Finite construction records around AlphaEvolve

• Domain: mathematics
• Role: discovery series
• Metric: cumulative record steps in five groups of finite construction and packing problems
• Coverage: 1949–2026, 22 record steps across the five groups; transcription current to 2026-08-12
• Data: alphaevolve-records.csv, with the sampling frame in ../math-alphaevolve-inventory/alphaevolve-inventory.csv
• Upstream: https://github.com/google-deepmind/alphaevolve_repository_of_problems and https://arxiv.org/abs/2511.02864, with per-step sources recorded in the CSV's ref and note columns
• Verdict: inconclusive — 1 record step in 2026 against 9 in 2025 and a 0.2/year mean over 1949–2024

Definition

Five groups of finite problems from the AlphaEvolve problem set, each with a standing record held by an explicit construction: the Erdős minimum-overlap type constant (6.5), the difference-basis constant (6.7), Heilbronn-style triangle packing (6.48) and convex packing (6.49), and a max-min packing ratio (6.50). In each case the record is a finite object — a step function, a difference set, a packing of $n$ shapes — that can be scored exactly, and the incumbent record often lives on a community page maintained by continuous computer search rather than in a journal. A "discovery" in this series is one such record being improved.

The sampling frame is the inventory folder's 65 numbered problems, of which 31 carry a live numeric record under the companion repository's classification; the five groups are drawn from those 31.

Facts

• steps: 22 record steps across the five groups, 1949–2026; 7 AI-set, all in 2025
• by-year: 1949: 1 · 1955: 1 · 1956: 2 · 1972: 1 · 1996: 1 · 2006: 1 · 2007: 2 · 2009: 2 · 2016: 1 · 2025: 9 · 2026: 1
• by-group: minimum-overlap 5 steps (1955–2025) · difference basis 4 steps (1949–2025) · triangle packing 2 steps (2006–2025) · convex packing 4 steps (2007–2025) · max-min packing 7 steps (2009–2026)
• ai-step sizes: +1.44% on triangle packing · +0.98% and +0.47% on convex packing · +0.69% on difference basis · +0.052% and +0.0057% on the two max-min slices · +0.00075% on minimum-overlap
• minimum-overlap levels: the 2025 AI step moves the upper bound from 0.380926853433087 (Haugland, 2016) to 0.380924
• pooled step-size medians (whole file): AlphaEvolve +0.98% over 28 steps · AI-guided search +3.90% over 2 · agent platform +1.85% over 1 · human computer search +2.52% over 13 · human by hand +2.83% over 9

The difference-basis record had stood since Golay in 1972. On that group the paper reports:

"AlphaEvolve by itself, with no expert advice, was not able to beat the 2.6571 upper bound. In order to get a better result we had to show it the correct code for generating Singer difference sets [258]. Using this code AlphaEvolve managed to find a substantial improvement in the upper bound from 2.6571 to 2.6390." — Georgiev, Gómez-Serrano, Tao and Wagner, arXiv 2511.02864, 2025 [@georgiev2025mathexploration]

Both max-min AI records were retaken in 2025 by FICO's Xpress global solver, run with no custom algorithm and verified with DeepMind's verification tool, per the FICO blog of 2025-06-13 and arXiv:2601.05943 recorded in the CSV's ref column. The one 2026 step, on the max-min $(3,14)$ slice, is credited by the community records page to M. Sun and E. Samanta.

The collection-wide cumulative index redraws this series as cumulative record steps to date:

Method

The CSV is built by fetch.py, which carries the hand transcription itself: the values, the quoted sentence each was read from, the reference, and the agent coding are all literals in that file, and the relative gains are computed from consecutive values as it writes. It is the only place any of these records is maintained, and it also writes the slice that sums and autoconvolution plots, so that child dataset cannot drift from the file it is drawn out of. This folder keeps the full transcription rather than a slice of it because alphaevolve-record-steps.png pools every record step in the frame, across all the problems the paper numbers, not only the five plotted in the first figure. check.py recomputes the fact lines from the CSV.

figure.py draws all three figures from alphaevolve-records.csv. For the first it holds the five problem ids in a hardcoded selected set, keeps rows with a non-empty year and is_record equal to yes, groups by problem, sorts each group by year then quantity then step, and plots the cumulative step count per group; the group labels in the legend are hardcoded in the function. The y-axis is a count because the values are not commensurable: an area, a squared radius, and two dimensionless constants cannot share a numeric axis, so this figure plots discoveries rather than levels, as the sphere-packing ladder does for the same reason. A callout names the Singer-code hint behind the difference-basis step.

Step markers are colour-coded by who set them: record_marker() reads agent, draws red for anything beginning ai_ and blue otherwise, and draws an unfilled marker where date_certain is no. The FICO solver steps and the 2026 step are blue: the coding distinguishes AI from non-AI rather than automated from manual, and much of the blue here is computer search.

The second figure's left three panels plot the best known value against record step rather than year, because several steps share a year: problem 6.44's sums-and-differences bound over eight steps, problem 6.3's autoconvolution constant over four, and problem 6.8's kissing number in dimension 11 over five. Marker colour is who made the step — AlphaEvolve, AI-guided search, an agent platform, human computer search, human work by hand, or a community records page — and each point is annotated with its year. The right panel pools every record step in the file that has a computable size and an is_record of yes, one row per kind of agent on a symmetric log axis, with a vertical bar at each row's median and the median printed above it.

Two rows in the same CSV are flagged is_record: no and are excluded from every figure. Both are spherical-design constructions the paper described as improving on the literature bounds it cited, where a 2016 result the paper does not cite was already better.

Limitations

• selection. The five groups are in the data because an AI system was pointed at them; the 2025 cluster follows from that selection, so no discovery rate can be read from this series.
• counts, not sizes. The axis counts steps, and the AI steps are among the smallest in the series, several in the fourth or fifth decimal place.
• groups mix quantities. Convex packing pools $n=13$ and $n=14$, and max-min pools two slices, so a group reaching seven steps is not one record improving seven times.
• overlapping markers. Two groups both take their second step in 1956, and the minimum-overlap one — whose date rests on secondary sources only, and which is therefore drawn as an open marker — is hidden underneath the other group's filled marker.
• the human baseline is itself continuous computer search. Several incumbent records come from Erich Friedman's packing pages, whose "+" convention truncates values and so slightly understates prior records and overstates AI gains; one page shows AlphaEvolve's value for the $n=13$ convex case while still crediting Cantrell in 2007, so page and paper disagree about who holds it.
• one value is missing. The 2026 max-min step has an empty value, because the community page truncates the figure and the exact number could not be established; the step is counted, the level is not recorded.
• hint-dependence. The 2025 difference-basis step, on a record standing since Golay in 1972, followed a supplied Singer difference-set code hint; the paper reports the unaided system did not beat the prior bound (quoted above).

AI attribution

Seven of the twenty-two steps in the five groups are AI-set, all AlphaEvolve, all in 2025 [@novikov2025alphaevolve; @deepmind2025problems]. Their sizes run from +1.44% down to +0.00075%; two were retaken in 2025 by FICO's Xpress global solver, and the difference-basis step followed the Singer-code hint quoted above. The paper's own summary of the system's reach:

"We have found that AlphaEvolve excels at discovering constructions that were already within reach of current mathematics, but had not yet been discovered due to the amount of time and effort required to find the right combination of standard ideas that works well for a particular problem." — Georgiev, Gómez-Serrano, Tao and Wagner, arXiv 2511.02864, 2025 [@georgiev2025mathexploration]

Tao's account of the same run, on named conjectures:

"For well-known open conjectures (e.g., Sidorenko’s conjecture, Sendov’s conjecture, Crouzeix’s conjecture, the ovals problem, etc.), AlphaEvolve generally was able to locate the previously known candidates for optimizers (that are conjectured to be optimal), but did not locate any stronger counterexamples" — Terence Tao, terrytao.wordpress.com, 2025-11-05 [@tao2025exploration]

The dated precedent for AI-set records on recognised open mathematical problems is FunSearch's cap-set improvement of December 2023 [@deepmind2023funsearch].

Sources

• [@georgiev2025mathexploration] — the mathematics paper whose appendix the 2025 steps and the bounds they were measured against are transcribed from; quoted above.
• [@novikov2025alphaevolve; @deepmind2025problems] — the AlphaEvolve system paper and the companion repository supplying the status classification used as the frame.
• [@tao2025exploration] — Tao's account of the mathematics paper, quoted above.
• [@deepmind2023funsearch] — FunSearch, the line of work's first recorded AI step.
• [@arxiv2026horizonmath] — HorizonMath, a benchmark comparing AI output to the current best published results, with no historical dimension.
• [@sherry2021fast] — records in algorithm families arrive in bursts with long gaps.
• The sampling frame is the inventory; the individually plotted child series from the same frame is sums and autoconvolution.