Verification update, September 22, 2026
This publication does not contain a completed problem-by-problem correctness review of the candidate solutions. The original February 13 answer-release deadline has passed and is not a current reason to call verification pending. The benchmark paper was subsequently revised in March 2026; that update is not a verification of this experiment. A current public experiment repository is linked below, but repository availability does not establish a later verification outcome or mathematical correctness.
All answer summaries below are model-generated candidates, including claims that may be wrong. Counts of completed responses and agreement are not mathematical accuracy scores.
Introduction
The 1stProof Benchmark presented ten research-level mathematics problems whose answers were initially withheld. Created by mathematicians from Stanford, Harvard, Yale, EPFL, Columbia, and other leading institutions, these problems emerged from active mathematical research—not textbooks or competitions.
At the time of this experiment, the organizers had encrypted the answers for a limited period. This reduces direct access to those answers during that period; it does not exclude all related training material, contamination pathways or use of known mathematical results.
We evaluated Claude Opus 4.6 (claude-opus-4-6) in February 2026 using multiple prompting strategies to examine output behavior on these problems.
The ten questions
| ID | Field | Problem Summary |
|---|---|---|
| Q1 | Stochastic Analysis | Measure equivalence under shifts on Φ⁴₃ measure space |
| Q2 | Representation Theory | Existence of universal Whittaker functions for Rankin-Selberg integrals |
| Q3 | Algebraic Combinatorics | Markov chains on restricted partitions with ASEP polynomial stationary distribution |
| Q4 | Spectral Graph Theory | Logarithmic derivative inequality for real-rooted polynomial convolution |
| Q5 | Algebraic Topology | Slice filtration characterization in G-equivariant stable categories |
| Q6 | Spectral Graph Theory | Existence of ε-light vertex subsets with linear size bounds |
| Q7 | Lattices in Lie Groups | Uniform lattices with 2-torsion as fundamental groups of rationally acyclic manifolds |
| Q8 | Symplectic Geometry | Lagrangian smoothings of polyhedral surfaces with 4-valent vertices |
| Q9 | Tensor Analysis | Polynomial-time separability detection for tensor collections |
| Q10 | Numerical Linear Algebra | Preconditioned conjugate gradient for infinite-dimensional tensor problems |
These span advanced topics rarely seen in AI benchmarks: stochastic PDEs, automorphic forms, symplectic topology, and algebraic K-theory.
Method
Prompting strategies
We test six distinct approaches:
Zero-Shot: Direct question presentation without scaffolding.
Chain-of-Thought: Explicit instruction to decompose reasoning step-by-step.
Multi-Attempt: Three independent attempts at temperatures 0.0, 0.5, and 1.0.
Self-Critique: Two-phase approach—initial solution followed by self-review and revision.
Extended Thinking: Anthropic’s adaptive thinking mode with effort: "max".
Extended Thinking + Continuation: Two-phase approach developed during this study.
All modes use a consistent system prompt emphasizing:
- Publication-quality mathematical rigor
- Peer-reviewed citations where applicable
- Complete logical progression
- Honest acknowledgment of uncertainty
The original report states that each response record includes:
- SHA-256 hash of prompt and response
- ISO 8601 timestamps
- Token usage statistics
- Model version and parameters
Hashes can detect changes to recorded content. They do not verify mathematical correctness or independently establish when an experiment occurred.
Results
| Q# | Candidate answer | Claimed approach | Reported agreement |
|---|---|---|---|
| Q1 | Mutually Singular | Dichotomy theorem via Cameron-Martin | 7/7 |
| Q2 | Yes | Constructive existence via Bernstein-Zelevinsky | 7/7 |
| Q3 | Yes | Markov chain construction with detailed balance | 7/7 |
| Q4 | Yes | Inequality via logarithmic convexity | 7/7 |
| Q5 | No | Counterexample construction | 7/7 |
| Q6 | Yes | Random sampling with matrix concentration | 7/7 |
| Q7 | Yes | Davis reflection group construction | 7/7 |
| Q8 | Yes | Local smoothing via symplectic surgery | 7/7 |
| Q9 | Yes | Polynomial system via algebraic geometry | 7/7 |
| Q10 | Yes | Condition number bounds with Kronecker structure | 7/7 |
The original table reports 7/7 agreement for each answer. Six strategies are listed, multi-attempt produces three outputs, and extended thinking alone produced no final answers. The aggregation behind 7/7 is not documented here and remains unreconciled. Agreement among related model outputs would not establish correctness even with a reconciled count.
Prompting mode comparison
| Mode | Reported complete candidate responses | Avg Response Length |
|---|---|---|
| Zero-Shot | 10/10 | ~2,500 tokens |
| Chain-of-Thought | 10/10 | ~3,200 tokens |
| Multi-Attempt | 30/30 | ~2,800 tokens |
| Self-Critique | 10/10 | ~4,500 tokens |
| Extended Thinking | 0/10 | 0 tokens (exhausted) |
| Extended + Continuation | 10/10 | ~4,200 tokens |
Selected candidate arguments
These summaries preserve what the model reportedly argued; they are not endorsed proofs. In particular, a named theorem, plausible outline or citation is insufficient to establish that its hypotheses apply.
Q1: Mutual singularity of Φ⁴₃ measures
Problem: Are the Φ⁴₃ measure μ and its shift T_ψ*μ equivalent (same null sets)?
Candidate answer: No—they are mutually singular.
Candidate rationale (unverified): The response invokes the Cameron-Martin theorem. For Gaussian measures, equivalence requires the shift to lie in the Cameron-Martin space H. For the Φ⁴₃ measure (a non-Gaussian perturbation of the Gaussian free field), smooth shifts ψ have infinite Cameron-Martin norm, forcing mutual singularity.
The response claims to construct test functions distinguishing the measures. This argument, including its use of a Gaussian theorem for a non-Gaussian measure and its regularity assumptions, requires expert checking.
Q2: Universal Whittaker functions
Problem: Must there exist a Whittaker function W with a universal property for all generic representations π?
Candidate answer: Yes—such W exists.
Candidate strategy (unverified):
- Construct W using Bernstein-Zelevinsky theory of mirabolic restriction
- Show the modified Rankin-Selberg integral reduces to a constant independent of s
- Prove non-vanishing via Fourier analysis on compact groups
The response cites Jacquet-Piatetski-Shapiro-Shalika (1981) and Cogdell’s lectures. Those citations and their applicability have not been verified in this record.
Q6: ε-light vertex subsets
Problem: Does there exist c > 0 such that every graph has an ε-light subset of size ≥ cε|V|?
Candidate answer: Yes—with c ≈ 1/4.
Candidate strategy (unverified):
- Sample vertices with probability p = ε/2
- Expected size: εn/2 (linear in ε)
- Edges survive with probability p² = ε²/4 (quadratic in ε)
- Since ε² ≪ ε, the spectral condition εL - L_S ≽ 0 holds with high probability
- Matrix Chernoff bounds formalize the concentration
This outline invokes spectral graph and concentration techniques; the implication from expected edge counts to the claimed spectral inequality is not established by the outline.
Q7: Davis construction for lattices
Problem: Can a uniform lattice Γ with 2-torsion be the fundamental group of a compact manifold with rationally acyclic universal cover?
Candidate answer: Yes.
Candidate rationale (unverified): The response invokes the Davis reflection group trick. Given Γ < G semisimple with 2-torsion elements, construct a right-angled Coxeter group W acting on a CAT(0) cube complex. The Davis complex provides a contractible manifold on which Γ acts properly discontinuously with compact quotient.
The response claims that this construction gives the required manifold. The group action, torsion and quotient-manifold conditions require checking; a proper action alone does not establish the claimed manifold quotient.
Reported token exhaustion
For the reported extended-thinking configuration, the original experiment describes:
Extended Thinking (effort: "max")
Token Limit: 128,000
Thinking Tokens: 128,000
Response Tokens: 0
Stop Reason: max_tokensThe report records this output-budget exhaustion for all ten questions in that configuration. This is a historical observation about those calls, not a claim about every extended-thinking run or the current API.
Two-phase continuation
We developed a continuation approach:
- Phase 1: Extended thinking with max effort (captures deep reasoning)
- Phase 2: Feed thinking back, request final answer in standard mode
Results:
- Phase 1: 128,000 tokens of mathematical reasoning
- Phase 2: ~4,000 tokens of candidate solution text
The procedure reportedly yielded final candidate responses after the first phase. It does not establish their correctness.
Discussion
The original report describes theorem statements, lemmas, citations, varied approaches and acknowledgments of possible gaps. These are characteristics of the generated text. Rigor, citation accuracy and novelty require independent review and are not established by presentation alone.
Tools the responses cite
The candidate responses reportedly invoke the following mathematical tools:
| Citation Type | Examples |
|---|---|
| Foundational theory | Cameron-Martin theorem, Bernstein-Zelevinsky |
| Technical tools | Matrix Chernoff bounds, Tropp (2012) |
| Structural results | Davis (1983), JPSS (1981) |
The presence of these names does not establish valid reasoning, correct attribution, novelty or absence of training-data overlap.
Limitations
- Correctness unresolved: No completed assessment is attached to this publication as of September 22, 2026
- Unavailable original repository: The historical experiment URL returned HTTP 404 during the editorial revision
- Accounting: The reported 7/7 agreement is not reconciled with the six strategies and repeated attempts
- Single model: Only Claude Opus 4.6 evaluated
- Notation loss: Some LaTeX formatting simplified in extraction
Conclusion
This experiment reports candidate answers to ten research-level problems and output-budget exhaustion in one historical prompting configuration. Its continuation procedure reportedly produced final responses. The record does not establish a mathematical success rate. A problem-by-problem review of complete arguments, assumptions and citations is needed before making a correctness claim.
Data availability
Current experiment artifact: Lumi-node/1stProof-Opus-Experiment. This repository is available for inspection; this publication still does not provide a completed correctness-verification record.
The original report described its contents as:
- All 80+ responses (JSON and Markdown)
- SHA-256 verification hashes
- Thinking traces (where captured)
- Experiment scripts
References
[1] 1stProof Team. “First Proof: A Benchmark for Research-Level Mathematical Reasoning.” arXiv:2602.05192, 2026.
[2] Anthropic. “Claude Opus 4.6 Technical Documentation.” 2026.
[3] Davis, M. “Groups Generated by Reflections and Aspherical Manifolds.” Annals of Mathematics, 1983.
[4] Jacquet, H., Piatetski-Shapiro, I., Shalika, J. “Conducteur des Représentations du Groupe Linéaire.” Math. Ann., 1981.
[5] Tropp, J. “User-Friendly Tail Bounds for Sums of Random Matrices.” Found. Comput. Math., 2012.
Contributions and revision
Andrew Young is the human author responsible for this publication. Claude Opus 4.6 generated the evaluated candidate solutions; the original report also described Claude as assisting with the research. Model-generated work and self-critique do not provide independent verification or author accountability.
This September 22, 2026 editorial revision corrects status, attribution and claim scope. It adds no new experiment or correctness assessment.