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Stable Menus of Public Goods: AI-Enabled Progress
Sara Fish
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%
Last extracted: 6/20/2026, 9:13:19 AM
Summary
This paper evaluates AI-driven research workflows using an open problem from the 'Stable Menus of Public Goods' (FGH'25) as a benchmark. The study investigates three research questions: the impact of human intuition in prompts, the effectiveness of multi-turn automated interactions (the 'autoresearcher' pipeline), and the comparative performance of LLMs (specifically GPT-5.5 Pro) against a first-year PhD student. Findings suggest that human intuition helps LLMs achieve 'better taste' or more ambitious results in upper bound proofs, and that multi-turn pipelines with a 'Supervisor' agent (Claude 4.7 Opus) can improve lower bounds, though the LLM remains slightly less effective than a human PhD student in certain complex tasks.
Entities (8)
Relation Signals (5)
Claude 4.7 Opus → actsas → Supervisor
confidence 100% · a 'Supervisor' agent (Claude 4.7 Opus) repeatedly encourages a 'Researcher' agent
Sara Fish → authored → Stable Menus of Public Goods
confidence 100% · Stable Menus of Public Goods: AI-Enabled Progress * Sara Fish
FGH'25 → containsproblem → Stable Menus of Public Goods
confidence 100% · revisit an open problem from the EC 2025 paper 'Stable Menus of Public Goods' (Fish, Gonczarowski, and Hart, 2025).
GPT-5.5 Pro → usedin → Autoresearcher Pipeline
confidence 100% · the 'Researcher' agent (GPT-5.5 Pro with 'xhigh' effort)
GPT-5.5 Pro → islesseffectivethan → First-year PhD Student
confidence 90% · find that the LLM is slightly less effective [than the first-year PhD student]
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Abstract
Abstract:Using an open problem from the EC 2025 paper "Stable Menus of Public Goods" as a testbed, we conduct experiments to understand the effectiveness of different AI-for-EconCS research workflows. Specifically, we study three questions: Does providing human intuition in the prompt help? Does automated multi-turn interaction help? And, does an LLM outperform a first-year PhD student? Regarding the first two questions, we provide evidence for the following workflow suggestions: (1) prompting with human intuition can encourage the LLM to have better "taste", (2) multi-turn workflows help when the pipeline encourages "ambitious" steps. Regarding the third question, using an unpublished manuscript written by the paper's senior authors prior to collaborating with the first-year PhD student, we compare the effectiveness of the LLM with that of the first-year PhD student, and find that the LLM is slightly less effective.
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- Source: https://arxiv.org/abs/2606.16989v1
- Canonical: https://arxiv.org/abs/2606.16989v1
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Stable Menus of Public Goods: AI-Enabled Progress ∗ Sara Fish † June 15, 2026 Abstract Using an open problem from the EC 2025 paper “Stable Menus of Public Goods” as a testbed, we conduct experiments to understand the effectiveness of different AI- for-EconCS research workflows. Specifically, we study three questions: Does providing human intuition in the prompt help? Does automated multi-turn interaction help? And, does an LLM outperform a first-year PhD student? Regarding the first two questions, we provide evidence for the following workflow suggestions: (1) prompting with human intuition can encourage the LLM to have better “taste”, (2) multi-turn workflows help when the pipeline encourages “ambitious” steps. Regarding the third question, using an unpublished manuscript written by the paper’s senior authors prior to collaborating with the first-year PhD student, we compare the effectiveness of the LLM with that of the first-year PhD student, and find that the LLM is slightly less effective. 1 Introduction Inspired by recent AI-driven leaps in combinatorics (OpenAI, 2026; Bloom et al., 2026), we revisit an open problem from the EC 2025 paper “Stable Menus of Public Goods” (Fish, Gonczarowski, and Hart, 2025). FGH’25 consider a novel public goods model that exhibits a rich combinatorial structure. For the question of the existence of stable menus in this model, they prove non-matching lower and upper bounds, and leave closing this gap as an open problem. ∗ Accepted to the EC’26 Workshop on AI-Driven Research in EconCS. † School of Engineering and Applied Sciences, Harvard University. Fish was supported by an NSF Grad- uate Research Fellowship and a Kempner Institute Graduate Fellowship. — E-mail : sfish@g.harvard.edu. Thanks to Yannai Gonczarowski and Pras Ramakrishnan for helpful comments. 1 arXiv:2606.16989v1 [cs.GT] 15 Jun 2026 In this paper, we use this open problem from FGH’25 as a testbed for evaluating the effec- tiveness of different AI-for-EconCS research workflows. Specifically, we conduct experiments to shed light on three research questions: (RQ1) Does providing human intuition in the prompt help? (RQ2) Does automated multi-turn interaction help? (RQ3) Does an LLM outperform a first-year PhD student? To study (RQ1), in Section 3, we conduct experiments in which an LLM (GPT-5.5 Pro with Extended Thinking) is asked to improve on the lower or upper bounds from FGH’25 in a single query. We compare two prompt types: WithContext, in which human intu- ition about what directions might be most promising is provided, and NoContext, which includes no such information. We observe that providing human intuition per se does not appear to help much, but that (for upper bounds) it appears to encourage the LLM to be more “ambitious” / have better “taste”, which leads to improved results. To study (RQ2), in Section 4, we construct a multi-turn pipeline in which a “Supervisor” agent (Claude 4.7 Opus) repeatedly encourages a “Researcher” agent (GPT-5.5 Pro with “xhigh” effort) to improve on its own lower or upper bounds. We test three rollouts of this pipeline, each consisting of ten turns. One rollout, seeking lower bounds, successfully arrived at results better than those from the single-shot pipeline from Section 3. However, the other two rollouts got “stuck”, whereby the Supervisor often made low-quality, incremental suggestions. These findings suggest that Supervisor-type agents may be more effective when nudged to be more ambitious and high-level in their feedback. To study (RQ3), in Section 5, we conduct experiments in which the LLM (GPT-5.5 Pro with Extended Thinking) is instead asked to improve on GH’20, an unpublished manuscript documenting the senior authors G and H’s progress on the problem prior to F joining the collaboration. By comparing the LLM’s improvements on GH’20 with FGH’25, we can compare the relative effectiveness of the LLM with that of the author F (myself as a first- year PhD student). Overall, we find that the LLM is slightly less effective than the first-year PhD student; however, the PhD student’s advantage appears to be fragile. Finally, regarding the actual problem at hand, the autoresearcher pipeline from Sec- tion 4 improves FGH’25’s lower bound (necessary condition, higher is better) result of u−1 t−1 ⪆ 23/11 ≈ 2.09 to u−1 t−1 ⪆ 8/3 ≈ 2.67, and the single-shot WithContext prompt from Section 3 improves FGH’25’s upper bound (sufficient condition, lower is better) result of u−1 t−1 ≥ g− 2 to u−1 t−1 ⪆ g 2 + log 2 (g + 1). In my opinion, both of these directions constitute substantial improvements that are interesting in their own right: the LLM’s lower bound 2 additionally resolves a separate (minor) open question from FGH’25, and the LLM’s upper bound uses novel techniques that go beyond those used in FGH’25. Related Work. This work relates to the growing literature on AI for Science (Wang et al., 2023). Works including Georgiev et al. (2025), Bloom et al. (2026), Nagda, Raghavan, and Thakurta (2026), and OpenAI (2026) demonstrate the promise of using LLMs to advance combinatorics research (see also Wagner 2021; Charton et al. 2024, which use non-LLM ML methods). The multi-turn “autoresearcher” pipeline (Section 4) follows directly in the footsteps of more elaborate systems such as Novikov et al. (2025), Breen et al. (2026), Feng et al. (2026), Tsoukalas et al. (2026), and Zheng et al. (2026). This paper’s contribution is to document and ablate certain AI-for-Science workflow choices, using a specific open problem from FGH’25 as a benchmark. 2 Background In this section we briefly describe the open problem the AI is asked to make progress on. For a more fleshed-out exposition, see FGH’25. For a less fleshed-out exposition, it suffices to look at Fig. 1. Very briefly (for problem parameters g, t, u), the problem is solved for g ≤ 6; for g ≥ 7, lower bound (necessary condition) improvements involve providing constructions that beat u−1 t−1 ⪆ 23 11 (i.e., value exceeding 23 11 ≈ 2.1), and upper bound (sufficient condition) improvements involve providing a proof that beats u−1 t−1 ≥ g− 2 (i.e., value lower than g− 2). 2.1 Problem Statement A menu selection problem is a triplet N, G, (≻ i ) i∈N where N =1, . . . , n is a finite set of agents, G =1, . . . , g is a finite set of (public) goods, and agent i’s (possibly incomplete) preference relation ≻ i is a strict partial order over G. A menu O is a set of goods. Given a menu O, the agent assignment a O : N → O∪⊥ maps each agent to their favorite good in O (or an outside option ⊥ if the agent’s preference list does not contain any goods in O). The intuition is that a central planner selects a menu of goods O to offer, and then each (unit-demand) agent uses the good from O that they prefer most. Given parameters t, u ∈N, a menu O is t-feasible if every offered good is the assigned good of at least t agents, that is, a −1 O (j) ≥ t for every j ∈ O. A menu O is u-uncontestable if no unoffered good is preferred to all of O by u or more agents—that is, a −1 O∪j (j) < u for every j ∈ G\ O. The intuition behind t-feasibility is that every good offered in the menu O should “justify its existence” by being used by at least t agents; the intuition behind 3 u-uncontestability is that the menu O should not fail to include a popular good that at least u agents would have preferred over all other goods in O. A menu is (t, u)-stable if it is both t-feasible and u-uncontestable. One of the central questions of FGH’25 is: for which triples (g, t, u) does every menu selection problem on g public goods admit a (t, u)-stable menu? Example 2.1. Consider a menu selection problem with n = 9 agents and g = 3 goods (denote the set of goods G := 1, 2, 3), where three agents have preferences 1 ≻ 2 ≻ 3, three agents have preferences 2≻ 3≻ 1, and three agents have preferences 3≻ 1≻ 2. Fix t = 4 and u = 7. Then O := 1 is (t, u)-stable, because it is t-feasible (all 9 ≥ t agents use good 1) and u-uncontestable (only 6 < u agents prefer good 3 over good 1). However, if we instead set u = 6, then no (t, u)-stable menus exist. The menu O :=1 is u-uncontestable since 6 ≥ u agents prefer good 3 over good 1. (By symmetry, 2 and 3 are similarly u-uncontestable.) The menu O := 1, 2 is t-infeasible since only 3 < t agents use good 2. (By symmetry, 2, 3 and 1, 3 are similarly t-infeasible.) Finally, by monotonicity, O :=1, 2, 3 is t-infeasible and O :=∅ is u-uncontestable. More generally, (t, u)-stability as a condition becomes stronger as t increases and as u decreases (and vice versa). Thus, for a given g and t, the question is to find the “cutoff” u g (t) so that for u≥ u g (t), every menu selection problem on g goods has a (t, u)-stable menu, and for u < u g (t), there exists a menu selection problem on g goods with no (t, u)-stable menu. 2.2 Known Results The existence bounds obtained in FGH’25 are as follows (see Figure 1 for a visualization): • (Proposition 3.1 and 1.2) For g = 2 and t, u∈N, every menu selection problem has a (t, u)-stable menu if and only if u−1 t−1 ≥ 1. • (Proposition 1.1 and 1.2) For every g ∈ 3, 4, 5, 6 and t, u ∈N, every menu selection problem has a (t, u)-stable menu if and only if u−1 t−1 ≥ 2. • (Theorem 1.4) For every g ≥ 7, there exists a menu selection problem with no (t, u)- stable menu if u ≤ 23⌊ t−1 11 ⌋ (roughly u−1 t−1 < 23 11 ≈ 2.1). That is, u ≥ 23⌊ t−1 11 ⌋ + 1 is a necessary condition for the existence of (t, u)-stable menus. • (Theorem 1.5) For every g ≥ 7, every menu selection problem on complete preferences has a (t, u)-stable menu if u−1 t−1 ≥ g− 2. 4 • (Proposition E.4) For every g ≥ 5, every menu selection problem—including those with incomplete preferences—has a (t, u)-stable menu if u > (g− 1− 1 6 )(t− 1) (equivalently, u≥⌊(g− 7 6 )(t− 1)⌋ + 1). That is, for g ∈ 2, 3, 4, 5, 6, the paper provides a complete characterization, and for g ≥ 7, the paper provides (non-matching) necessary and sufficient conditions. u−1 t−1 0123g−2g−1 23 /11 g = 2: g ∈3, 4, 5, 6: g ≥ 7: Figure 1: Visualization of the existence bounds obtained in FGH’25. Red zigzag lines indicate values of u−1 t−1 for which (t, u)-stable menus are not guaranteed to exist. Green solid lines indicate values of u−1 t−1 for which (t, u)-stable menus exist for all menu selection problems. Green dotted lines indicate that existence is only guaranteed for menu selection problems with complete preferences. Notation. Many results, both in FGH’25 and proven by the LLM in this paper, arrive at bounds of the form u− 1 ≥ k 1 ⌊ t−1 k 2 ⌋, for k 1 , k 2 ∈N. For the rest of this paper, we use the notation u−1 t−1 ⪆ k 1 k 2 (or⪆ k 1 /k 2 for short) to refer to such inequalities. (When k 2 divides t− 1, they are equivalent when⪆ is replaced by≥, but otherwise, u− 1≥ k 1 ⌊ t−1 k 2 ⌋ is slightly weaker than u−1 t−1 ≥ k 1 k 2 .) 3 Does providing human intuition in the prompt help? We first investigate whether human intuition can be leveraged to elicit stronger results from the LLM. To do so, we compare two kinds of prompts. The first prompt type, NoContext, simply provides a link to the paper and asks the LLM to improve on the paper’s results. The second prompt type, WithContext, additionally includes a description of the paper’s results and some hints about potentially promising directions. 1 We write separate (but similar) prompts for the lower and upper bound cases, leading to four different prompts. (See Section B.1 for the full prompts.) 1 The WithContext hints were written based on deep familiarity with FGH’25, but had not been seriously pursued in advance. That is, they were written purely based on where the paper “left off”, without prior knowledge of what improvements were attainable. 5 For example, NoContextUpper reads (in its entirety): https://arxiv.org/abs/2402.11370 I would like you to work on deriving improvements for the upper bounds, that is, the sufficient condition bounds. Do not work on lower bound constructions, because another agent has that covered. Keep working until you have an actual upper bound proof better than the bounds in the paper. No need to report back with partial progress not yet constituting a proof. All of the upper bounds in the paper in fact are very loose, so progress is tractable. Whereas WithContextUpper includes context such as: [...] A good place to start would be to look at the upper bound proofs in the paper. They include the aforementioned (u-1)/(t-1) = g-2 bound, but also the simple u >= (g-1)(t-1) argument in Appendix E, as well as the more refined u >= (g-1-eps)(t-1), also in Appendix E. These arguments arrive at sufficient conditions for the existence of (t,u)-stable menus for certain g by uncovering some sort of mathematical structure about the problem. The (g-2) bound uncovers one kind of structure (this "no gaps" idea), and the (g-1-eps) bound uncovers a different kind of structure (that, when small (t,u)-stable menus fail to exist, for u large, this implies that the menu selection problem has a specific cyclic structure). One promising direction could be to push either of these observations, or both simultaneously. Or, you are welcome to pursue other techniques. [...] Some further remarks on the prompts: • The instruction “Do not work on lower bound constructions” was added because in early testing, when instructed to prove upper bounds, the LLM would often “give up” and work on lower bound constructions instead. • The instruction “All of the upper bounds in the paper in fact are very loose, so progress is tractable” was similarly added to “encourage” the LLM. • The prompt includes the paper URL, rather than the entire paper .tex or PDF, to allow the LLM to read the paper with the method it most “prefers”. (In all cases, the LLM has web browsing enabled.) In practice, the LLM appears to use a mix of PDF and HTML. 6 Using GPT-5.5 Pro with Extended Thinking via the ChatGPT web interface, we collect five samples each of NoContextLower, NoContextUpper, WithContextLower, and WithContextUpper. Due to monthly Pro query limits, the samples were collected across two accounts. Browsing was enabled. All personalization features were left blank or set to default values, and memory was disabled. Lower bounds. The lower bound results are displayed in Table 1. A link to the full LLM outputs is provided in Appendix A. All proofs (constructions) were checked by a human. Across all ten trials, the LLM improves on the g ≥ 7 lower bound from FGH’25. Provid- ing human intuition in the prompt (WithContext) reduces the time taken by the LLM by 18m11s on average (p = 0.023 < 0.05, two-sided Welch’s t-test). However, the NoCon- text and WithContext prompts produce equally strong bounds on u−1 t−1 (in both cases, improving to⪆ 2.5 four times and⪆ 7/3≈ 2.3 one time). All ten of the LLM’s proofs follow a similar blueprint. The LLM produces a construction on g = 9 goods, with a similar cyclic structure to the construction on g = 7 goods from FGH’25. The LLM finds this construction using a mix of LP/MILP solvers and heuristics (e.g. imposing cyclic structure). The proof that no (t, u)-stable menu exists also follows the methods of FGH’25: in their g = 7 construction, they identify a (3,4)-gap (no O ⊆ G with |O| = 3 is u-uncontestable, and no O ⊆ G with |O| = 4 is t-feasible), and in the LLM’s g = 9 constructions, it identifies a (4, 5)-gap (analogously, no O ⊆ G with |O| = 4 is u-uncontestable, and no O ⊆ G with |O| = 5 is t-feasible). Upper bounds. The upper bound results are displayed in Table 2. A link to the full LLM outputs is provided in Appendix A. All proofs were checked by a human. Across all ten trials, the LLM technically improves on the upper bounds from FGH’25, however, in eight out of these ten cases the improvements are near-trivial. The four⪆ g− 2 + 5 g+1 “improvements” are completely trivial: Proposition E.4 of FGH’25 bounds 5 g+1 by 5 6 (using g ≥ 5) to make the theorem statement cleaner, and the LLM’s improvement is simply to restore the 5 g+1 term in the bound. The four ≥ g − 2 improvements follow from observing that Lemma 5.7 from FGH’25, which was thought to only hold for complete preferences, can be extended to the incomplete preferences case (with a relatively simple argument, that admittedly was overlooked by FGH’25). Both of the two nontrivial improvements are achieved by WithContext prompts. 2 Trial 3 achieves a slight improvement (replacing 5 g+1 with 4 g ) by tightening the argument 2 Trial 3 of NoContextUpper additionally “proves” an upper bound of ≥ g − 3, but it relies on an assumption that makes the problem substantially easier, so we do not consider this progress. 7 TreatmentTrial Time taken Bound foundRegime NoContextLower186m 4s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 NoContextLower272m 21s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 NoContextLower384m 17s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 NoContextLower484m 8s u−1 t−1 ⪆ 7 3 ≈ 2.3 g ≥ 9 NoContextLower561m 8s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 WithContextLower157m 53s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 WithContextLower242m 59s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 WithContextLower364m 8s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 WithContextLower464m 11s u−1 t−1 ⪆ 7 3 ≈ 2.3 g ≥ 9 WithContextLower567m 52s u−1 t−1 ⪆ 5 2 = 2.5 g ≥ 9 FGH’25 (Theorem 1.4)— u−1 t−1 ⪆ 23 11 ≈ 2.1 g ≥ 7 Table 1: Lower bounds found and time taken by the LLM for each of the ten trials (five NoContextLower, five WithContextLower). The last row displays the g ≥ 7 lower bound from FGH’25. Across all ten trials, the LLM improves on the bound from the paper (when g ≥ 9). TreatmentTrial Time taken Bound foundPreferences NoContextUpper121m 0s u−1 t−1 ≥ g− 2incomplete NoContextUpper219m 58s u−1 t−1 ≥ g− 2incomplete NoContextUpper320m 34s u−1 t−1 ⪆ g− 2 + 5 g+1 incomplete NoContextUpper425m 39s u−1 t−1 ≥ g− 2incomplete NoContextUpper526m 34s u−1 t−1 ⪆ g− 2 + 5 g+1 incomplete WithContextUpper131m 26s u−1 t−1 ⪆ g 2 + log 2 (g + 1)incomplete WithContextUpper222m 8s u−1 t−1 ⪆ g− 2 + 5 g+1 incomplete WithContextUpper326m 8s u−1 t−1 ⪆ g− 2 + 4 g incomplete WithContextUpper425m 30s u−1 t−1 ⪆ g− 2 + 5 g+1 incomplete WithContextUpper530m 12s u−1 t−1 ≥ g− 2incomplete FGH’25 (Theorem 1.5)— u−1 t−1 ≥ g− 2complete only FGH’25 (Proposition E.4)— u−1 t−1 ⪆ g− 7 6 incomplete Table 2: Upper bounds found and time taken by the LLM for each of the ten trials (five NoContextUpper, five WithContextUpper). The last two rows display two upper bound results from FGH’25. The LLM makes trivial improvements in eight out of the ten trials; the remaining two nontrivial improvements are highlighted. All bounds hold for g ≥ 7. 8 in Proposition E.4 of FGH’25. Trial 1 achieves a substantial improvement, with a creative technique not applied anywhere in FGH’25. (Briefly, the LLM considers a dominating set of a tournament on the g goods, and uses it to construct a (t, u)-stable menu. For the full proof see the data release in Appendix A.) Regarding time taken, the LLM takes substantially less time to prove upper bounds than lower bounds (43m35s less on average, p < 0.001). This effect appears to be driven by the LLM writing more code (e.g. invoking MILP solvers) for Lower than Upper. Unlike in the lower bound case, there is no significant difference in time taken by the NoContext and WithContext prompts. Finally, the best upper bound also took the most time to produce (31m 26s). Conclusion. It is, of course, difficult to isolate causes of particular LLM behaviors, espe- cially in settings like this one in which the LLM’s full CoT is not even accessible (let alone the model internals). Still, we conclude with some informal observations consistent with the above data: (1) For problems for which the LLM can make progress by pushing known techniques (here: lower bounds), providing human intuition does not appear to be necessary: the LLM can identify such promising directions on its own. (2) The LLM is “lazy”: it is “satisfied” with any progress, even if it is clearly trivial. Providing human intuition may counteract this effect by encouraging the LLM to be more “ambitious” / have better “taste” (here: upper bounds). 4 Does automated multi-turn interaction help? So far, we have demonstrated that a single-shot prompt is sufficient for eliciting improvements on FGH’25 from the LLM. However, one limitation of this approach is the LLM’s “laziness” (see Item (2) above). For example, for the two lower bound trials that yielded⪆ 7/3≈ 2.3, it seems likely that a simple follow-up message such as “Can you improve this further?” might have yielded the⪆ 2.5 bounds found in other trials. In this section, we experiment with a simple multi-turn “autoresearcher” pipeline, in which a second LLM agent “supervises” the main LLM research agent. 4.1 Pipeline architecture See Fig. 2 for an illustration. The pipeline consists of three agents, described in more detail below. For the full prompts see Section B.2. 9 Researcher GPT-5.5 Pro web search run python Extractor GPT-5.5 Supervisor Claude Opus 4.7 Figure 2: Illustration of the “autoresearcher” pipeline. The first agent, Researcher, is given a prompt stating a research question, and (after an extended thinking period) outputs its result as freetext. It operates via a single GPT-5.5 Pro query, using the OpenAI API (as opposed to the ChatGPT web interface). We set the reasoning effort to xhigh and equip the LLM with two tools: OpenAI’s built-in web- browsing tool, and a custom run python tool. 3 The run python tool gives the LLM access to a Docker container with 2GB memory, 120s timeout, and Python 3.11 with standard packages (numpy, scipy, z3-solver, pulp, ortools). Typical Researcher queries use 1 million input tokens, 20,000 reasoning tokens, and 50,000 output tokens. The second agent, Extractor, is given the Researcher’s output and asked to return the bound obtained by the Researcher (if applicable). It is implemented via a single GPT-5.5 API call. For the lower bounds, the Extractor is additionally asked to extract from the Researcher’s response a specific instantiation of the construction (if applicable), which is fed into a verifier script. The verifier script takes as input a structured representation of a menu selection problem, alongside parameters t, u ∈N, and outputs whether a (t, u)-stable menu exists. The Extractor and Verifier are not particularly load-bearing; we include them in the spirit of prototyping (e.g. more complex workflows could involve verifying proofs with Lean). The third agent, Supervisor, is tasked with writing prompts to “encourage” the Re- searcher to make progress. It is implemented via a single Claude Opus 4.7 call. 4 The Su- pervisor is given: (1) information about the research question (2) the Researcher’s prompt, summarized CoT, and output, (3) the Extractor’s output, (4) memory notes written by the Supervisor in the previous turn. Then, the Supervisor’s task is to output a new prompt for the Researcher. To illustrate, below is an excerpt from the Supervisor’s (lower bound) prompt: 3 OpenAI does provide built-in tools for Python code execution—one Python code interpreter and one general shell environment. However, we encountered technical issues with both options, similar to what is described in this bug report: https://community.openai.com/t/container-is-expired-error-in-ope nai-responses-api-stream/1321773. 4 Claude Opus 4.7 was selected for its strong agentic and prompt-writing capabilities. Though this model is less strong at mathematics research, it is sufficiently capable to understand the content. 10 TurnTime taken Bound found(g, t, u, n) 1102m 22s u−1 t−1 ⪆ 23 11 ≈ 2.09 (7, 12, 23, 70) 282m 35s u−1 t−1 ⪆ 13 6 ≈ 2.17 (9, 19, 39, 54) 338m 31s u−1 t−1 ⪆ 7 3 ≈ 2.33 (9, 10, 21, 27) 438m 53s u−1 t−1 ⪆ 5 2 ≈ 2.50 (9, 9, 20, 18) 5104m 7s u−1 t−1 ⪆ 8 3 ≈ 2.67(9, 4, 8, 27) 6175m 25s u−1 t−1 ⪆ 8 3 ≈ 2.67 (9, 4, 8, 27) 7140m 12s u−1 t−1 ⪆ 8 3 ≈ 2.67 (9, 4, 8, 27) 8155m 16snone found— 9146m 56snone found— 10166m 30snone found— FGH’25 (Theorem 1.4) — u−1 t−1 ⪆ 23 11 ≈ 2.09 (7, 12, 23, 70) Table 3: Lower bounds found (with construction parameters g, t, u, n) and time taken by the autoresearcher, with initial prompt WithContextLower, for ten sequential turns. The last row displays the g ≥ 7 lower bound from FGH’25. The LLM improves on the results from the one-shot queries from Table 1 in turn 5 (highlighted). When the subagent shows a promising pattern, give it concrete next steps (perturb cohort sizes, try a different underlying symmetry, recheck a specific menu cardinality). When it is stuck or keeps regenerating the known g = 7 / 23/11 construction, pivot -- change g, the preference structure, or the combinatorial template. After a new ratio is verified, do not stop: record it as the new bar and direct the next attempt at exceeding it. And below is an excerpt from an example Supervisor-written prompt for the Researcher: A previous subagent proposed [...] This does not count as progress for our goal. [...] # Concrete directions to try [...] Direction 1 (recommended): Try g = 8 with a Z/8 cyclic structure. More goods give more potential orbit types. [...] 4.2 Results We conduct three rollouts of the autoresearcher pipeline, testing each of the three prompts as the initial Researcher prompt: NoContextLower, WithContextLower, and With- ContextUpper. Each rollout consists of 10 Researcher turns. 11 Positive results. Table 3 displays the results for WithContextLower. As intended, the Supervisor successfully “encourages” the Researcher to seek out successively better lower bounds. The bounds obtained in turns 2–5 outperform FGH’25, and the bound from turn 5 ( u−1 t−1 ⪆ 8/3≈ 2.67) outperforms the best bound from the single-shot pipeline ( u−1 t−1 ⪆ 5/2 = 2.5, see Table 1). Beyond providing improved lower bounds, the LLM’s u−1 t−1 ⪆ 8/3 result also resolves a minor open question from FGH’25. Recall from Section 3 that FGH’25’s⪆ 23/11 construc- tion, as well as the LLM’s⪆ 7/3 and⪆ 5/2 constructions, exhibit (k, k + 1)-gaps, for k = 2 and k = 3 respectively. Remark 5.6 of FGH’25 asks whether all such constructions must ex- hibit (k, k + 1)-gaps. The LLM’s⪆ 8/3 construction does not satisfy this property, thereby resolving this open question via counterexample. Negative results. In the NoContextLower and WithContextUpper experiments, the autoresearch pipeline gets “stuck”. For NoContextLower, the Researcher does not beat the bounds from FGH’25, and for WithContextUpper, the Researcher at best ex- tends u−1 t−1 ≥ g − 2 to incomplete preferences (a near-trivial improvement, see Section 3). With only one rollout per prompt type, it is unclear whether these results can be attributed to the prompt, or are simply noise. Still, we report some potential failure modes: (1) For NoContextLower, the Supervisor repeatedly encourages the Researcher to com- putationally search for constructions with cyclic structure, however, the pair fixates on using SMT/SAT solvers, rather than MILP/LP solvers (which appear to be more efficient in this context). (2) For NoContextLower, the (less mathematically capable) Supervisor appears to inject its own overly-specific hypotheses about promising constructions (e.g., asking for constructions based on AG(2, 3) or the Petersen graph). This does not happen in WithContextLower, where the Supervisor is a little more “high-level” (e.g., just asking for higher g values). (3) For both NoContextLower and WithContextUpper, in later turns, the Su- pervisor appears to anchor on the progress rate of earlier turns. For example, the Supervisor’s next steps in WithContextLower are more “ambitious” than those in NoContextLower. We reiterate that these effects could be driven by noise, rather than by the prompts or harness. Still, these results suggest that Supervisor-type agents may be more effective when nudged to be more ambitious and high-level in their feedback. 12 5 Does the LLM outperform a first-year PhD student? AI is changing the way we do research. We can ask it questions like we would a colleague, and we can delegate tasks to it like we would a coauthor. These dynamics may alter how we collaborate. To better understand such effects, in this section, we measure the counterfactual impact of adding an LLM to a collaboration, as opposed to a first-year PhD student, again using FGH’25 as a testbed. To do this, we leverage the specific way that the collaboration in FGH’25 unfolded. The project began as a collaboration between the senior authors G and H, which paused in 2020. They had written a five-page unpublished manuscript, GH’20, summarizing their progress. 5 F (a first-year PhD student) joined the collaboration after this point, eventually resulting in FGH’25. Thus, using GH’20, we can compare two workflows: first, the baseline workflow of collaborating with a first-year PhD student to improve on GH’20 (resulting in FGH’25), and second, the workflow of asking an LLM to improve on GH’20. 6 One potential concern is that the LLM might “cheat” at the task of improving on GH’20 by drawing a connection to FGH’25. Fortunately, GH’20 used sufficiently different language and notation to describe the problem so that this seems unlikely (but of course, not possible to rule out). Indeed, across all experiments from this section, none of the reasoning summaries or LLM outputs explicitly reference FGH’25, or seek it out. 7 We collect three samples each of NoContextLower and NoContextUpper, with the arXiv URL replaced by an attachment of GH’20. As in Section 3, we use ChatGPT-5.5 Pro with Extended Thinking via the web interface, with browsing enabled, and personaliza- tion and memory disabled. The results of the six trials are displayed in Tables 4 and 5. Upper bounds. For the upper bounds, in all three trials, the LLM matches the u−1 t−1 ≥ g−2 bound from FGH’25. (For the comparisons here, we assume complete preferences, because the problem statement in GH’20, while otherwise identical, requires complete preferences and nonempty menus.) That is, the LLM is as effective as the first-year PhD student. 5 GH’20 proved that u−1 t−1 ≥ 2 was a necessary condition (when g ≥ 3), and that u−1 t−1 ≥ g − 1 was a sufficient condition. Additionally, for g = 4, GH’20 obtained the upper bound u−1 t−1 ≥ g− 2 = 2, giving a complete characterization. Finally, GH’20 had conjectured that u−1 t−1 ≥ 2 was sufficient, which FGH’25 later disproved with the u−1 t−1 ⪆ 23/11 bound for g ≥ 7. 6 Here, we focus on the single-shot setting, which yields a lower bound of the LLM’s usefulness. A proper interactive “collaboration” with the LLM is likely more realistic, but also difficult to test experimentally. 7 Example excerpt: “I don’t need any current information or web results. The focus is on solving the math problem based on the uploaded paper.”. 13 TrialTime taken Bound found 158m 11s u−1 t−1 ≥ 2 262m 57s u−1 t−1 ≥ 2 379m 54s u−1 t−1 ≥ 2 GH’20— u−1 t−1 ≥ 2 FGH’25 (Theorem 1.4) — u−1 t−1 ⪆ 23 11 ≈ 2.1 LLM, one-shot (best)42m59s u−1 t−1 ⪆ 5 2 = 2.5 LLM, autoresearcher366m28s u−1 t−1 ⪆ 8 3 ≈ 2.67 Table 4: Lower bounds found and time taken by the LLM for each of the three trials of NoContextLower, with GH’20 provided instead of FGH’25. The final four rows display baselines (see Tables 1 and 3). TrialTime taken Bound found 129m 14s u−1 t−1 ≥ g− 2 234m 10s u−1 t−1 ≥ g− 2 327m 19s u−1 t−1 ≥ g− 2 GH’20— u−1 t−1 ≥ g− 1 FGH’25 (Theorem 1.5) — u−1 t−1 ≥ g− 2 LLM, one-shot (best)31m26s u−1 t−1 ⪆ g 2 + log 2 (g + 1) LLM, autoresearcher916m21s u−1 t−1 ≥ g− 2 Table 5: Upper bounds found and time taken by the LLM for each of the three trials of NoContextUpper, with GH’20 provided instead of FGH’25. The final four rows display baselines (see Table 2). Trial Time taken Bound found 190m 10s u−1 t−1 ≥ 2 285m 33s u−1 t−1 ≥ 2 388m 39sinvalid (flawed proof claiming u−1 t−1 ⪆ log g log log g for suff. large g) Table 6: Lower bounds found and time taken by the LLM for each of the three trials of NoContextLower, with GH’20 provided instead of FGH’25, and with an additional instruction to ignore a false conjecture from GH’20. For baselines see Table 4. 14 Lower bounds. For the lower bounds, the LLM does not make progress beyond the trivial bound presented in GH’20. In particular, it does not beat the u−1 t−1 ⪆ 23/11 lower bound from FGH’25. This may be due to the LLM “anchoring” on GH’20’s false conjecture that u−1 t−1 ≥ 2 ought to be a sufficient condition, in which case no lower bound improvements would be possible. To mitigate this effect, we run three additional trials with the following sentence appended to the prompt (Conjecture 3 in GH’20 is the aforementioned false conjecture): (In particular, ignore Conjecture 3, which is likely false.) The results are displayed in Table 6. In two out of the three trials, the LLM continues to fail to make progress. The reasoning summaries reveal attempts to use MILP/LP solvers, but the LLM does not push these techniques far enough to obtain an improved bound. In the third trial, the LLM claims an improved bound of u−1 t−1 ⪆ log g log log g (for sufficiently large g) using the probabilistic method, however, the proof does not appear to be correct. Overall, these results indicate that the LLM is slightly less effective than the first-year PhD student. That said, the PhD student’s advantage appears to be fragile; the lower bound reasoning summaries reveal that the LLM had the “idea” to search for constructions computationally, and Sections 3 and 4 reveal that the LLM is adept at producing viable constructions when it exerts sufficient “effort”. References Bloom, T. F. et al. (May 2026). The sum-product conjecture is false for real numbers. doi: 10.48550/arXiv.2605.28781. Breen, B. et al. (May 2026). Ax-Prover: A Deep Reasoning Agentic Framework for Theorem Proving in Mathematics and Quantum Physics. doi: 10.48550/arXiv.2510.12787. Charton, F. et al. (Nov. 2024). PatternBoost: Constructions in Mathematics with a Little Help from AI. doi: 10.48550/arXiv.2411.00566. Feng, T. et al. (Feb. 2026). Towards Autonomous Mathematics Research. en. Fish, S., Y. A. Gonczarowski, and S. Hart (2025). “Stable Menus of Public Goods: A Match- ing Problem”. In: Proceedings of the 26th ACM Conference on Economics and Compu- tation, p. 348. doi: 10.1145/3736252.3742543. Georgiev, B. et al. (Dec. 2025). Mathematical exploration and discovery at scale. doi: 10. 48550/arXiv.2511.02864. Nagda, A., P. Raghavan, and A. Thakurta (Apr. 2026). Reinforced Generation of Combina- torial Structures: Ramsey Numbers. doi: 10.48550/arXiv.2603.09172. 15 Novikov, A. et al. (June 2025). AlphaEvolve: A coding agent for scientific and algorithmic discovery. doi: 10.48550/arXiv.2506.13131. OpenAI (May 2026). An OpenAI model has disproved a central conjecture in discrete geom- etry. Tsoukalas, G. et al. (May 2026). Advancing Mathematics Research with AI-Driven Formal Proof Search. doi: 10.48550/arXiv.2605.22763. Wagner, A. Z. (Apr. 2021). Constructions in combinatorics via neural networks. doi: 10. 48550/arXiv.2104.14516. Wang, H. et al. (Aug. 2023). “Scientific discovery in the age of artificial intelligence”. In: Nature 620.7972, p. 47–60. doi: 10.1038/s41586-023-06221-2. Zheng, D. et al. (2026). AI co-mathematician: Accelerating mathematicians with agentic AI. arXiv: 2605.06651 [cs.AI]. A Code and data The code is publicly available here: https://github.com/sara-fish/stable_menus_a i_enabled_progress. The LLM-written results can be viewed in this dashboard: https: //sara-fish.github.io/stable_menus_ai_enabled_progress/. Data collection. For Section 3, the data was collected between May 24, 2026 and May 26, 2026. For Section 4, the data was collected between May 25, 2026 and May 27, 2026. For Section 5, the data was collected on May 30, 2026 and May 31, 2026. B Prompts B.1 Single-turn prompts These prompts were entirely human-written. B.1.1 NoContextLower https://arxiv.org/abs/2402.11370 16 I would like you to work on deriving improvements for the lower bound constructions, that is, the necessary condition bounds. Do not work on upper bound proofs, because another agent has that covered. Keep working until you have an actual lower bound construction better than the bounds in the paper. No need to report back with partial progress not yet constituting an improved lower bound. All of the lower bounds in the paper in fact are very loose, so progress is tractable. B.1.2 NoContextUpper https://arxiv.org/abs/2402.11370 I would like you to work on deriving improvements for the upper bounds, that is, the sufficient condition bounds. Do not work on lower bound constructions, because another agent has that covered. Keep working until you have an actual upper bound proof better than the bounds in the paper. No need to report back with partial progress not yet constituting a proof. All of the upper bounds in the paper in fact are very loose, so progress is tractable. B.1.3 WithContextLower https://arxiv.org/abs/2402.11370 This paper presents a variety of bounds that aim to understand necessary and sufficient conditions for the existence of stable menus. When the number of goods g satisfies g <= 6, the paper provides a complete characterization. For g= 2, it shows that (t,u)-stable menus exist for all menu selection problems if and only if u >= t. For g = 3,4,5,6, it shows that (t,u)-stable menus exist for all menu selection problems if and only if u >= 2t-1. For g >= 7, the lower and upper bounds do not match. The paper shows using a lower bound construction that it is necessary for u >= 23 (t-1) / 11 to have (t,u)-stable menus exist for all menu selection problems. The paper also proves, for all g >= 7, that (u-1)/(t-1) >= g-2 is a sufficient condition (with complete preferences). 17 I would like you to work on deriving improvements for the lower bound constructions, that is, the necessary condition bounds. Do not work on upper bound proofs, because another agent has that covered. Because the paper already provides a complete characterization for g <= 6, progress for lower bounds involves looking at g >= 7. A good place to start would be to look at the lower bound constructions in the paper. They include the construction consisting of cycles 1>2>3, 2>3>1, 3>1>2 that show that u >= 2t-1 is a necessary condition for the existence of stable menus for g >= 3, and then the more complex construction in Section 5 showing that u >= 23 (t-1)/11 + 1 is necessary for the existence of stable menus for g >= 7. You'l notice in both cases, the constructions have a specific cyclic structure. One promising direction could be to continue to look for constructions of this form. Of course, you are welcome to pursue other techniques. Keep working until you have an actual lower bound construction better than the bounds in the paper. No need to report back with partial progress not yet constituting an improved lower bound. All of the lower bounds in the paper in fact are very loose, so progress is tractable. B.1.4 WithContextUpper https://arxiv.org/abs/2402.11370 This paper presents a variety of bounds that aim to understand necessary and sufficient conditions for the existence of stable menus. When the number of goods g satisfies g <= 6, the paper provides a complete characterization. For g= 2, it shows that (t,u)-stable menus exist for all menu selection problems if and only if u >= t. For g = 3,4,5,6, it shows that (t,u)-stable menus exist for all menu selection problems if and only if u >= 2t-1. For g >= 7, the lower and upper bounds do not match. The paper shows using a lower bound construction that it is necessary for u >= 23 (t-1) / 11 to have (t,u)-stable menus exist for all menu selection problems. The paper also proves, for all g >= 7, that (u-1)/(t-1) >= g-2 is a sufficient condition (with complete preferences). 18 I would like you to work on deriving improvements for the upper bounds, that is, the sufficient condition bounds. Do not work on lower bound constructions, because another agent has that covered. Because the paper already provides a complete characterization for g <= 6, progress for upper bounds involves looking at g >= 7. A good place to start would be to look at the upper bound proofs in the paper. They include the aforementioned (u-1)/(t-1) = g-2 bound, but also the simple u >= (g-1)(t-1) argument in Appendix E, as well as the more refined u >= (g-1-eps)(t-1), also in Appendix E. These arguments arrive at sufficient conditions for the existence of (t,u)-stable menus for certain g by uncovering some sort of mathematical structure about the problem. The (g-2) bound uncovers one kind of structure (this "no gaps" idea), and the (g-1-eps) bound uncovers a different kind of structure (that, when small (t,u)-stable menus fail to exist, for u large, this implies that the menu selection problem has a specific cyclic structure). One promising direction could be to push either of these observations, or both simultaneously. Or, you are welcome to pursue other techniques. Keep working until you have an actual upper bound proof better than the bounds in the paper. No need to report back with partial progress not yet constituting a proof. All of the upper bounds in the paper in fact are very loose, so progress is tractable. B.2 Autoresearcher prompts These prompts were written by asking Claude Opus 4.7 to flesh out a human-written outline. To avoid AI detector false positives, the prompts can be found here: https://github.com /sara-fish/stable_menus_ai_enabled_progress/blob/e12e54863e397c4b23e0cff1c0 e7d21468d60855/autoresearcher/prompts.py 19