Mathematical exploration and discovery at scale

1. INTRODUCTION

• AlphaEvolve combines LLM creativity, evolutionary computation, and automated evaluation to discover constructions matching or improving long-standing mathematical bounds at scale.It targets problems where complex objects must satisfy quantitative properties, but does not match or exceed prior results in every case.Source: §1 · p. 1

1.3. Evolving Algorithms to Find Constructions.

• Its search mode evolves heuristic programs that improve the best construction found so far, chaining exploratory and problem-specific refinements within fixed computational budgets.This shifts expensive LLM generation toward designing search strategies that can cheaply explore millions of candidate constructions.Source: §1.3 · p. 3, §2 · p. 5

1.4. Generalizing from Examples to Formulas: the generalizer mode.

• The generalizer mode seeks programs valid across input sizes, producing interpretable constructions that inspired new work on the Nikodym and arithmetic Kakeya problems [281] [282].It evaluates candidate programs over multiple n values and attempts to infer formulas from often-optimal small cases.Source: §1.4 · p. 4

1.5. Building a pipeline of several AI tools.

• For the finite-field Kakeya problem, AlphaEvolve’s construction was proved by Deep Think, converted into a closed-form size formula, and fully formalized in Lean using AlphaProof.This demonstrates a pipeline from empirical pattern discovery to symbolic proof generation and formal verification.Source: §1.5 · p. 4

1.6. Limitations

• AlphaEvolve is most effective for large portfolios of optimization problems with hill-climbable score functions, but struggles when progress requires genuinely new deep insights.The study tested 67 literature problems, often reformulating them as construction-based upper or lower bound searches and discretizing infinite-dimensional domains when necessary.Source: §1.6 · p. 4, §4 · p. 8, §6 · p. 9

3.1. The Trade-off Between Speed of Discovery and Evaluation Cost.

• 20 threads found the state-of-the-art bound faster than 2 threads, but higher parallelism increased total LLM calls and compute cost.The experiments on Problem 6.2 averaged 100 runs with 2 threads and 10 runs with 20 threads, making the speed-cost trade-off explicit.Source: §3.1 · p. 5, Figure 1 · p. 6

3.2. The Role of Model Choice: Large vs. Cheap LLMs.

• For the simple autocorrelation problem, repeated runs with the cheapest LLM beat the literature bound for a few USD, whereas that model failed to produce the most elaborate Nikodym constructions.More capable models often achieved better scores with fewer evolutionary steps, but model diversity could sometimes outperform exclusive use of high-end models.Source: §3.2 · p. 6

4. CONCLUSIONS

• Verifier design, continuous scoring, expert prompting, and correlated problem families materially improve discovery, while leaky verifiers can induce cheating rather than genuine solutions.Generalization also improved when AlphaEvolve received fewer inputs, such as prior best solutions only for small n values.Source: §4 · pp. 7–8

6. MATHEMATICAL PROBLEMS WHERE AlphaEvolve WAS TESTED

• AlphaEvolve matched or nearly matched existing bounds in many cases, sometimes with interpretable descriptions of the extremizers, and improved the state of the art in several cases.Some matched bounds are known or conjectured to be sharp.Source: §6 · p. 9
• AlphaEvolve failed to match literature bounds in some cases, which the authors document alongside successful experiments to characterize what works and what does not.The authors also include problems attempted only very briefly.Source: §6 · p. 9
• Further improvements may come from better prompting and setup, theory- or numerics-guided customization, or hybrid methods, as occurred in some previously announced results.The authors hope these findings stimulate progress using diverse methods.Source: §6 · p. 9

Search and verification

• AlphaEvolve used generalizer-mode scoring over many large primes or prime powers, while Deep Think derived size formulas and AlphaProof formalized proofs when the required mathematics was supported by mathlib.The dimension-4 proof was too difficult for AlphaProof and lacked an exact size formula, requiring manual verification; guided experiments also reproduced constructions from after a few hundred iterations.Source: §1 · pp. 12–13

Kakeya results

• 1/4p^3 + 7/8p^2 − 1/8 bounds C_K(3,p) for every prime p ≡ 1 mod 4, slightly refining the previous 1/4p^3 + 7/8p^2 + O(p) bound from [49].The construction is explicit, and its size formula was cross-checked across several primes and fully formalized in Lean via AlphaProof.Source: §1 · p. 12
• For primes p ≡ 1 mod 4, AlphaEvolve found Kakeya constructions with sizes 19/32p^3 + 11/16p^2 + O(p^3/2) in dimension 4 and 177/256p^3 + O(p^5/2) in dimension 5, refining lower-order terms relative to [49].In dimension 4, the leading two coefficients match the best-known construction, while Deep Think connected the proof to elliptic curves; the dimension-5 proof was checked by hand.Source: §1 · pp. 12–13

Nikodym results

• p^d−(((d−2)/log 2) + 1 + o(1))p^(d−1) log p upper-bounds C_N(d,p) for fixed d ≥ 3, improving the best known Nikodym construction after human simplification of AlphaEvolve’s proposal.The original high-degree-surface construction was difficult for Deep Think to analyze, whereas the simplified lower-degree and probabilistic approach yielded the stronger result; details appear separately [281].Source: §1 · p. 13
• q^2 − q^3/2 + O(q log q) constructions emerged for two-dimensional Nikodym sets over square-order fields only after AlphaEvolve was told that this target was attainable.Without the hint, it found only q^2 − O(q log q) constructions; with the hint, it first reached q^2 − c q^3/2 and eventually recovered the sharper form.Source: §1 · p. 13

Problem 6.2

• C6.2 ≤1.5032 improved on the π/2 = 1.57079 … bound after AlphaEvolve used cubic-backtracking heuristics informed by a Newton-type optimization method.The search used piecewise-constant functions on finite partitions, with n=10 already sufficient to improve the π/2 bound.Source: §2 · p. 14
• A later meta-analysis using different optimization methods did not improve AlphaEvolve’s Problem 6.2 result, illustrating the difficulty caused by highly irregular numerical optimizers.The analysis is reported in, and Figure 3 displays AlphaEvolve’s constructions with scores 1.5053, 1.5040, and 1.5032, where smaller is better.Source: §2 · p. 14, Figure 3 · p. 15

Problem 6.3

• C6.3 ≥0.961 was discovered using a 50,000-part step function after AlphaEvolve combined gradient-based methods with additional heuristics.The result followed an earlier quick-experiment bound of C6.3 ≥0.8962 and several hours of further search; the authors expect more parts could improve it further.Source: §2 · pp. 14–15

Problem 6.4

• For Problem 6.4, AlphaEvolve improved the prior upper bound to C6.4 ≤1.4688 using the same piecewise-constant setup as the preceding autocorrelation problems.The prior bound was C6.4 ≤1.4993.Source: §2 · p. 16

Problem 6.5

• For the minimum-overlap constant, AlphaEvolve slightly improved the known upper bound to C6.5 ≤0.380924.The previously known bounds were 0.379005 ≤C6.5 ≤0.3809268534330870; the constant controls the asymptotics of Erdős’s “minimum overlap problem”, [118, Problem 36].Source: §2 · p. 16

Search methodology

• The initial direct-function attack on another autocorrelation problem performed poorly because AlphaEvolve was constrained to predefined transformations and exploited numerical integration with highly irregular functions.The authors identify bounded step functions with fixed step sizes and search mode as a better setup, which had improved all previous bounds in this section.Source: §2 · p. 17

3. Difference bases.

• AlphaEvolve improved the difference-bases upper bound from 2.6571 to 2.6390 after being given code for generating Singer difference sets [260].Without expert advice, it did not beat 2.6571; the resulting construction is provided in the Repository of Problems.Source: §3 · p. 17

4. Kissing numbers.

• AlphaEvolve raised the known lower bound for the kissing number C6.8(11) from 592 to 593.The system generated finite sphere configurations and optimized a pairwise-distance penalty; losses below O(10^-20) were sufficient for conversion to exact solutions using simple mathematical results.Source: §4 · p. 18, Table 2 · p. 18

5. Kakeya needle problem.

• For fixed inputs in the 2D Kakeya problem, AlphaEvolve found constructions with better union area than Keich’s within 5 to 10 evolution steps.Longer runtimes and guided prompting were expected to improve results further; the evolved configurations were generally irregular and non-self-similar compared with earlier schemes.Source: §5 · p. 19, Figure 5 · p. 20
• AlphaEvolve also improved performance relative to Keich’s construction when directly optimizing the S_T score.These results are summarized in Figure 6.Source: §5 · p. 19, Figure 6 · p. 20
• Generalizable Kakeya programs initially underperformed Keich sets for large n, including variants based on perturbed uniform meshes and geometric progressions.The experiments evaluated concise, fast, reproducible, human-readable algorithms across mixtures of small and large inputs.Source: §5 · p. 19
• Prompting AlphaEvolve with Keich’s construction produced bitwise programs for non-powers of 2, and further guidance mitigated performance jumps near powers of 2.The initial programs added offsets and weighting but showed suboptimal behavior around n = 2^k + 1; later prompting encouraged denser packing.Source: §5 · p. 19, Figure 9 · p. 22, Figure 10 · p. 23
• In the 3D Kakeya problem, AlphaEvolve found volume improvements over Keich’s construction for inputs n ≤ 64.The search used dense Monte Carlo volume estimates implemented in JAX and run on GPUs; the lower-bound result was not feasible to reproduce with AlphaEvolve.Source: §5 · pp. 22–23
• The Kakeya experiments remain focused on finding better generalizable constructions and asymptotic insight as n → ∞.This ongoing work covers both two-dimensional and higher-dimensional cases.Source: §5 · p. 23

6. Sphere packing and uncertainty principles.

• The uncertainty-principle search constrained self-Fourier test functions through Hermite coefficients or Laguerre double-root positions and scored candidates by their last sign change.Fast floating-point and slower rational-arithmetic verifiers exposed speed–accuracy tradeoffs.Source: §6 · p. 25
• AlphaEvolve obtained C_6.11 ≤0.3521 from a three-term Hermite construction, refining the configuration in [145] but not surpassing the C_6.11 ≤0.3102 state of the art announced in.The Hermite coefficients were [0.32925, −0.01159, −8.9216 × 10−5].Source: §6 · pp. 24–25
• The sphere-packing experiments were competitive with prior bounds such as but did not produce qualitatively new state-of-the-art results, while reducing experimental preparation effort.AlphaEvolve required a verifier and prompt, leaving optimization-scheme design to the system.Source: §6 · pp. 25–27
• For sphere-packing linear-programming bounds, AlphaEvolve matched Cohn–Elkies baselines up to four digits in lower dimensions and improved them in higher dimensions.The method optimized spherically symmetric Laguerre-polynomial combinations with prescribed single and double roots, using extended precision and rational arithmetic.Source: Figure 13 · p. 26, §6 · p. 26

7. Classical inequalities.

• Classical-inequality benchmarks tested whether AlphaEvolve could obtain efficient lower bounds using explicitly truncated, discretized step-function evaluators for scenarios with known theoretical optimal bounds.They included the Hausdorff–Young, Gagliardo–Nirenberg, Young’s, and Hardy–Littlewood maximal inequalities.Source: §7 · p. 27
• AlphaEvolve consistently recovered the correct extremizer structures for Hausdorff–Young, Gagliardo–Nirenberg, and Young’s inequalities, usually immediately or within two iterations.It identified the Gaussian for Hausdorff–Young, the hyperbolic secant for Gagliardo–Nirenberg, and Gaussian parameter couplings for Young’s inequality.Source: §7 · pp. 27–28
• C_6.18 ≥1.5080 surpassed the 3/2 barrier without reaching the optimal value, while additional hints produced improvements up to ∼1.533 for Hardy–Littlewood.Search used heuristic randomized mutations and large-scale exploration; generalization recovered the 3/2 bound but needed further guidance for improvement.Source: §7 · p. 29

8. The Ovals problem.

• For the Ovals problem, AlphaEvolve obtained the circle in most runs, matching the conjectured lower bound C6.19 = 1, but found no additional class of optimal ovals.The search modeled three normalized one-dimensional functions with spline interpolation points and evaluated the objective using quadratures and exact derivatives.Source: §8 · p. 30

9. Sendov’s conjecture and its variants.

• For Sendov’s conjecture, AlphaEvolve rediscovered the expected maximizers p(z) = z^n − e^(iθ) and near-maximizers such as p(z) = z^n − z, without finding additional maximizers.The search optimized over root sets, projecting out-of-disk roots to the unit circle and computing derivative roots numerically.Source: §9 · p. 31
• For Schmeisser’s stronger conjecture, AlphaEvolve found no counterexample; its best constructions placed roots and critical points near the circle boundary.Forcing some roots away from the boundary produced the suboptimal constructions shown in Figure 14.Source: §9 · p. 31
• For Borcea’s conjecture at p = 1, AlphaEvolve generated z^n − nz and z^n − nz^(n−1)-type constructions but found no counterexample, even after penalizing similarity to known examples.The conjecture asserts C6.22(p,n) = 1 for all 1 ≤ p < ∞ and n ≥ 2, and is stronger than Sendov’s conjecture.Source: §9 · p. 32
• For Smale’s problem, AlphaEvolve matched the best known lower bound by finding the z^n − nz optimizer, while also finding similarly scoring constructions but no counterexample.Figure 15 shows the optimizer z^12 − 12z and another discovered construction.Source: §9 · p. 32, Figure 15 · p. 33
• AlphaEvolve exposed a larger feasible region for the de Bruin–Sharma problem than the human analysis predicted, revealing an oversight caused by the zero-sum constraint.The discovered intercepts arose from real-rooted polynomials with nearly balanced multiplicities and from (z−a)^(n−1)(z+(n−1)a).Source: §9 · p. 33
• Tang [278] subsequently established the two conjectured de Bruin–Sharma inequalities proposed from AlphaEvolve’s analysis and numerical evidence.The proof used a new interpolation-based approach.Source: §9 · pp. 33–34

Problem statement

• Crouzeix’s conjecture asserts that the lower bound 2 is sharp for all polynomials, with equality cases also posed as an open question.The problem is formulated for operator norms of polynomial matrix evaluations relative to the numerical range.Source: §10 · p. 34

Known cases

• Known cases include monomials, matrices of size 2 or minimum-polynomial degree 2, and matrices whose numerical range is a disk.The passage notes that only a few cases have been solved in general.Source: §10 · p. 34

Numerical maximizers

• Numerical investigations conjectured that, modulo specified transformations, the only maximizer has a monomial polynomial and a structured matrix form.The transformations include scaling, shifting, unitary similarity, and permitted replacement of a zero block.Source: §10 · p. 34

AlphaEvolve experiment

• AlphaEvolve optimized the ratio of the operator norm to the supremum over the numerical range using the Kippenhahn–Johnson characterization, but found no example exceeding 2.The search used matrices of variable sizes and matched, rather than improved, the literature bound.Source: §10 · pp. 34–35

11. Sidorenko’s conjecture.

• Sidorenko’s conjecture asks whether every finite bipartite graph has the Sidorenko property, namely C6.26(H) = |E(H)|.The constant C6.26(H) is defined through a homomorphism-density inequality over all graphons, with constant graphons establishing the lower bound.Source: §11 · p. 35
• Prior work proves the conjecture for several families, including complete bipartite graphs, even cycles, trees, hypercubes, and other structured bipartite graphs [72] [197] [273].These results include bipartite graphs with at most four vertices on one side and graphs containing a vertex complete to the other side.Source: §11 · p. 35
• AlphaEvolve searched for a graphon violating Sidorenko’s inequality on the smallest bipartite graph whose Sidorenko property remains unknown, but found no counterexample.The search penalized graphons close to constant because constant graphons trivially give equality.Source: §11 · p. 35

12. The prime number theorem.

• The prime number theorem addresses asymptotic quantities involving π(x), and its classical conclusion is established by the theorem itself and, in the cited discussion, by elementary means.The supplied passage introduces this as an initial test of AlphaEvolve’s applicability to analytic number theory.Source: §12 · pp. 35–36
• A lower-bound method uses finitely supported weights satisfying a specified summation condition, and can recover Sylvester’s result.The method derives a lower bound from the admissible weight function.Source: §12 · p. 36
• Good weight choices are often truncated Möbius functions, where μ(n) equals (−1)^j for products of j distinct primes and zero otherwise.This structure motivated the constructions tested by AlphaEvolve.Source: §12 · p. 36
• AlphaEvolve’s blind search recognized Möbius-function structure and generated composite-factor and truncated-Möbius constructions, but its final score was 0.938, below the best known lower bound.The system received only a hidden number-theoretic evaluation objective and was asked to infer general forms from prior candidates.Source: §12 · p. 36

• The underlying problem asks whether large-degree ±1-coefficient polynomials can be flat on the unit circle, while ultraflat polynomials remain an open possibility [200].Ultraflatness is equivalent to lim inf_n→∞ C^w_6.28(n)=0, whereas Erdős conjectured that ultraflat Littlewood polynomials do not exist.Source: §13 · pp. 36–37
• Odlyzko’s extensive numerical calculations suggested limiting values C+_6.28(n)≈1.27, C−_6.28(n)≈0.64, and C^w_6.28(n)≈0.79.These calculations used roughly 30 CPU-years and reached n as large as 100.Source: §13 · p. 37
• AlphaEvolve searches for lower bounds on C−_6.28 and upper bounds on C+_6.28 by estimating boundary extrema on a dense unit-circle mesh.For n≤100, the experiments use K=6, 7 mesh resolution as a balance between evaluator accuracy and evolutionary search speed, then validate constructions with larger K.Source: §13 · p. 38
• For degrees larger than 40, AlphaEvolve’s polynomial constructions deteriorate and show wider score variation, without definitive convergence as n increases.For n≤40, the constructions can appear comparable in magnitude to some prior results, including Odlyzko’s calculations.Source: §13 · p. 38

• AlphaEvolve produced the correct general solution after one or two iterations in generalizer mode.The result was obtained from an LLM-generated argumentation for a recursive relation.Source: §14 · p. 39
• The derived construction computes x_i backward from x_{n-1}=0 using the average of suffix values minus 0.5.The recurrence is x_i = (sum_{j=i+1}^{n-1} x_j) / (n - i - 1) - 0.5.Source: §14 · p. 39
• The method selects P just below its upper bound by 2 * FLOAT_TOLERANCE to satisfy the strict scoring inequality.The upper-bound target is 0.5 - (sum_k x_k) / n, while P must also respect lower-bound constraints.Source: §14 · pp. 39–40
• The refined construction sets positions[k] = 0.5 * (H_n - H_{n-k-1}) - 2 * FLOAT_TOLERANCE and satisfies all get_positions_score constraints.This formulation leverages harmonic numbers and was implemented as the next program produced by AlphaEvolve.Source: §14 · pp. 40–41

15. The arithmetic Kakeya conjecture.

• The arithmetic Kakeya conjecture asserts that C6.30({r1, …, rk}; r∞) can be made arbitrarily close to 1.The constants govern entropy inequalities for projections of discrete random variables.Source: §15 · p. 41
• AlphaEvolve obtained the lower bound 1.668 ≤ C6.30({0, 1, 2, ∞}; −1), improving the previous result; its generated distributions resembled discrete Gaussians.The system also improved another bound only in the eighth decimal and inspired a rigorous asymptotic for C6.30({0, 1, ∞}; s), to appear in [282].Source: §15 · p. 41

16. Furstenberg–Sárközy theorem.

• The Furstenberg–Sárközy theorem [136], gives C6.31(k, N) = o(N) for fixed k, and consequently C6.31(k, Z/MZ) = o(M).The problem seeks the strongest possible upper and lower bounds for sets avoiding differences equal to perfect kth powers.Source: §16 · pp. 41–42
• For square and cube differences, AlphaEvolve quickly reproduced the known lower bounds using moduli 205 and 91, but found nothing better.These searches targeted improved lower bounds for C6.31(2, N) and C6.31(3, N).Source: §16 · p. 42

17. Spherical designs.

• AlphaEvolve optimized a nonnegative spherical-harmonic error that is zero exactly when the points form a t-design, accepting configurations below 10^-8.The search found the sanity-check constructions C6.32(1, t) = t + 1 instantly.Source: §17 · p. 43
• Sizes 198, 200, 202, and 204 for C6.32(2, 19), and 234 and 236 for C6.32(2, 21), improved the literature bounds.AlphaEvolve also found new, non-improving sizes 122, 124, 126, 128, and 130 for C6.32(2, 15).Source: §17 · p. 43
• The spherical-design configurations provide high-precision solution candidates, not certified solutions; certification with Arb [171] or FLINT and a fixed-point method is left for future work.The paper notes that a computer-assisted proof could certify a solution near each approximation.Source: §17 · p. 44

18. The Thomson and Tammes problems.

• For the Thomson problem, AlphaEvolve searched point positions on the sphere using only the Coulomb-energy objective and matched reported results for N<50 to within 10^-8 within one hour.Searches covered N up to 300; larger values required O(10) hours to reach saturation, and the resulting upper bounds matched state-of-the-art numerics to high precision.Source: §18 · p. 44, Table 4 · p. 45
• Generalized Thomson programs evaluated energies across selected N values up to 500, commonly using uniform sampling followed by a few refinement steps.These programs had reasonable runtime performance, but formal asymptotic analysis remained non-trivial because of optimization components such as gradient descent.Source: §18 · p. 45, Figure 20 · p. 46
• For the Tammes problem, AlphaEvolve matched theoretically known small-N configurations, while larger-N searches combined case-by-case checks with optimization procedures.The search used the Tammes energy as its scoring function and ran for O(10) hours; shorter prompts or reduced search time helped avoid obscure or overly long programs.Source: §18 · p. 46, Table 5 · p. 46, Figure 21 · p. 47

19. Packing problems.

• For general packing in a dilated shape, AlphaEvolve searched dozens of two- and three-dimensional instances using intersection penalties, binary search over scale, and fixed orientations.The score combined the bounding scale with penalties for intersecting copies, designed so locally optimal configurations contain no intersections.Source: §19 · p. 47
• AlphaEvolve improved the best known hexagon packings for n=11 and n=12, although human analysis revealed that the resulting constructions were not completely symmetric.The packings were subsequently analyzed and refined by Johann Schellhorn, highlighting the benefit of human–AI collaboration.Source: §19 · p. 47, Figure 22 · p. 48
• For cubes, AlphaEvolve matched known arrangements for n=9, 10, and 12 and improved the n=11 upper bound from 2 + 3/5 ≈2.912096 to 2.894531.The new construction packs 11 unit cubes into a larger cube of side length approximately 2.895.Source: §19 · p. 47, Figure 23 · p. 48
• AlphaEvolve also produced circle-packing constructions that improved existing bounds, with later experiments surpassing subsequently refined numerical results.The problem supports continued numerical improvement as computational investment increases, and full-precision constructions are provided in the Repository of Problems.Source: §19 · p. 48

20. The Turán number of the tetrahedron.

• On the Turán tetrahedron problem, AlphaEvolve quickly rediscovered the 5/9 construction but found no better lower bound for n = 10, 15, 20, 25, 30.The conjectured sharp value is C6.37 = 5/9, with the upper bound obtained by Razborov.Source: §20 · p. 49
• For factorial factoring, AlphaEvolve improved the benchmark in several cases and matched it in all others tested for 80 ≤ N ≤ 600, often approaching integer-programming optima.The benchmark was Sutherland’s algorithm; later versions of [5] introduced exact integer programming for the tested range.Source: §20 · p. 50, Table 6 · p. 50

22. Beat the average game.

• AlphaEvolve obtained the lower bound C6.39 ≥ 0.389 within a few hours without expert hints, while revealing sparse, patterned discrete measures that could guide later proofs.This bound did not outperform the final results of [21].Source: §22 · pp. 50–51

23. Erdős discrepancy problem.

• With a multiplicativity hint, AlphaEvolve found an Erdős-discrepancy-2 sequence of length 380, versus length 200 without guidance, but remained below the known value 1160.The experiments used the same amount of time, and possible SAT-solver guidance was not explored.Source: §23 · p. 51

24. Points on sphere maximizing the volume.

• For the sphere-volume problem, AlphaEvolve matched the first approximately 60 results in to all 13 reported digits and improved none of them.The remaining approximately 70 reported results were not attempted.Source: §24 · pp. 51–52

25. Sums and differences problems.

• 1.1584 ≤ C6.44 was achieved by AlphaEvolve, improving the prior bound; subsequent work raised the lower bound further to C6.44 ≥ 1.173077.AlphaEvolve found sets of 2,003 and 54,265 integers, yielding successive bounds of 1.1479 and 1.1584.Source: §25 · p. 53

26. Sum-product problems.

• For primes p with N=⌊p^1/2⌋, AlphaEvolve generated constructions whose max(|A+A|, |A⋅A|) was slightly below N^3/2 when p∼10^9.The construction intersected arithmetic and geometric progressions and was evaluated across multiple primes to encourage generalization.Source: §26 · p. 53
• The construction arose from smooth Gaussian-integer sets with small sum and product sets, then projection to F_p when p=1 mod 4.AlphaEvolve selected norm and smoothness parameters, including thresholds such as 13 or 17.Source: §26 · pp. 53–54
• Human analysis judged the apparent exponent improvement a finite-size artifact, reflecting slow double-logarithmic convergence and favorable Gaussian-integer features rather than asymptotic progress.The authors therefore caution against interpreting experiments at non-asymptotic sizes as reliable asymptotic evidence.Source: §26 · p. 54

27. Triangle density in graphs.

• For triangle density, AlphaEvolve evolved 100 rank-1 step-function graphons without being told the underlying problem, representing each as a list of real numbers.Their edge and triangle densities formed points whose lower envelope was intended to approximate C6.46(ρ).Source: §27 · p. 54
• The initial area-under-chords objective produced an invalid construction with a better-than-possible score because the feasible region was nonconvex.AlphaEvolve learned to place points to cut off concave regions, exposing the scoring flaw.Source: §27 · p. 55
• Using horizontal extensions to the left and slope-3 extensions to the right corrected the objective, and AlphaEvolve converged to the correct solution curve.The resulting area measured points lying under all induced piecewise-linear bounds; Figure 25 shows approximation improving over time.Source: §27 · p. 55, Figure 25 · p. 55

28. Matrix multiplications and AM-GM inequalities.

• The matrix problem asks for C6.47(n, m, d) over positive-semidefinite matrices and unitarily invariant norms, including operator and Schatten p-norms.The setting includes any m ≤ n and infimizes over matrices and invariant norms.Source: §28 · p. 56
• For matrix AM-GM inequalities, AlphaEvolve found constructions attaining C6.47(n, m, d) ≤ 1 but none improving the bound or refuting Duchi’s conjecture.Duchi conjectured C6.47(n, m, d) = 1 for all n, m, d; the case m ≥ 4 remains open.Source: §28 · p. 56

29. Heilbronn problems.

• The Heilbronn searches optimized the smallest triangle area after projecting proposed points onto the boundary of the convex body when necessary.The experiments considered many combinations of n and K, including the unit square, equilateral triangle, and arbitrary convex regions.Source: §29 · p. 56
• 11 points in a unit-area equilateral triangle achieved a minimum formed-triangle area of 0.0365, improving the reported result.AlphaEvolve did not beat records for the unit square but improved the n = 11 equilateral-triangle case; Figure 26 shows the construction.Source: Figure 26 · p. 57, §29 · p. 57
• For arbitrary convex bounding boxes, AlphaEvolve matched numerical candidates for n ≤ 12 and improved Cantrell’s constructions for n = 13 and n = 14.Figure 26 depicts the new best bounds, with minimum triangle areas 0.0309 for 13 points and 0.0278 for 14 points.Source: §29 · p. 57, Figure 26 · p. 57

30. Max to min ratios.

• C6.50(2, 16) ≈ 4.165849767 was improved by AlphaEvolve in an experiment lasting only a few hours, before later improvement in.The search minimized the maximum-to-minimum distance ratio across several (d, n) pairs and matched the best known results for most tested parameters.Source: §30 · p. 57
• Figure 27 shows improved low max-to-min-ratio configurations for 16 points in 2 dimensions and 14 points in 3 dimensions.The method searched multiple correlated (d, n) pairs simultaneously, exploiting the expectation that successful heuristics would transfer across parameters.Source: §30 · p. 57, Figure 27 · p. 58

31. Erdős–Gyárfás conjecture.

• AlphaEvolve found no counterexample to the Erdős–Gyárfás conjecture after searching graphs with up to 40 vertices, while the conjecture remains open.The search penalized minimum degree below 3 and cycles whose lengths are powers of 2; prior work proves the claim for sufficiently large minimum degree.Source: §31 · p. 58

32. Erdős squarefree problem.

• For the Erdős squarefree problem, AlphaEvolve readily recovered the residue-class construction but found no better one, while showed the lower bound is sharp for all sufficiently large N.A positive score encoded a valid construction by rewarding set density and penalizing pairs with squarefree ab + 1.Source: §32 · pp. 58–59

33. Equidistant points in convex polygons.

• For convex polygons, AlphaEvolve constructed examples where every vertex has at least three equidistant others, but none with four, leaving the stated problem unresolved.The score normalized diameter and penalized degeneracy through the minimum side length and equidistance error.Source: §33 · p. 59

34. Pairwise touching cylinders.

• The system reached loss O(10^-23) for seven mutually touching unit cylinders in two hours, whereas it could not improve the eight-cylinder loss beyond 0.003; it also found the optimal nine-cylinder unequal-radius configuration.The seven-cylinder construction is shown in Figure 28, and the eight-cylinder result supports the conjecture that seven is optimal.Source: §34 · pp. 59–60, Figure 28 · p. 60

35. Erdős squares in a square problem.

• For squares in a square, AlphaEvolve matched the best known constructions for n∈{10, 12, 14, 17, 26, 37, 50} but did not find them for some larger n.The score was the sum of side lengths, with invalid configurations assigned minus infinity.Source: §35 · p. 60

36. Good asymptotic constructions of Szemerédi–Trotter.

• In initial Szemerédi–Trotter experiments, restricting to rational lattice configurations let AlphaEvolve recover grid constructions with greedily selected rich lines, while broader searches remained ongoing.The lattice restriction avoided infinite-precision incidence tests but was not without loss of generality.Source: §36 · p. 61

37. Rudin problem for polynomials.

• For the Rudin problem, preliminary searches on S^2 minimized exact L4/L2 scores for spherical harmonics up to degree 30, using Wigner 3-j symbols and results summarized in Figure 29.The work began with C^4 because it is cheaper to compute than the supremum-based quantity, and further exploration in two and higher dimensions was planned.Source: §37 · pp. 61–62, Figure 29 · p. 62

Scope

• The studied tasks ask for extremal constructions or bounds: convex polygons among planar points, triangle-free grid subsets, sphere-avoiding lattice subsets, ring-loading discrepancy, and maneuverable sofas.These problems span combinatorics, geometry, optimization, and number-theoretic-style discrete construction search.Source: §38 · p. 62, §39 · p. 63, §40 · p. 63

38. Erdős–Szekeres Happy Ending problem.

• For the Happy Ending problem, AlphaEvolve found 2^k−2 points with no convex k-gons and 2^k−2+1 points with exactly one convex k-gon for every k≤8, but did not improve the lower bound.The loss counted convex k-gons, with invalid near-coincident or nearly collinear configurations assigned negative infinity.Source: §38 · pp. 62–63

39. Subsets of the grid with no isosceles triangles.

• AlphaEvolve found the previously elusive 112-point configuration on the 64 × 64 grid and improved the 100 × 100-grid record from 160 to 164 points.The search used advice favoring near-fourfold symmetry, slightly offset symmetry axes, and points concentrated near the grid boundary; the 164-point construction was believed nonoptimal.Source: §39 · p. 63, Figure 30 · p. 64

40. The “no 5 on a sphere” problem.

• AlphaEvolve improved the no-5-on-a-sphere lower bounds to C6.60(7) ≥21, C6.60(8) ≥23, C6.60(9) ≥26, C6.60(10) ≥28, C6.60(11) ≥31, and C6.60(12) ≥33.The gains came from exploring thousands of exotic local-search methods, despite using slower CPU evaluators than the GPU-based setup in [56].Source: §40 · p. 64

41. The Ring Loading Problem.

• Compared with the original work, which required checking hundreds of millions of specially structured instances, AlphaEvolve reduced the effort from weeks to hours, although clever humans could eventually discover the same constructions.Source: §41 · p. 65

42. Moving sofa problem.

• For the sofa search, AlphaEvolve proposed motion paths and computed the largest shape surviving each corridor traversal, using point clouds and rigorous cell verification in 3D.Mesh-based polytope-intersection libraries were infeasible because of performance costs and mathematically difficult accumulated errors.Source: §42 · pp. 65–66
• AlphaEvolve found a 3D sofa with rigorous volume lower bound 1.81 and estimated volume 1.84, exceeding the two candidates proposed in.It also recovered Gerver’s and Romik’s sofas within 0.02% and 1.5% for the classic and ambidextrous problems, respectively.Source: §42 · p. 66
• For the classic and ambidextrous sofa problems, AlphaEvolve recovered Gerver’s and Romik’s constructions nearly exactly, while the classic optimum is known as C6.62 = 2.2195 … and the ambidextrous optimum remains open.Gerver’s bound was proved sharp by Baek [10], whereas Romik’s lower bound remains unresolved.Source: §42 · pp. 65–66

43. International Mathematical Olympiad (IMO) 2025: Problem 6.

• The IMO problem asks for the minimum number of nonoverlapping rectangular tiles leaving exactly one uncovered unit square in every row and column.The grid may be 2025 × 2025 or, more generally, n × n.Source: §43 · p. 67
• The experiment targeted construction rather than proof: establishing optimality remained beyond AlphaEvolve, and even the constructive component exceeded Deep Think’s capabilities.A further experiment would test whether AlphaEvolve can solve the problem for all n rather than only perfect squares.Source: §43 · pp. 68–69
• AlphaEvolve searched tiling programs by scoring rectangle count plus penalties for overlaps or incorrect uncovered-square counts, using small-instance constructions to extrapolate general patterns.Programs were evaluated across small and large inputs, with average normalized scores guiding evolution.Source: §43 · p. 68
• AlphaEvolve found the optimal tiling solution for every perfect-square n, although its performance was inconsistent on non-perfect-square inputs.The system was evaluated only on perfect squares, and the construction was not generally optimal for all n.Source: §43 · p. 68, Figure 35 · p. 69

44. Bonus: Letting AlphaEvolve write code that can call LLMs.

• AlphaEvolve was extended to evolve code that could call LLMs, enabling experiments on function guessing and Smullyan-type logic puzzles.These examples tested whether generated programs could use external language-model interactions as part of their strategies.Source: §44 · p. 69, §44.2 · p. 70

44.1. The function guessing game.

• In function guessing, LLM-enabled programs outperformed point-query baselines, but noisy oracle answers, unreliable generated search code, and cheap-model leakage made the result unreliable.The oracle was restricted to yes/no answers, yet the system sometimes exploited weak models to reveal hidden functions.Source: §44.1 · pp. 69–70

44.2. Smullyan-type logic puzzles.

• The logic-puzzle setup was inefficient because the second question could be unnecessary after a “yes” answer, despite the evolved program retaining a placeholder question.Evaluation simulated all 36 door-and-guard permutations using cheap LLMs as guards.Source: §44.2 · pp. 70–72
• In the guard puzzle, AlphaEvolve evolved a perfect $110 strategy, then improved practical performance by rephrasing questions and using prompt injection against unreliable guard LLMs.The logically perfect strategy initially averaged $83 because cheap guards mishandled convoluted questions; a prompt-injection formulation reached $110.Source: §44.2 · p. 71