R-2969B0: the 2013 construction beats the AlphaEvolve value, under the repository's own definition

Verdict: option 1 of the three the brief names. Penman and Wells (2013) reach a strictly higher value of exactly the quantity DeepMind's Problem 42 defines, and they do it with an explicit finite set that is smaller than the AlphaEvolve one. The normalisations agree; this is not apples to oranges. The challenger issue is right on its prior-art half, including its citation.

Everything below is exact finite-set arithmetic over the integers. No floating point enters the verdict: the comparison g(Q_36) > g(AlphaEvolve) is decided at 120 decimal digits (probe.py:g_beats), and the two sides differ in the sixth digit.

1. The two statements, side by side (VERIFIED)

DeepMind, problems/42.html, title "42. Sum-difference problem I":

Let $C$ be the least constant such that $|A+A|/|A| \leq (|A-A|/|A|)^{C}$ for any non-empty finite set $A$ of integers. Establish upper and lower bounds for $C$ as strong as possible.

So C = sup_A g(A) with g(A) = ln(|A+A|/|A|) / ln(|A-A|/|A|).

Penman and Wells, INTEGERS 13 (2013) A57, section 4, page 22:

Some authors, e.g., Granville in [2], prefer to use g(A) = ln(|A+A|/|A|)/ln(|A-A|/|A|) for which the analogous bounds are 1/2 <= g(A) <= 2.

and their Theorem 21 (page 22), quoted whole:

Given eps > 0, there is a set C of integers for which g(C) = ln(|C+C|/|C|)/ln(|C-C|/|C|) > ln(32/5)/ln(26/5) - eps ~= 1.125944426. Proof. Take Q_j for j sufficiently large.

The normalisations are identical, character for character. This is the step the brief said everyone skips, and it is the one that decides the outcome. It was worth checking, because the same paper carries a different quantity in its abstract, f(A) = ln|A+A|/ln|A-A|, which is where the paper's headline "previous record" claim lives, and which is not Problem 42's C. Confusing the two is the obvious way to get this wrong in either direction. On the AlphaEvolve set, f = 1.0229 and g = 1.1219; on Q_1000, f = 1.0212 and g = 1.1258. Ranking by f would reverse the answer, so the distinction is load-bearing and is stated here rather than assumed.

That we are computing the paper's g and not something adjacent is pinned by four numbers the paper prints and this probe recomputes, none of which were used to build anything:

2. The AlphaEvolve construction, evaluated (VERIFIED)

Taken from the repository's own notebook, experiments/sums_differences_problems/sums_differences_problems.ipynb, cell 1 (best_list), and re-counted by enumeration:

which is the 1.1219 the issue attributes to it. All 309 listed values are distinct.

One thing found on the way, MEASURED and not part of the verdict: the notebook carries three different score functions for this problem, and two of them are not the quantity the problem page defines.

• Cell 2 (the prompt AlphaEvolve was given) returns log(|A-A|/|A|) / log(|A+A|/|A|): the reciprocal, 0.8913 on best_list. Maximising it would minimise g.
• Cell 5 ("Verification function used") returns the right ratio plus a size bonus (1 - 1/|A|)/100, i.e. 1.1319 on best_list, inflated by 0.00998.
• Cell 1 (the "Data and verification" cell, run last) returns the clean ratio.

The published 1.1219 is the clean one, so the bonus did not leak into the reported record. But a bonus of up to 0.01 sitting inside a scorer whose whole scale of interest is 1.06 to 1.13 is worth naming, since the search was steered by it.

3. The Penman-Wells family, instantiated (VERIFIED)

Q_j is Theorem 12, page 14, transcribed into probe.py:Q:

Q_j = {0,2,4,12} u {1,5,...,1+4(4j+8)} u {24,40,...,8+16j}
      u {4+16(j+1), 12+16(j+1), 14+16(j+1), 16(j+2)},   j >= 1

Corollary 13 (page 17) states |Q_j| = 5j+17, |Q_j+Q_j| = 32j+63, |Q_j-Q_j| = 26j+61, |Q_j (+) Q_j| = 32j+56 for the restricted sumset. Recomputed by enumeration for 64 values of j (all of 1..60, plus 80, 120, 200, 400) and again for every j <= 36 inside the search loop: zero mismatches, all four counts, every j. The corollary is not being quoted here, it is being reproduced.

g(Q_j) increases monotonically toward the limit and does not attain it, which is why Theorem 21 is stated with an eps. That costs nothing: C is defined as the least constant valid for all A, so C >= g(Q_j) for every j and hence C >= 1.125944426.

4. The head-to-head (VERIFIED)

Q_36 is the smallest member of the 2013 family that beats the 2026 machine search: 197 integers against 309, margin +1.48e-05, decided exactly. Every Q_j with j >= 36 also beats it, so this is a family and not a coincidence. The supremum clears the AlphaEvolve value by +0.00401, about 33× the Q_36 margin.

For the record's other end: Granville's bounds 1/2 <= g <= 2, quoted by Penman and Wells, put C <= 2, so nothing here is near closing the problem. The finding is about the lower bound only.

5. What this settles, and what it does not

Settled. The world_record marking on Problem 42 in status.json does not hold as a record for the lower bound on C: a 2013 paper, published, human, and cited as best known by Staps (INTEGERS 15 (2015) A42, per the issue, a claim this hunt did not independently verify), gives a strictly better value under DeepMind's own definition, and gives it as a theorem about an explicit infinite family rather than as one searched set. The challenger issue's prior-art half is correct, and its attribution to Theorem 21 is exact.

Not settled here, deliberately. The same issue's C = 2 claim, which is the large one. It is untouched, per the brief. Nothing in this hunt supports or weakens it; anything read off this page about it would be INFERRED at best.

Not checked. That Staps (2015) records the value as best known, taken from the issue, not read. That no construction published between 2013 and now beats ln(32/5)/ln(26/5); this hunt establishes a floor for the 2013 result, not the current state of the art. And whether the AlphaEvolve authors were aware of the Penman-Wells paper; the repository cites arXiv:2511.02864 section 6.25, which was not read.

A note on what "beats" means. The comparison is between a number reported as a record and a number provable from a 2013 theorem. AlphaEvolve's search found a 309-element set achieving 1.12194 in 1000 seconds; that is a real artifact and this hunt reproduces it exactly. What fails is the classification, not the search.

Loose threads

1. The stated baseline 1.0598 does not match anything in the 2013 paper. Penman and Wells list the suprema of g over their four families on page 22: ln(16/3)/ln(14/3) ~= 1.0867 (T_j, T'_j), ln(23/4)/ln(11/2) ~= 1.0261 (R_j), ln(11/2)/ln(5) ~= 1.0592 (M_j). The issue reports AlphaEvolve's baseline as 1.0598, six units in the fourth decimal off the M_j value and not equal to any of the others. Why it might matter: if the baseline was taken from this paper at all, then the paper was in hand and the 1.12594 on page 22 was passed over; if it came from elsewhere, the provenance of the baseline is a separate unknown. First step: read section 6.25 of arXiv:2511.02864 and find what 1.0598 is cited to.

1. The reciprocal scorer in the AlphaEvolve prompt (cell 2). The prompt AlphaEvolve was given asks it to maximise log(|A-A|/|A|)/log(|A+A|/|A|), the reciprocal of the target, while the final verification cell uses the right ratio. Why it might matter: Problem 43 in the same notebook genuinely is the reciprocal quantity (|A-A| <= |A+A|^C), and it is classified worse_than_record. A crossed scorer between two adjacent problems is a cheap explanation for one of those two outcomes. First step: re-score the Problem 43 published set under both ratios and see which one its reported value matches.

1. The size bonus (1 - 1/|A|)/100 inside the search scorer. It rewards larger sets by up to 0.01 on a scale where the whole contested range is 0.06 wide. Why it might matter: it may explain why the machine landed on a 309-element set when a 197-element member of a 2013 family does better. First step: re-run a short local search on the clean ratio and see whether it prefers smaller sets.

1. f and g rank these sets in opposite directions. Q_1000 beats best_list on g and loses to it on f. Why it might matter: both quantities appear in this literature under the same words "sum-difference", and Problem 42's page does not say which. It happens to define g unambiguously in its formula, but a reader going by the title alone could pick either. First step: nothing needed here; it is a caution for whoever drafts an upstream reply.

1. The upper bound side is untouched. C <= 2 is what Granville's bounds give and what the issue's C = 2 claim would make sharp. This hunt says nothing about it. First step: it is the issue's part 1, and it is the expensive half; treat it as its own hunt.

Reproduce: python3 hunts/r_2969b0/probe.py (standard library only, ~4 s). Nothing here is evidence for or against RH (docs/08).