Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop
Paper • 2602.05394 • Published
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Research Math Rollouts
Agent trajectories from two systems that attack informal research mathematics: 55 solveall research prompts (Rethlas and Danus) and 14 numerical-linear-algebra problems (Danus). All files passed a strict SFT validator (reasoning present, tool calls matching their declared schemas, no degenerate repetition, no leaked control markers).
Each record is {conversation, …, token_count, token_count_answer, token_count_think}; in conversation, assistant turns carry content, think and tool_calls.
Problems
The agents work on two problem sets.
solveall — 55 research problems from solveall.org. The set is weighted toward open problems posed at COLT 2018–2025 (improper learning of mixtures of Gaussians, average-case hardness of hypergraphic planted clique detection, whether single-shuffle SGD can beat reshuffling SGD and GD, properly learning decision trees in polynomial time, tight characterization of instance-optimal identity testing, the complexity of joint differential privacy in linear contextual bandits, instance-dependent sample complexity for finding Nash equilibria in two-player zero-sum matrix games, …) and also includes standalone open problems such as the KLS conjecture, the smoothed complexity of the simplex method, the tensor PCA detection threshold, the capacity of the binary deletion channel, and log-concave sampling complexity.
NLA — 14 open problems in numerical linear algebra (p1–p7), precise statements of open questions from Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop (arXiv:2602.05394v3, Amsel et al., 2026):
| id | topic |
|---|---|
| p1 | global convergence of shifted QR iterations (degree-k shift strategies, uniform decoupling rate) |
| p2a | GECP growth factor: polynomial upper bound |
| p2b | GECP growth factor: superpolynomial lower bound |
| p3a | convergence of restarted GMRES |
| p3b | worst-case vs ideal GMRES: maximal residual ratio |
| p3c | Forsythe conjecture for restarted CG |
| p4 | sparse SPD solvers beyond Laplacians |
| p5a | matvec query complexity of low-rank approximation |
| p5b | algorithmic Kadison–Singer: polynomial-time partition |
| p5c | matrix Spencer conjecture |
| p6a | maximal vs generic tensor rank |
| p6b | decidability of tensor rank over the rationals |
| p6c | complexity of approximating the tensor spectral norm |
| p7 | numerical range as a complete 2-spectral set (Crouzeix inequality) |
Coverage
rethlas-gen.jsonlcovers 27 of the 55 solveall problems, one generator session each. Across all 55, the generator runs ended 35 solved, 13 errored, 4 failed. Rethlas also attempted three NLA problems (p2a, p5b, p5c), but no session completed a turn, so NLA contributes no Rethlas records.
The 27 solveall problems with generator records
- asymptotic-normality-of-spectrum-aware-debiasing-beyond-right-rotationally-invar
- characterize-adaptive-distribution-classes-where-two-point-rates-are-attainable
- colt-2018-improper-learning-of-mixtures-of-gaussians
- colt-2019-do-good-algorithms-necessarily-query-bad-points
- colt-2020-average-case-hardness-of-hypergraphic-planted-clique-detection
- colt-2020-tight-convergence-of-sgd-in-constant-dimension
- colt-2021-can-single-shuffle-sgd-be-better-than-reshuffling-sgd-and-gd
- colt-2021-tight-online-confidence-intervals-for-rkhs-elements
- colt-2022-properly-learning-decision-trees-in-polynomial-time
- colt-2023-is-there-a-first-order-method-that-only-converges-to-local-minimax-optima
- colt-2023-learning-sparse-linear-concepts-by-priming-the-features
- colt-2023-polynomial-linearly-convergent-method-for-g-convex-optimization
- colt-2024-tight-characterization-of-instance-optimal-identity-testing
- colt-2024-what-is-the-complexity-of-joint-differential-privacy-in-linear-contextual-bandits
- colt-2025-optimal-instance-dependent-sample-complexity-for-finding-nash-equilibrium-in-two-player-zero-sum-matrix-games
- critical-snr-for-outlier-emergence-at-fixed-summary-statistics
- does-the-score-matched-optimal-convex-estimator-attain-the-full-semiparametric-e
- finite-sample-error-rate-control-and-power-guarantees-under-the-linear-subspace
- finite-sample-optimal-fdr-fnr-frontier-under-the-two-group-model
- from-low-degree-hardness-to-unconditional-polynomial-time-hardness
- full-multiple-break-theory-for-latent-group-structure-and-coefficients
- generalization-error-estimation-beyond-gaussian-designs
- information-theoretic-limits-under-heavy-tailed-non-sub-gaussian-noise
- optimal-adaptation-beyond-compact-manifolds
- optimal-guarantees-in-the-low-sample-tensor-regime-below-constant-frobenius-thre
- remove-logarithmic-gaps-in-minimax-sample-complexity-for-matrix-normal-covarianc
- robust-confounder-selection-under-imperfect-primary-set-elicitation
rethlas-ver.jsonlcovers 9 solveall problems, heavily skewed — the binary deletion channel alone accounts for 144 of the 287 sessions, because the generators' MCP verify calls timed out and they called the verify service directly, sending proof fragments for verification:
| problem | sessions |
|---|---|
| capacity-binary-deletion-channel | 144 |
| does-the-score-matched-optimal-convex-estimator-attain-the-full-semiparametric-e | 36 |
| remove-logarithmic-gaps-in-minimax-sample-complexity-for-matrix-normal-covarianc | 30 |
| optimal-guarantees-in-the-low-sample-tensor-regime-below-constant-frobenius-thre | 23 |
| colt-2024-tight-characterization-of-instance-optimal-identity-testing | 16 |
| optimal-adaptation-beyond-compact-manifolds | 15 |
| asymptotic-normality-of-spectrum-aware-debiasing-beyond-right-rotationally-invar | 12 |
| colt-2024-black-box-reductions-and-adaptive-gradient-methods-for-nonconvex-optimization | 10 |
| colt-2021-tight-online-confidence-intervals-for-rkhs-elements | 1 |
danus.jsonlcovers 16 projects: 12 of the 14 NLA problems (all but p6c and p7) and the first four solveall projects —asymptotic-normality-of-spectrum-aware-debiasing-beyond-right-rotationally-invar,capacity-binary-deletion-channel,characterize-adaptive-distribution-classes-where-two-point-rates-are-attainable, andcolt-2018-improper-learning-of-mixtures-of-gaussians. 13,509 records come from the NLA run (Aug 21–Sep 9) and 1,062 from the solveall run (Sep 9–14, stopped early). Filter per problem withproblem_id(Rethlas files) ormetadata.project(Danus).
Files
rethlas-gen.jsonl — 27 records
Generator sessions from Rethlas (Kimi K3 via the Kimi Code CLI, thinking effort max): each session writes a markdown proof "blueprint" for one solveall problem, alternating no-search deep-thinking rounds with search-allowed rounds, until the blueprint passes verification. Skills include immediate conclusions, toy examples, counterexamples, subgoal decomposition (with a subgoal-prover subagent), failure analysis and literature search; MCP tools cover arXiv theorem search, memory, branch updates and proof verification. The role signal is the lane field (gen). 38.6M tokens across the two Rethlas files.
rethlas-ver.jsonl — 287 records
Verifier sessions from Rethlas: a fresh Kimi session per verification request checks each blueprint statement in sequence, then referenced results (arXiv search first, then the web), and writes a verification.json report; "correct" iff both the critical-error and gap lists are empty. One problem (the binary deletion channel's capacity) accounts for 144 of these sessions. The role signal is the lane field (ver).
danus.jsonl — 33,174 records, 3.91B tokens
Sessions from the Danus swarm working the same linalg and solveall problems: a main agent publishing elaborations and consulting a stronger model on strategy, 7 workers per project proving lemmas round by round via fact_submit, a fresh verifier judging every submitted fact (facts enter the shared graph only on a correct verdict), and paper roles turning verified facts into amsart papers. The role signal is metadata.role (worker, verify, authoring; main records are health-check pings and the orchestrator sessions are not included). 33,174 of 33,776 candidates passed the gate (98.2%); the most common drops are model-side violations (string-encoded array arguments, invented arguments, repetition loops). Mean 117.8k tokens, median 92.4k.
| Role | Records | Tokens |
|---|---|---|
| verify | 18,520 | 1.40B |
| worker | 14,571 | 2.50B |
| authoring | 61 | 5.0M |
| main | 13 | 0.3M (health pings only) |
| other | 9 | 0.25M |
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