First Proof

Introduction

• The set contains 10 naturally arising research-level questions across diverse mathematical fields, each answered by its authors with a proof roughly five pages or less.The answers are not yet posted online; the length limit reflects technical limitations of current publicly available AI systems, and some questions are smaller components of future publications.Source: §1 · p. 2
• The experiment tests proof-writing once questions and frameworks are understood, not question selection, definition-making, or theory development.The authors note that important research also involves formulating questions and developing frameworks, and say evaluating those higher-level tasks remains essential.Source: §1 · pp. 2–3
• The authors invite community trials of the encrypted questions to learn about prompting, answer formats, grading, and contamination before developing a more formal benchmark.Answers are scheduled for release on February 13, 2026; the authors hope to finalize a second question set a few months later.Source: §1 · pp. 2–3
• The question list is not yet a benchmark because it is small, costly to produce, and lacks a formal grading scheme for potentially non-unique answers.Assessment must currently be done by a human expert.Source: §1 · p. 3
• The proposed assessment uses questions from mathematicians’ current research whose answers have not been publicly posted, while allowing unrestricted access to outside resources.The authors describe this combination as distinguishing their approach from previous assessments; proof answers currently require human grading.Source: §1 · p. 3
• The authors’ one-shot preliminary tests indicate that the best publicly available AI systems struggle with many of the questions.They used GPT 5.2 Pro and Gemini 3.0 Deepthink, without iterative interaction or rerunning queries.Source: §1 · p. 3

Questions 1–2

• Questions whether a nonzero smooth shift preserves equivalence of the Φ4^3 measure and whether a GL(n+1) Whittaker function yields a finite, nonzero Rankin–Selberg integral for every s.Source: §2 · p. 4

Question 3

• Question 3 asks whether a nontrivial Markov chain can have the specified ratio of interpolation ASEP and interpolation Macdonald polynomials as its stationary distribution.Source: §2 · pp. 4–5

Question 4

• Question 4 asks whether 1/Φ_n(p ⊞_n q) ≥ 1/Φ_n(p) + 1/Φ_n(q) for monic real-rooted polynomials p and q.Source: §2 · p. 5

Question 5

• Question 5 asks for an O-slice filtration adapted to an incomplete transfer system and a characterization of O-slice connectivity for connective G-spectra using geometric fixed points.Source: §2 · p. 5

Question 6

• Question 6 asks whether every graph has an ε-light vertex subset of size at least cε|V| for a universal c > 0 and every 0 < ε < 1.Source: §2 · p. 5

Questions 7–8

• Question 7 asks whether a uniform lattice with 2-torsion can be the fundamental group of a compact rationally acyclic-cover manifold; Question 8 asks whether a four-face-at-each-vertex polyhedral Lagrangian surface admits smoothing.Source: §2 · pp. 5–6

Question 9

• Question 9 asks for a polynomial map, independent of the generic matrices and with degree bounds independent of n, that detects whether tensor coefficients factor as four vectors.Source: §2 · p. 6

Question 10

• Question 10 asks for an efficient preconditioned conjugate-gradient method for the infinite-dimensional RKHS mode subproblem, including preconditioning, matrix-vector products, complexity, and avoiding O(N) computation.The setup assumes n, r < q ≪ N, with q observed tensor entries.Source: §2 · pp. 6–7

3 Related work

• FrontierMath contains private expert-level problems with integer or symbolic final answers, enabling automatic grading and reinforcement-learning post-training.OpenAI has access to a subset of its problems and solutions, while EpochAI has access to all solutions; 12 examples are public.Source: §3 · p. 7
• IMProofBench evaluates research-level proof creation; subquestions allow automatic grading, but experts must verify full correctness, and its questions are private.Source: §3 · p. 7
• RealMath scrapes post-training-cutoff arXiv papers, emphasizes constructive fields, and uses public short-answer questions designed for automatic grading and periodic refresh.Its source categories are mathematics and computer science; the refresh is intended to reduce data contamination.Source: §3 · p. 7

4 Implementation details

• The authors tested roughly 20 research-level math questions with Gemini 3 Pro, GPT-5.1 Pro, and later GPT-5.2 Pro.Testing more than 10 questions was intended to probe the boundary between questions the models could solve and those beyond their reach.Source: §4 · pp. 7–8
• Questions were selected only if no unknown previous answer was revealed, a one-page statement sufficed for model reformulation, authors agreed to provide a human proof, and no team member contributed more than one problem.Source: §4 · p. 8
• To minimize contamination, the team turned off data sharing for model training and improvement, while noting retention periods of 3 days by Google and 30 days by OpenAI.Source: §4 · p. 8
• The team endeavored to keep answers private, uploaded encrypted answers to a private repository, and planned public release about a week after releasing the questions.Source: §4 · p. 8

5 Discussion

• The “first proof” experiment assesses the answer-finding stage of mathematical research, not AI’s ability to create research questions or develop novel theories.The paper identifies question creation and selection, theory development, and finding and rigorously proving answers as components of mathematical research.Source: §5 · p. 8
• The authors plan a second set of similar questions in the coming months and are open to agreements for testing frontier AI systems before release, with the set serving as a benchmark.Source: §5 · p. 8
• Beyond the next release, the authors plan to remove some artificial constraints, such as length, and explore measuring other aspects of mathematical research.Source: §5 · p. 8

• On February 4 and 5, 2026, the authors tested the questions on Gemini 3.0 Deep Think and ChatGPT 5.2 Pro.Source: A First Proof Solutions and comments · p. 9
• Prompt 1 requested a rigorous, complete answer as a compilable LaTeX document meeting mathematical-literature standards.Source: A First Proof Solutions and comments · p. 9
• Prompt 2 discouraged web search and asked for a rigorous, complete answer as a compilable LaTeX document grounded in reasoning.Source: A First Proof Solutions and comments · p. 9

A.10 Question 10: Tammy Kolda

• The best LLM solution was correct, lowered computational complexity versus Kolda’s solution, and revealed an insight she had not yet seen; its main idea also appears in an existing solution.Kolda found the main idea through a search for “subsampled kronecker product matvec.”Source: §A.10 · p. 13
• The LLM answer lacked appropriate citations and omitted Kolda’s problem transformation, which was unnecessary for the open-ended task; she plans to borrow aspects with better attribution.Source: §A.10 · p. 13

• The solutions match those encrypted on February 5, 2026, and released on February 13, 2026.Source: B The human-generated solutions to our… · p. 13

Proof of the lemmas

• Lemma B.2 follows using a Sobolev embedding, Gaussian-polynomial moment equivalence, and a standard Borel–Cantelli argument.Source: Proof of the lemmas · p. 17
• Lemma B.3 gives a lower bound on γ that ensures Aγ (or Bγ) is distinguishable under the shifted measure from the non-shifted one.The proof expands cN,2 and bounds a sum by an integral to obtain a lower bound by a multiple of Z_N.Source: Proof of the lemmas · p. 18

These are compatible with the natural nullification functors

• The stated natural map for a connective G-spectrum is always a weak equivalence.The surrounding argument relates natural transformations of nullification functors and factors the nullification functor through an inclusion.Source: These are compatible with the natural… · p. 40
• Ordinary suspension yields maps between O-slice subcategories, and the ∞-category of O-k-slices is discrete for every k ≥ 0.The suspension argument uses that adjoining a fixed point preserves admissibility while increasing cardinality by one.Source: These are compatible with the natural… · p. 41

The O-slices of geometric spectra

• For a geometric Mackey functor M, HM is a k[G : χO(G)]-slice.The bound uses the largest cardinality of an admissible G-set with k orbits.Source: These are compatible with the natural… · p. 43

Rewriting O-slice connectivity

• The full O-slice connectivity criterion follows from geometric fixed-point connectivity, with the converse proved by isotropy separation and induction on the subgroup lattice.The forward direction is supplied by Lemma B.14; the converse combines the geometric-spectrum bound with the localizing-category argument.Source: These are compatible with the natural… · pp. 42, 44

B.10.1

• The subsection considers several approaches to solving Equation (B.1), including a direct method for the symmetric linear system with an additional regularization term.Source: §B.10 · p. 62

B.10.2

• A second approach transforms the symmetric system using the eigendecomposition of K.Source: §B.10 · p. 62

B.10.4

• An iterative method operates on the transformed symmetric system and adds regularization akin to the symmetric direct method.Source: §B.10 · p. 62

B.10.5

• The subsection accounts for costs and compares the direct and iterative methods in Table 1 and Section B.10.5.Source: §B.10 · p. 62

• Adding a regularization term parameterized by ρ > 0 makes the indefinite symmetric system of size rn × rn positive definite.Source: §B.10.1 · p. 62
• The right-hand side is F = S^T(Z ⊗ K), with K pulled inside the vectorization.Source: §B.10.1 · p. 62
• To compute F without explicitly forming the N × nr Kronecker product, the method constructs ˆK and ˆZ from rows corresponding to known entries of T.These matrices represent the subsets of rows of K and Z associated with those known entries.Source: §B.10.1 · p. 62

B.10.2 Transforming the UI Subproblem

• Factoring out I_r ⊗ U using K = UDU^T recasts Equation (B.1) as an equivalent system that may be better conditioned.The transformed variable is W̄ = U^T W, and W is recovered as U W̄ after solving.Source: §B.10.2 · pp. 62–63
• Because D is indefinite, it cannot be absorbed into W̄; the transformed system instead defines F̄ := S^T(Z ⊗ UD) and B̄ := D U^T B, with regularization including λ(I_r ⊗ D).The modified system also includes the term ρI_rn.Source: §B.10.2 · p. 63

Lemma 1

• Lemma 1 computes Cx as (A ∗BX)1r in O(q(r + n)) operations, reducing the explicit-matrix cost of O(qrn) and memory to O(q(r + n)).Here x = vec(X), with C defined row-wise by C(ℓ, :) = A(ℓ, :) ⊗B(ℓ, :).Source: §B.10.3 · p. 63

Lemma 2

• Lemma 2 computes CTv as vec(BT diag(v)A) without forming C, keeping cost at O(qrn) while reducing memory from O(qrn) to O(q(r + n)).The input is v ∈Rq.Source: §B.10.3 · p. 63

Lemma 3

• Lemma 3 computes the diagonal of CTC in O(q(r2 + n2)) operations, down from O(qr2n2), and reduces memory from O(qrn) to O(q(r + n)).The passage describes this computation but does not provide the diagonal formula.Source: §B.10.3 · p. 64

B.10.4 PCG Solution of Transformed UI Subproblem

• The transformed operator F̄ is formed analogously to F, with H = UD ∈ R^{n×n} and Ĥ ∈ R^{q×n}.Source: §B.10.4 · p. 64
• The conjugate-gradient matrix-vector products can be computed without forming any Kronecker products, using the matrix representation of the vector and results from Section B.10.3.For x with vec(X) = x, the passage gives a way to compute F̄ᵀF̄x; it also states that the preconditioner's last step follows from Lemma 3 in Section B.10.3.Source: §B.10.4 · p. 64
• A diagonal preconditioner is proposed for the conjugate-gradient iterations.Source: §B.10.4 · p. 64

Factorizing the kernel matrix K for the transformed system

• PCG alone requires an eigendecomposition of K, costing O(n^3) flops, performed once before the outermost alternating optimization iterations.The comparison assumes d < n and r ≪ q, without generally assuming a relative size for n and r.Source: §B.10.5 · p. 65

Shared costs of all methods

• Both methods compute ˆZ at O(qrd) cost, and using it reduces the incomplete-tensor MTTKRP B computation from O(qrd) to O(qr).The MTTKRP is relatively cheap and never dominates the cost because the tensor is incomplete.Source: §B.10.5 · p. 65

Direct solve of symmetric regularized system

• The direct solve costs O(qr^2n^2) to form the system matrix and O((rn)^3) to solve it, with storage O(rnq + r^2n^2).Computing F costs O(qrn) and requires O(qrn) storage; computing vec(KB) costs O(n^2r).Source: §B.10.5 · p. 66

PCG iterative solve of transformed system

• PCG iterations cost O(pqnr) operations, with each matrix-vector product costing O(qnr) and each preconditioner application O(rn).The transformed-system setup includes O(n^2) to form H and O(qn^2 + qr^2) to form the diagonal preconditioner; recovering W costs O(n^2r), and storage is O(qn + qr).Source: §B.10.5 · p. 66

Summary and Comparison

• PCG costs orders of magnitude less than the direct method, with significantly lower storage requirements.The comparison depends on the iteration count p and the relative sizes of n, r, and p; the authors expect a well-conditioned problem so p is not too large. Assuming r < n < rn < q, storage is qrn for direct methods versus qn for PCG.Source: Table 1 · p. 65, §B.10.5 · p. 66