency hub nor a premier-list problem.
52	Perfect cuboid problem	Exceptional recreational fame and age, but little machinery depends on its answer.
50	Moving sofa problem	Famous and technically rich, though more isolated and less mathematically consequential than the 60 threshold requires.

Sources referenced (I supplied most of these)

Source	Role
VibeMathed v2 scoring prompt	Fixed scale, anchors, and scoring procedure.
VibeMathed public dataset	Current catalog and the existing 68, 65, and 60 entries.
Clay Millennium Prize Problems	Authoritative status and premier-list signal for the six open Millennium problems.
Smale, Mathematical Problems for the Next Century	Premier cross-field list and direct source for the Jacobian and strongly-polynomial-LP comparisons.
Hilbert, Mathematical Problems	Original Hilbert-list signal.
Landau’s four prime problems	Recognized list signal for Goldbach, twins, Legendre, and n²+1.
Barry Simon, Fifteen Problems in Mathematical Physics	Recognized-list evidence for QFT and the statistical-mechanics candidates.
Arthur, The Principle of Functoriality	Primary expository evidence for the standing and reach of Langlands functoriality.
Richard Taylor, Modular Arithmetic	IAS exposition identifying global Langlands reciprocity as the ultimate reciprocity law and separating it from functoriality.
Journal of Open Mathematical Problems	Institutionally curated surveys of famous open problems, especially generalized Hodge and its place in the algebraic-cycle programme.
AMS monographs on algebraic cycles and rational points	Book-scale evidence for Bloch–Beilinson, Manin, Brauer–Manin, and related arithmetic-geometry programmes.
AIM problem lists	Community-maintained field lists, especially Sarnak, Tate, prime gaps, and arithmetic geometry.
Wikipedia’s composite list	Broad discovery index and first-pass status screen, followed by stronger sources for included problems.

Reflections on the exercise

My overall view of the prompt

The significance prompt is unusually good for a task that is intrinsically judgmental. Its best decision is to define significance as a historical and sociological counterfactual: how much mathematics cared about the problem before its resolution. That excludes several powerful distortions, including the glamour of a surprising proof, the identity of a solver, and publicity created by the solution itself. The instruction to score the problem rather than the solution is essential for a catalog about AI involvement, where fascination with the agent could otherwise leak directly into the score.

The comparative spine is another strong design choice. An unaided request for a score from 0 to 100 invites each evaluation to invent its own scale. “Is this above Jacobian 65 and below RH 100?” is much more stable than “how important does this feel?” The direction to allow ties, take the lower value when torn, and justify the strongest signal also counteracts score inflation. The prompt even acknowledges that one-point precision has meaning only as a comparison among neighbors. That is honest and much better than pretending that 73 and 74 are natural scientific quantities.

The main weakness is that the fixed spine becomes sparse exactly where this census operates. There are detailed anchors from 5 through 65, then no fixed catalog anchor between Jacobian 65 and RH 100. The prose supplies approximate landmarks such as Collatz around 80 and Goldbach, twin primes, and Navier–Stokes around 85 to 90, but the hardest distinctions in this table are precisely the distinctions among 70, 80, and 90. Fixed high-end anchors for BSD, Hodge, P versus NP, Collatz, and Goldbach would improve inter-rater consistency more than another dozen low-end anchors.

There is also a mild ambiguity at the threshold. The prompt says that conjectures with textbooks and subfields organized around them occupy roughly 50 to 60. Taken literally, 60 is the very top of that band; taken in the spirit of the anchors, 60 is already sofic-level standing and should require an extra signal. I used the stricter interpretation. This explains why KLS remains at 58 here even though it is explicitly named in the 50 to 60 description.

What a single score compresses

“Significance” combines several properties that do not naturally move together. At least six axes are hiding inside each integer:

Axis	What it tries to capture	Characteristic distortion
Institutional recognition	Millennium, Hilbert, Smale, Simon, Yau, Erdős, and other recognized lists	Committee choices and inherited prestige become self-reinforcing.
Dependency centrality	Conditional theorems, standard hypotheses, and machinery waiting on the answer	Informal or difficult-to-index dependencies are undercounted.
Cross-field breadth	The number and distance of communities that use or attack the problem	Large fields and fashionable interfaces receive an advantage.
Cultural fame	Recognition outside the originating specialty or outside mathematics	Elementary statements such as Collatz gain heavily from accessibility.
Program maturity	Decades of partial results, monographs, workshops, and named techniques	Age and institutional resources can masquerade as intrinsic importance.
Statement identity	Whether there is a stable, precise target with its own literature	Grand programs are either penalized for breadth or inconsistently counted as one problem.

The prompt deliberately aggregates these signals because a public ranking needs a single sortable field. The resulting order is useful as triage. It should not be mistaken for a total ordering of intellectual value. The contrast between Collatz at 80 and the standard conjectures on algebraic cycles at 77 makes this plain: Collatz dominates in fame and accessibility, while the standard conjectures dominate in structural integration with a large body of mathematics.

There is a close analogy with responsible research assessment. The Leiden Manifesto recommends that metrics support expert judgment, account for variation among fields, avoid false precision, and be scrutinized as their use changes behavior. A problem-significance score faces the same hazards on a smaller scale. If the score becomes a target for what mathematicians, funders, or AI labs should work on, it begins to shape the attention it was meant to describe. That is the Goodhart pressure the threshold audit is trying to resist.

For a richer database, I would keep the integer and add three fields: a plausible score interval, a confidence level, and an overlap family. Generalized Hodge might then read “79, plausible 77 to 80, medium confidence, Hodge and motives family.” The integer preserves sorting; the interval exposes judgment; the family prevents a user from treating RH, GRH, and special L-function variants as independent units.

Significance and difficulty are different coordinates

There is no reliable monotone relationship between significance and difficulty. A problem can be significant because it is a dependency hub, culturally famous, or institutionally canonized. None of those properties determines how far the shortest proof or counterexample lies from current mathematics. Conversely, an extraordinarily difficult specialist problem may remain at 35 because only one community knows it and few external theorems depend on it.

Difficulty is especially treacherous for an unsolved problem because the decisive missing idea is invisible by definition. Long resistance is evidence, but it mixes several causes: genuine conceptual depth, a poorly chosen formulation, weak feedback from failed attempts, a tiny expert community, or the absence of adjacent technology. A problem can look impossible until a new bridge makes it routine. It can also look approachable for decades because many partial results imitate progress without touching the obstruction.

A useful difficulty profile would itself be multidimensional. Relevant components include known barrier theorems, the amount of conceptual novelty apparently required, the size and continuity of the search space, the availability of decisive computational experiments, the length and modularity of plausible proofs, the cost of verification, and sensitivity to small changes in formulation. Complexity lower bounds have explicit barriers such as relativization, natural proofs, and algebrization. Diophantine conjectures may permit massive computation without creating a path to a proof. Geometric PDE can offer numerical evidence while leaving singularity formation analytically inaccessible. These are different kinds of hard.

AI tractability is a third coordinate. It depends on formalizability, dense intermediate feedback, searchable examples, reusable libraries of lemmas, and the possibility of independently checking candidate steps. A 55-point finite or algebraic problem with excellent verification loops might fall before an 85-point grand program, even if human experts consider the former technically difficult. High significance can help AI because it produces a rich literature and many partial results. It can hurt because generations of experts have already removed the obvious approaches. A sensible AI watchlist would therefore combine significance, estimated tractability, verification quality, and spillover value from partial progress.

Candid audit of this table

The top of the list is more stable than the boundary. I found no credible omitted 90-plus problem in the second sweep. Global Langlands reciprocity was the one serious omitted 80-plus candidate: the IAS describes it as Langlands’s “ultimate reciprocity law,” applying far beyond the original modularity setting, and emphasizes that much remains open. Its score of 85, tied with functoriality, is defensible because reciprocity and functoriality are distinct central pillars with comparable reach. Generalized Hodge at 79 is the next-highest addition; Claire Voisin’s 2025 survey explicitly presents it as Hodge’s companion involving Hodge structures, algebraic cycles, and motives.

Several new scores could reasonably move by two or three points. Generalized Hodge might be 77 or 78 if independent fame is weighted more heavily than logical strength. P versus PSPACE at 72 could be 69 or 70 because it is foundational but attracts less broad attention than P versus NP or NP versus coNP. BQP versus BPP at 69 and classical factoring at 68 depend on treating theoretical computer science, quantum information, and cryptography as full participants in mathematical significance. I think that inclusion is right, though a pure-mathematics editorial culture might lower both. Manin’s conjecture at 68 has a monograph-scale program and wide arithmetic-geometric reach, yet 65 to 68 is a plausible band.

The exact-60 additions are the most fragile. Weinstein in higher dimensions, finiteness of central configurations, Brauer–Manin sufficiency, and large-data Vlasov–Maxwell regularity all have a real case: famous or premier-list status plus a substantial cross-field program. Each could also be placed at 58 or 59 by an editor who requires greater fame outside its research complex. Their inclusion says that sofic groups at 60 is a genuine comparator, not merely a verbal threshold.

I also mixed units of analysis. Some rows are crisp propositions, such as Legendre’s conjecture. Others are families or programs, such as global Langlands reciprocity, Stark conjectures, cosmic censorship, Bloch–Kato, and rigorous four-dimensional QFT. GRH and grand RH are combined in one row even though they can be separated in several ways. This inconsistency is partly inherited from mathematical practice: communities really do name and attack both individual statements and conjectural packages. It nevertheless creates score and row-count ambiguity. A stricter database should distinguish “atomic problem,” “named family,” and “research program,” then let users filter them.

Overlapping rows are another source of inflation. Hardy–Littlewood contains the twin-prime prediction; Bateman–Horn contains Bunyakovsky; full BSD contains Tate–Shafarevich finiteness; Langlands contains Artin, Ramanujan, and Sato–Tate cases. Keeping the rows is appropriate for a watchlist because each target has an independent mathematical life. Counting 112 rows as 112 unrelated mysteries would be wrong. The overlap-family field proposed above would make that limitation machine-readable.

There was one wording error worth correcting during this audit. “Polynomial-time discrete logarithms” was too broad because discrete-log problems depend on the family of groups. The row now specifies finite fields and elliptic-curve groups, the standard cryptographic cases. Status-sensitive additions survived a current check: a 2025 survey still treats the general Weinstein conjecture as active; a 2025 monograph still centers the ¹¹⁄₈ conjecture; recent work continues to prove Brauer–Manin sufficiency only for significant classes rather than all rationally connected varieties; and the basic critical-percolation questions in intermediate dimensions remain open in modern surveys.

The prompt’s examples are not chiefly responsible for these uncertainties. They expose the intended value judgments quite clearly. Collatz around 80 announces that cultural fame can outrank structural dependence. Jacobian at 65 announces that a Smale-list problem central to a major field is still far below Millennium territory. Sofic groups at 60 announces that cross-links among group theory, dynamics, and operator algebras can clear the threshold without public fame. The remaining variance comes from compressing legitimate plural values into one dimension and from the sparse high-end spine.

What I find most interesting

The list is partly a map of mathematics and partly a map of mathematical memory. Named conjectures, English-language surveys, famous problem lists, and fields with strong expository traditions are easier to discover and easier to justify. Logic, foundations, parts of category theory, non-Western problem traditions, and important questions whose statements resist a memorable name are likely underrepresented. List membership is a meaningful signal, but it is also a record of who had the standing to make lists and which communities preserved them.

The exercise also reveals that “problem identity” is an editorial act. Global Langlands reciprocity can be treated as one problem, hundreds of local and global cases, or a guiding program. Smooth four-dimensional topology can be represented by Poincaré, Schoenflies, ¹¹⁄₈, and many Kirby-list questions. The chosen granularity changes both the count and the apparent distribution of significance. Avoiding Goodharting therefore requires restraint at the level of ontology, before scoring begins.

For thinking about AI successors, the most valuable output may be the ecosystem structure rather than the rank. A model that contributes a reusable lemma to a 75-point ecosystem may create more value than one that settles an isolated 60-point statement. Partial progress can improve formal libraries, generate counterexamples, connect fields, or eliminate broad strategy classes. A future catalog might track these spillovers alongside solved and unsolved status.

The adjacent questions are rich: Should an AI lab allocate effort by significance, expected probability of progress, verification cost, or social value? How should negative results and barrier theorems be valued? Can conditional-theorem graphs estimate dependency centrality without reproducing citation bias? Should a problem’s significance be time-indexed as fields rise and fall? How should independently undecidable statements fit beside ordinary open conjectures? Can expert pairwise comparisons produce a stable partial order, or will context effects create cycles? What level of problem decomposition best predicts AI progress? These questions do not need a single ranking, but this ranking is a useful way to make them visible.

I then asked it to look over the list it compiled and ballpark how far it currently seems from being autonomously solvable as of today (14th Aug 2026):

Sol’s examples-derived tiering of estimated difficulty of problems above

The reference system is a best-available model with web and literature access, code, computer algebra, formal-proof tools, persistent memory, a high inference budget, parallel explorers, and adversarial referee agents. It receives a precise statement and may decide whether to prove or refute it. After kickoff it receives no substantive mathematical steering from a human. “Solved” means a publishable resolution of the exact row in the census that survives expert review. Progress on a nearby variant, an improved bound, or one implication does not count as solving the row.

These are ordinal capability-distance tiers, not estimates of years, compute, or success probability. A problem’s unknown truth value matters. A false universal conjecture may have a small counterexample even when every proof route would be forbidding. Each placement therefore averages over two very different worlds and should be read with a band of at least one neighboring tier.

The significance score follows each name so that the size of the reordering is visible. Within a tier the problems are unranked.

Difficulty tier, high to low	Count	Problems, with significance score	Commentary on the tier
D8: beyond the current map	5	Riemann hypothesis, 100 Generalized and grand Riemann hypotheses, 97 P versus NP, 97 Navier–Stokes existence and smoothness, 90 Yang–Mills existence and mass gap, 87	The known programs do not expose a narrow closing lemma. Complexity has a barrier zoo, the zeta campaign repeatedly reached RH-equivalent positivity, and the PDE or QFT rows require global control at the scale where current estimates lose it. An enormous campaign could produce important byproducts while leaving the exact target untouched.
D7: a new architecture appears necessary	14	Birch and Swinnerton-Dyer conjecture, 89 Hodge conjecture, 88 Langlands functoriality, 85 Global Langlands reciprocity, 85 General Boolean circuit lower bounds, especially NP not in P/​poly, 79 Rigorous construction of a nontrivial four-dimensional quantum field theory or QCD, 78 NP versus coNP and superpolynomial proof lower bounds, 77 Grothendieck’s standard conjectures on algebraic cycles, 77 Bloch–Beilinson filtration and conjectures on Chow groups, 76 Existence of one-way functions, 75 P versus PSPACE, 72 VP versus VNP, 70 Grothendieck period conjecture, 64 Permanent versus determinant and unbounded determinantal complexity, 63	Each is a foundation-sized target whose solution would organize or create machinery rather than merely complete a mature local argument. Several complexity rows inherit known lower-bound barriers. The motives, cycles, Langlands, and QFT rows are broad enough that even specifying the right intermediate theorem is part of the discovery.
D6: grand, multi-bottleneck problems	18	Hilbert’s sixteenth problem, 84 Smooth four-dimensional Poincaré conjecture, 83 Cosmic censorship, 82 Three-dimensional Euler regularity and blow-up, 81 Tate conjecture, 80 Generalized Hodge conjecture, 79 Bloch–Kato and Tamagawa-number conjectures, 76 Schanuel’s conjecture, 76 Hilbert’s twelfth problem, 76 Hilbert’s tenth problem over the rationals, 75 Beilinson conjectures on special values of L-functions, 73 The polynomial hierarchy does not collapse, 66 Rigorous three-dimensional Ising critical behaviour, 65 Global regularity for three-dimensional magnetohydrodynamics, 62 Global smooth solutions of the Boltzmann equation for general large data, 60 Anderson delocalization in three dimensions, 60 Crystallization conjecture, 60 Global classical regularity for the three-dimensional relativistic Vlasov–Maxwell system, 60	These have plausible subprograms and many partial theorems, yet closing the full statement requires several interfaces to work simultaneously. Four-dimensional topology and real dynamics offer sparse formal feedback; PDE, statistical mechanics, and many-body problems couple local estimates to global or multiscale behavior; the arithmetic rows span several layers of conjectural infrastructure.
D5: several locked interfaces	22	Goldbach’s conjecture, 90 Twin prime conjecture, 89 Hardy–Littlewood prime-tuples conjecture, 86 Collatz conjecture, 80 abc conjecture, 79 Bombieri–Lang conjecture, 74 Resolution of singularities in positive characteristic, 74 Vojta’s conjecture, 73 Generalized Ramanujan conjecture, 73 Artin holomorphy conjecture for L-functions, 72 BPP equals P and full derandomization, 72 Homological mirror symmetry, 70 BQP versus BPP, 69 Novikov conjecture, 68 Finiteness of Tate–Shafarevich groups, 68 Abundance conjecture, 67 Baum–Connes conjecture, 66 Farrell–Jones conjecture, 66 Fontaine–Mazur conjecture, 65 Full nonlinear stability of the subextremal Kerr family, 64 Quantum PCP conjecture, 62 Mandelbrot set local connectivity conjecture, 60	A frontier campaign can attack meaningful modules here, and several rows have very mature technical ecosystems. Full resolution still requires more than one independent leap, such as parity plus distribution for prime patterns, local-to-global plus height control in Diophantine geometry, or structural and quantitative control together in topology, dynamics, or quantum complexity.
D4: one dominant wall inside rich machinery	28	Kakeya conjecture in dimensions at least four, 77 Invariant subspace problem, 75 Inverse Galois problem over the rationals, 74 Bateman–Horn conjecture, 70 Lindelöf hypothesis, 70 Matrix-multiplication exponent ω equals 2, 68 Classical polynomial-time integer factorization, 68 Manin’s conjecture on rational points of bounded height, 68 Nonexistence of Landau–Siegel zeros, 66 Littlewood conjecture, 65 Deterministic polynomial-time polynomial identity testing, 65 General Sato–Tate conjecture, 64 Infinitely many primes of the form n²+1, 64 Mumford–Tate conjecture, 64 Zilber–Pink conjecture, 64 Schinzel’s hypothesis H, 64 Critical exponents and scaling limits of self-avoiding walk in dimensions two and three, 63 Strominger–Yau–Zaslow conjecture, 63 Grothendieck section conjecture, 62 Shub–Smale τ-conjecture, 62 11⁄8 conjecture for smooth spin four-manifolds, 62 Stark conjectures, 60 Leopoldt’s conjecture, 60 Smooth four-dimensional Schoenflies conjecture, 60 Legendre’s conjecture, 60 Bunyakovsky conjecture, 60 Infinitely many Mersenne primes and even perfect numbers, 60 Brauer–Manin sufficiency for rationally connected varieties, 60	The live obstruction is comparatively recognizable and the literature offers many attack surfaces. That makes serious autonomous progress more plausible. The wall can still encode something enormous, such as the parity problem, a circuit lower bound, a missing height inequality, or a four-manifold classification step, so this tier remains well beyond the demonstrated base rate.
D3: plausible target for a very large successor campaign	21	Existence of odd perfect numbers, 72 Montgomery pair-correlation conjecture, 66 Strongly polynomial algorithm for linear programming, 65 Simplicity of all nontrivial zeta zeros, 65 Artin’s primitive-root conjecture, 64 Cramér’s conjecture on prime gaps, 63 Absence of an infinite cluster at criticality in three-dimensional percolation, 63 Classical polynomial-time discrete logarithms in finite fields and elliptic-curve groups, 63 Asymptotic sphere-packing density in high dimensions, 63 Hadwiger’s conjecture, 62 Asymptotic growth of diagonal Ramsey numbers, 62 Unique Games conjecture, 62 Density of hyperbolicity for complex polynomials, 61 Selberg’s 1⁄4 conjecture, 60 Chowla conjecture on correlations of the Möbius or Liouville function, 60 Hilbert–Smith conjecture, 60 Weinstein conjecture in higher dimensions, 60 Volume conjecture for knots, 60 Grothendieck–Katz p-curvature conjecture, 60 Atiyah conjecture on L²-Betti numbers, 60 Finiteness of central configurations in the Newtonian n-body problem, 60	These expose denser feedback through explicit algorithms, finite groups or graphs, low-dimensional dynamics, computable statistics, algebraic reductions, or sharply stated intermediate conjectures. They resemble domains where present systems have found real bridges. The Astra sphere-packing and Ramsey advances raise the value of campaigns on the corresponding full problems without making those full rows close. “Plausible target” means worth an Astra-scale campaign; it does not imply that current models are more likely than not to solve them.
D2: compact-certificate asymmetry	2	Kaplansky zero-divisor conjecture, 61 Shannon zero-error capacity of graphs, 60	Both are tool-rich and have a sharp status split. An explicit torsion-free group-ring counterexample or a finite graph-power construction can be compact and exactly checkable. Universal nonexistence or optimality statements may sit several tiers higher, so placement here is driven by the possibility of a short disproof or witness.
D1: nearest the demonstrated frontier	2	Hadwiger–Nelson problem, 62 Graph isomorphism in polynomial time, 60	Both invite explicit, cheaply testable artifacts: a finite unit-distance obstruction or coloring construction, and a concrete algorithm with a conventional correctness and complexity proof. Their remaining branches can still demand global arguments, so this is “nearest among this 60+ survivor set,” never “easy.”

Sources referenced to ground “autonomous solvability difficulty” (I supplied them all)

What the observed solve processes actually show

Evidence examined	Process detail that mattered for this table	Difficulty lesson
Gowers’s account of GPT-5.5 Pro	For the diameter-two problem the model recognized that Rajagopal’s powers-of-two choice could be replaced by a quadratic-diameter Sidon set. For general diameter it retained Rajagopal’s proof frame and introduced an h²-dissociated construction that compressed an exponential interval to a polynomial one. Gowers judged the result a reasonable PhD-chapter-sized contribution and stressed that target-driven combinatorics supports backward search especially well.	A mature frame, a crisp quantitative target, and replaceable discrete components can make a problem tractable even when the final idea is original. Forward exploration that must decide which observations deserve a theory remains a different and less tested task.
VibeMathed’s public dataset and repository	The 561 recorded entries comprise 338 arguments, 200 constructions, and 23 computations. Combinatorics and number theory account for 265 entries, while differential equations and mathematical physics account for 19. There are 346 unreviewed records, and the catalog has no denominator showing how often each field or problem type was attempted.	The catalog supports claims about successful representations, especially short reductions, explicit constructions, finite certificates, and formalizable arguments. It does not support a field-by-field success rate. Corpus exposure, the Erdős-problem source, and verification policy strongly affect the observed mix.
Claude’s zeta-zero research account, transcript, and paper	The campaign used 60 subagents and roughly 31 million output tokens after an earlier ledger of hundreds of ideas. Many routes reduced to Weil positivity in disguise. A proposed Hilbert-space route was found to be indefinite, described in the log as “Krein, not Hilbert.” The coordinator reused that failure, hostile referees caught a false mass-matrix premise, numerical tests rejected another prompt premise, and a von Neumann rank-trace inequality emerged during the push toward two thirds. The full RH attempt ended in a major 67.25% byproduct.	Massive search can map an obstruction and extract a strong neighboring theorem when the headline target remains out of reach. Cheap numerical falsification, exact finite-dimensional reformulation, an explicit do-not-repeat ledger, and independent refereeing materially improve autonomous search.
VibeMathed’s Jacobian entry and independent verification record	The result is an explicit three-variable polynomial map with constant nonzero Jacobian and a directly checkable collision. That makes verification tiny compared with the conjecture’s historical depth. I found no complete public prompt-and-search transcript, so the discovery path cannot be audited at zeta-campaign resolution.	Conjecture polarity can dominate age and prestige. Falsity plus a compact algebraic certificate creates an AI-shaped target; opacity about the search prevents stronger conclusions about autonomy or reproducibility.
OpenAI’s ten Astra results, papers, and reasoning walkthroughs	The successful routes repeatedly changed representation. The quantum parallel-repetition attempt found that the obvious tensor argument pointed in the wrong direction, then moved through an oscillatory gauge to resolvent purification. The Ehrhart proof spent a long time on harmonic symmetrization, which identified the simplex while losing the decisive factorial, then moved to a toric Monge–Ampère potential and jet counting. The Connes counterexample turned a factor-isomorphism question into explicit acting-group, torsion, and binary-carry data.	Current systems can bridge distant mature theories and repair substantial dead ends. Their strongest cases expose a finite or modular bottleneck after the right change of setting. The walkthroughs are retrospective reconstructions, and the total search and selection cost is not reported, so they calibrate proof style more reliably than base rates.
Aletheia, DeepMind’s research account, and OpenAI’s First Proof report	Aletheia autonomously solved four questions in a semi-autonomous sweep of 700 Erdős problems and uses generator, verifier, reviser, and restart loops with an explicit ability to admit failure. For First Proof, OpenAI reported at least five of ten attempts as likely correct, later withdrew one initially favored proof, and described human redirection of retries and selection among several attempts.	Self-critique and restart improve reliability, while specialist semantics and end-to-end verification remain real bottlenecks. Reported “autonomy” spans materially different protocols, so this table uses the stricter protocol defined above.

Reflections on the difficulty tiering

Discrete mathematics often supplies exact counterexamples, finite reductions, small parameter tests, and independently checkable lemmas. Analytic number theory can also become agent-friendly when explicit formulae reduce the live issue to matrices, traces, or optimized test functions. Nonlinear PDE, kinetic theory, and mathematical physics frequently demand global a priori estimates at a critical scale; numerical behavior can guide intuition without certifying the estimate that a proof needs. Arithmetic geometry has highly formal objects and a rich literature, yet its grand conjectures span a very deep conceptual stack and often require forward creation of machinery. Operator algebras looked similarly forbidding until Astra found a representation in explicit group and module data. Problem representation is a better predictor than the field label.

Operational difficulty profile

Axis	Features that lower AI-relative difficulty	Features that raise AI-relative difficulty
Search geometry	Finite objects, explicit constructions, algorithm design, a small counterexample, or a narrow quantitative optimization	A universal theorem over infinite-dimensional objects, a diffuse classification programme, or no stable candidate space
Feedback density	Cheap numerical tests, exact symbolic identities, SAT or SDP bounds, small cases, and formalizable intermediate lemmas	Failed attempts return little information, numerics sit far from proof, or correctness depends on a long analytic tail
Barrier structure	A known proof frame with one replaceable component or a precise local bottleneck	Relativization, natural-proofs or algebrization barriers, critical-scaling obstructions, or every route restating the conjecture
Conceptual distance	The missing bridge joins mature theories already present in the literature	A solution appears to require new language, new invariants, or a new global architecture for the subject
Verification	A compact certificate or modular proof that independent agents and proof assistants can audit	Delicate limits, genericity, regularity, equality cases, or specialist conventions that can hide a fatal gap
Autonomy burden	Backward reasoning from a crisp target and measurable subgoals	Long forward exploration whose central task is judging which observations are mathematically fruitful

The strongest lesson is that AI-relative difficulty attaches to a representation of a problem. “Operator algebras are hard for AI” would have predicted the Connes result badly. The successful construction became tractable after the target was recoded into groups, torsion, actions, and a binary carry. “Analytic number theory is amenable” would be equally misleading. The zeta campaign was highly productive once the partial-density problem became a finite quadratic form and a trace inequality, while the full RH repeatedly returned as its own positivity obstruction. A useful forecast asks what objects an agent can manipulate and what feedback those objects provide.

The Jacobian case makes difficulty status-dependent. If a conjecture is false and the counterexample fits on a page, discovery can look like constrained program synthesis followed by exact checking. If the same conjecture were true, a proof might require control of every polynomial map in every dimension. Hadwiger–Nelson, Kaplansky, Shannon capacity, and several algorithmic rows have a related split. A single tier hides a probability distribution over proof-world and counterexample-world. Future versions should record separate “proof if true” and “disproof if false” tiers.

The zeta transcript made me less impressed by raw branch count and more impressed by research-state design. The earlier attempts became valuable only after the coordinator classified them into known theorem, finite check, tautology, or RH-equivalent step and kept a do-not-repeat ledger. The decisive progress also came from agents violating the local brief productively: one repurposed an indefinite form, another used computation to reject the premise it had been given, and referees forced the argument to abandon a false mass matrix. An autonomous system needs memory of failed strategy classes, permission to contradict its task decomposition, and referees with genuinely different failure models.

Gowers’s example and Astra point to two complementary strengths. Gowers’s model worked inside a nearly complete human architecture and replaced the expensive combinatorial component. Astra sometimes searched much farther before a representation change exposed the right invariant. In both settings the agent could recognize success locally. That feature weakens in cosmic censorship, crystallization, grand Langlands statements, and parts of four-manifold topology, where a novel observation may not announce whether it is central, peripheral, or already implicit in a different formalism. Gowers’s distinction between backward, target-driven work and forward mathematical taste belongs directly in a difficulty model.

The tier counts are 5, 14, 18, 22, 28, 21, 2, and 2 from hardest to least hard. They initially broaden in the direction the user anticipated, then contract sharply near the current frontier. I do not read the contraction as evidence against a heavy-tailed distribution of mathematical difficulty. This census is conditioned twice: every row cleared an extreme significance threshold, and every row is still open after the first wave of AI-assisted results. VibeMathed already contains some of the unusually exposed high-significance cases, including Jacobian and sofic groups. The low-difficulty tail of an open-only table is therefore depleted by construction.

The evidence also argues against treating VibeMathed’s field counts as capability rankings. Its 148 combinatorics records and nine differential-equations records combine source selection, attempt volume, what users submit, what can be checked, and actual model skill. A controlled forecast would sample unsolved problems by field and significance, fix budgets and autonomy rules, publish every run, and have blinded experts grade progress as well as complete solutions. It would also distinguish compact disproofs, theorem completion inside a supplied frame, cross-field synthesis, and autonomous creation of a new program. Those are different capabilities.

Formal verification changes the search landscape without solving the discovery problem. Lean can make a candidate branch cheap to trust and allows aggressive decomposition, but many frontier gaps occur before a formal statement of the decisive lemma exists. First Proof’s withdrawn problem-two attempt is a useful warning: fluent end-to-end mathematics can survive internal confidence and still fail on a specialist point. Expert review, adversarial agents, numerical controls, and proof assistants cover different error surfaces. A reliable research system needs all four.

This table will age faster than the significance ranking. Better context windows, mathematical retrieval, theorem libraries, verifier models, experiment design, and persistent multi-agent memory can move a problem even when the base model changes little. One new human theorem can also expose a bottleneck that makes a whole family suddenly AI-shaped. The tiers are best treated as an August 2026 map of where feedback and obstruction appear to lie, with the source record kept alongside the judgments so later errors can be diagnosed rather than merely rescored.

Finally, I asked it to look at problems with very different estimated significance vs difficulty ratings:

Sol thinks significance ≫ estimated difficulty

Problem	Significance	Difficulty	Approximate rank gap	Why it separates
Odd perfect numbers	72	D3	+58 places	Enormous cultural longevity and an elementary statement; highly constrained arithmetic search space and cheap computation made it look relatively AI-shaped.
Kakeya in dimensions at least four	77	D4	+50	It is a major dependency hub across harmonic analysis, while recent progress and a mature incidence-geometric toolkit expose concrete intermediate targets.
Goldbach	90	D5	+44	Household-name significance is much greater than its structural centrality; the problem is still extremely hard, though more sharply represented than D8 foundational problems.
Invariant subspace problem	75	D4	+43	A crisp yes-or-no target with an immense operator-theory literature and possible constructive routes.
Twin primes	89	D5	+42	Cultural fame and its status as a canonical prime problem raise significance much more than they raise AI difficulty.
Montgomery pair correlation	66	D3	+42	Explicit formulae, matrices, optimization, and numerical zero data provide unusually dense feedback for a major analytic-number-theory conjecture.
Inverse Galois over the rationals	74	D4	+40	The universal problem is formidable, while individual group families invite explicit constructions and computational checking.
Hardy–Littlewood prime tuples	86	D5	+39	It inherits extraordinary prominence from the prime-pattern programme, despite having a fairly legible central obstruction in the sieve parity barrier.
Strongly polynomial linear programming	65	D3	+37	A proposed algorithm can be run, attacked on examples, and accompanied by conventional correctness and complexity proofs.
Simplicity of all nontrivial zeta zeros	65	D3	+37	It lives near RH culturally, while its quantitative statistics and finite-dimensional reformulations create more graded feedback than RH itself.

Hadwiger–Nelson at 62 and D1 deserves separate mention. Its rank residual is smaller because 62 is near the bottom of this already elite significance census. In absolute terms, it may be the clearest high-significance, low-AI-difficulty opportunity: either an explicit coloring or a finite unit-distance obstruction could settle it.

Graph isomorphism at 60 and D1 has the same basic shape. A candidate algorithm is executable, falsifiable on adversarial instances, and supported by an ordinary complexity proof. The quasipolynomial algorithm also provides an unusually advanced starting architecture.

Sol thinks estimated difficulty ≫ significance

Problem	Significance	Difficulty	Approximate rank gap	Why it separates

Anderson delocalization in three dimensions

60

D6

−73 places

The metallic phase lacks the monotone multiscale machinery available on the localization side; numerical transport evidence does little to certify a proof.

Crystallization

60

D6

−73

Local energy minimization and finite simulations must somehow become global statements over arbitrary configurations, defects, sizes, and thermodynamic limits.

Large-data Boltzmann global regularity

60

D6

−73

Infinite-dimensional nonlinear evolution, weak compactness, tails, collisions, and global a priori control produce little cheap intermediate feedback.

Relativistic Vlasov–Maxwell global regularity

60

D6

−73

Transport, electromagnetic fields, momentum growth, and continuation criteria interact globally; computation cannot certify the needed bounds.

Permanent versus determinant

63

D7

−64

The score reflects specialist visibility, while the theorem requires a major algebraic lower bound and runs into the deepest obstacles of geometric complexity theory.

Grothendieck period conjecture

64

D7

−56

It is a universal account of every algebraic relation among periods, with almost no effective feedback for deciding what new motivic or transcendence machinery is missing.

Three-dimensional MHD regularity

62

D6

−56

It combines the Navier–Stokes obstruction with magnetic coupling and additional nonlinear structure.

Mandelbrot local connectivity

60

D5

−53

Enormous computation produces pictures and parameter evidence, while the proof requires uniform control over infinitely renormalizable and other delicate regimes.

Quantum PCP

62

D5

−36

Local Hamiltonians look finite and combinatorial, yet entanglement prevents many classical PCP decompositions and makes intermediate claims difficult to verify.

Rigorous three-dimensional Ising critical behaviour

65

D6

−33

Simulations know the answer extremely well; converting renormalization and universality into a complete proof is a qualitatively different task.

Reflections

What the outliers reveal

1. I mixed two meanings of “difficulty”

There is a crucial distinction between:

1. Difficulty of proving the conjectured answer.

2. Difficulty of obtaining either a proof or a disproof.

These can differ by several tiers.

Global MHD regularity contains Navier–Stokes as the zero-magnetic-field special case. Therefore, a universal MHD regularity proof would also settle Navier–Stokes. Its D6 placement below Navier–Stokes at D8 is inconsistent if we mean “difficulty conditional on regularity being true.”

A singular MHD solution could refute regularity without settling Navier–Stokes. Under the “either answer wins” definition, lower placement remains possible.

The same issue appears throughout the table:

Broader problem	Contained or implied problem	Current tiers	Tension
Bateman–Horn	Twin primes and primes of the form n²+1	D4 versus D5	A proof of Bateman–Horn cannot plausibly be easier than proving its famous cases.
Schinzel’s hypothesis H	Bunyakovsky and many prime-pattern cases	D4 versus D4–D5	A compact counterexample could be easier; a proof should inherit every contained obstruction.
Polynomial hierarchy does not collapse	P versus NP	D6 versus D8	Proving noncollapse proves P≠NP.
NP not in P/​poly	P versus NP	D7 versus D8	The stated lower bound is strictly stronger than P≠NP.
Existence of one-way functions	P versus NP	D7 versus D8	Proving existence implies P≠NP, although proving nonexistence would not establish P=NP.

I handled truth-value asymmetry inconsistently. Jacobian encouraged me to reward problems that might have compact counterexamples, while some other rows were ranked as though their standard conjectural direction were true.

A better table needs two difficulty columns:

Difficulty estimate	Meaning
Proof-if-true difficulty	Distance to proving the standard conjectural answer
Disproof-if-false difficulty	Distance to a decisive counterexample or contrary theorem

An optional third field could combine them using an explicit prior over truth values. That combined estimate would expose the assumption currently hidden inside each tier.

2. Logical strength and significance are strongly non-monotone

A generalization may be logically harder and culturally less significant than its canonical special case.

Navier–Stokes is the iconic core problem. MHD contains that core plus magnetic coupling, yet its significance is only 62. Bateman–Horn contains many famous prime conjectures while receiving much less public attention than Goldbach or twin primes. General formulations often lose fame because their statements become less elementary, less memorable, and less historically canonical.

This is a genuine difference between the two axes. Significance measures what mathematics cared about. Difficulty inherits logical containment even when mathematical culture does not.

3. Elementary fame produces one characteristic positive gap

Goldbach, twin primes, odd perfect numbers, and Collatz receive large significance bonuses from accessibility, age, and fame outside their fields.

That does not make them easy for AI. It makes them easier relative to their cultural ranking. In fact, Collatz and odd perfect numbers may be hostile to current AI because vast computation supplies abundant facts with almost no indication of a closing argument.

Odd perfect numbers being the largest numerical outlier makes me distrust its D3 placement. Its many necessary conditions create dense activity, though they may be a deceptive form of feedback: the agent can always derive one more restriction without approaching nonexistence. I would now move it to D4.

This suggests distinguishing:

Feedback type	Example	Value
Directional feedback	A failed matrix inequality identifies the exact term that must be controlled	Often useful
Accumulative feedback	Another divisibility condition is added to a long list	May create little convergence
Falsifying feedback	A numerical experiment kills the premise of a proposed lemma	Extremely useful
Confirmatory computation	Another trillion cases satisfy Collatz	Psychologically impressive, mathematically weak

My original difficulty framework called too many of these simply “dense feedback.”

4. Graded objectives are unusually AI-friendly

Montgomery pair correlation and zeta-zero proportions have intermediate numerical targets. One can improve a constant, optimize a test function, bound a matrix, or discover that a proposed inequality tops out at a specific value.

RH is brittle. Either every nontrivial zero lies on the line or the theorem is absent.

This helps explain how the zeta campaign could fail completely at RH and still obtain a major theorem. The research landscape around a problem may be smooth even when the final endpoint is a cliff.

For attention allocation, this is important. Montgomery pair correlation at 66 may offer more expected mathematical value per campaign than many D1 or D2 problems because failed attempts can produce reusable quantitative progress.

5. Executable artifacts align discovery with verification

The most exposed problems tend to have candidate solutions that can do something:

Artifact	Examples
Explicit finite object	Hadwiger–Nelson obstruction, Shannon-capacity construction
Algorithm	Graph isomorphism, strongly polynomial linear programming, discrete logarithms
Polynomial or algebraic certificate	Jacobian, Kaplansky if false
Optimized test function or finite matrix	Zeta-zero density, pair correlation, sphere-packing bounds
Small combinatorial construction	Many Astra and VibeMathed cases

The artifact supplies immediate feedback and later becomes part of the proof certificate. Search and verification point in the same direction.

Crystallization and large-data kinetic PDE have the opposite structure. A simulation can display the expected phenomenon while remaining far removed from the universal estimate.

6. Some apparent difficulty gaps are granularity errors

“Permanent versus determinant” is a clean theorem, though it packages an enormous lower-bound challenge.

“Crystallization conjecture” can refer to several potentials, dimensions, densities, boundary conditions, and notions of ground state.

“Global Langlands reciprocity” is closer to a research programme than one atomic statement.

Broadly packaged rows acquire enormous difficulty because “solving the row” means closing a whole ecosystem. Their significance can be depressed because no single stable statement attracts all the fame. Splitting a programme into atomic conjectures would reduce difficulty and often increase the apparent number of significant problems.

7. Recent AI successes create a dangerous halo

Astra’s restricted permanent circuit and formula lower bounds do not make permanent-versus-determinant close. Its high-dimensional sphere-packing result concerns the Cohn–Elkies linear programme, rather than the actual asymptotic packing density. Its multicolor Ramsey advance does not settle diagonal Ramsey growth.

I tried to account for this, though D3 for full asymptotic sphere packing and diagonal Ramsey may still be too optimistic. D4 is probably more defensible for both.

The important predictor is whether the missing step has the same representation as the AI advance. Sharing a field or famous noun is weak evidence.

8. The open-only list is missing its most revealing positive outliers

Jacobian and sofic groups would sit near the extreme high-significance, relatively-low-difficulty corner, but they have already been removed from the open census because AI helped solve them.

That creates survivorship bias. The more successful AI becomes, the remaining open list will make AI look progressively worse at significant mathematics, since the exposed cases disappear first.

A longitudinal catalog should retain resolved problems on the difficulty plot at their estimated pre-resolution difficulty. Otherwise the apparent relationship between significance and AI tractability will drift mechanically.

My revised view

The most credible positive gaps are Kakeya, Montgomery pair correlation, strongly polynomial linear programming, Hadwiger–Nelson, and graph isomorphism. They have concrete machinery or artifacts that generate informative feedback.

Goldbach and twin primes have large statistical gaps mainly because their significance is culturally enormous. I would still expect them to resist frontier AI for a long time.

The strongest negative gaps are permanent-versus-determinant, Grothendieck periods, Anderson delocalization, crystallization, large-data Boltzmann, and Vlasov–Maxwell. They combine modest public visibility with weak feedback, universal quantifiers, and missing conceptual machinery.

The exercise also uncovers likely tiering corrections: odd perfect numbers to D4; full sphere packing and diagonal Ramsey to D4; Bateman–Horn and Schinzel H to at least D5 conditional on truth; MHD, polynomial-hierarchy noncollapse, and NP not in P/​poly to D8 conditional on proving their standard conjectured direction.

The central conclusion is that “AI difficulty” should become a profile rather than one tier. Truth polarity, logical containment, feedback quality, proof verification, and opportunity for valuable partial progress each deserve their own field. The single tier is still useful for orientation, but the largest residuals are often where its hidden assumptions break.

The table below seems like a neat mid-Aug 2026 snapshot of AI x math jaggedness:

Axis	Features that lower AI-relative difficulty	Features that raise AI-relative difficulty
Search geometry	Finite objects, explicit constructions, algorithm design, a small counterexample, or a narrow quantitative optimization	A universal theorem over infinite-dimensional objects, a diffuse classification programme, or no stable candidate space
Feedback density	Cheap numerical tests, exact symbolic identities, SAT or SDP bounds, small cases, and formalizable intermediate lemmas	Failed attempts return little information, numerics sit far from proof, or correctness depends on a long analytic tail
Barrier structure	A known proof frame with one replaceable component or a precise local bottleneck	Relativization, natural-proofs or algebrization barriers, critical-scaling obstructions, or every route restating the conjecture
Conceptual distance	The missing bridge joins mature theories already present in the literature	A solution appears to require new language, new invariants, or a new global architecture for the subject
Verification	A compact certificate or modular proof that independent agents and proof assistants can audit	Delicate limits, genericity, regularity, equality cases, or specialist conventions that can hide a fatal gap
Autonomy burden	Backward reasoning from a crisp target and measurable subgoals	Long forward exploration whose central task is judging which observations are mathematically fruitful