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EvoDuet: Bilevel Co-Evolution of Web Searching and Task Solving for Scientific Discovery

Young-Jun Lee, Jinheon Baek, Soyeong Jeong, Minki Kang, Seungyeon Jwa, Jonghyun Choi, Seungho Han, Dongyeop Kang

arXiv:2609.40340v1cs.CL

TL;DR

LLM-driven evolutionary search can stall when progress requires external knowledge, and in-loop search can repeatedly retrieve the same pages. EvoDuet co-evolves queries and solutions; across 21 tasks it improves OpenEvolve with GPT-5.6-Luna and Gemini-3.8-Flash, but not Qwen3.5-9B.

  • Problem

    Evolutionary-search scaffolds can stall when progress requires external knowledge absent from the run’s history and the LLM’s parametric knowledge.

    Source: §1 · p. 1

  • Method

    EvoDuet co-evolves solutions and web queries with fixed LLM parameters, using a knowledge-gap gate and inner query and outer solution loops.

    Source: §1 · p. 2

  • Results

    Across 21 tasks, NDG rises from 74.1% to 78.0% for GPT-5.6-Luna and 61.3% to 82.3% for Gemini-3.8-Flash, but Qwen3.5-9B does not benefit.

    Source: §1 · p. 2, §7 · p. 9

  • Takeaways & Limitations

    Best runs surpass previously reported scores on eight tasks and match them on three, while EvoDuet also improves Top-K and EvoX on Sums/Diffs and Denoising.

    Source: §7 · p. 9

    Reported gains reflect the study’s task evaluators, and programs require independent validation before use in other settings.

    Source: Ethics statement · p. 10

Abstract

from arXiv · show

Evolutionary search with large language models (LLMs) can stall when progress requires external knowledge the model lacks. Supplying relevant documents helps, but simply adding web search tool can keep returning the same pages as solutions change. We introduce EvoDuet, a bi-level optimization method that co-evolves solutions and search queries with fixed model parameters. At each iteration, a retrieval gate lets the LLM assess its knowledge gap and choose to retrieve new documents, reuse stored ones, or proceed without them. An inner loop refines queries and ranks documents by the solution scores they are predicted to yield; an outer loop generates candidates in parallel from these documents and records the evaluated outcomes for later searches. Across 21 optimization tasks with one candidate per iteration, EvoDuet raises OpenEvolve's normalized discovery gain from 74.1% to 78.0% with GPT-5.6-Luna and from 61.3% to 82.3% with Gemini-3.8-Flash, whereas Qwen3.5-9B does not benefit. Our best runs surpass the previously reported best scores on eight tasks, including Swap Reduction on Q20 and Rosetta, and match them on three more. EvoDuet also improves with other scaffolds (e.g., Top-K, EvoX) on Sums/Diffs and Denoising, demonstrating its applicability across evolutionary search scaffolds.

1. Introduction

LLM-driven evolutionary search can stall when a solution requires external knowledge that the model and its search history do not contain. EvoDuet co-evolves queries and solutions, improving OpenEvolve with some models while showing that retrieval benefits depend on the model and search setting.

1. Introduction

  • Existing loop-packed scaffolds still draw on evolutionary history and the LLM’s parametric knowledge, so they can stall when progress requires external knowledge absent from both.Evolutionary search iterates by proposing candidates, scoring them with an evaluator, and using high-scoring candidates to seed later iterations.Source: §1 · p. 1
  • Oracle documents raised average NDG by 10.3% for Qwen3.5-9B and 4.0% for GPT-5.6-Luna, but Sums/Diffs required eight candidates and in-loop Denoising search repeated 88.1% of URLs.These observations motivate evolving queries alongside solutions rather than relying on a search tool alone.Source: §1 · pp. 1–2
  • EvoDuet co-evolves solutions in an outer loop and search queries in an inner loop, using a knowledge-gap gate to retrieve new documents, reuse stored ones, or proceed without them.The inner loop ranks documents by predicted solution scores; the outer loop generates candidates in parallel and records evaluated outcomes to inform later searches.Source: §1 · p. 2
  • With one candidate per iteration, EvoDuet raises OpenEvolve’s overall NDG from 74.1% to 78.0% with GPT-5.6-Luna and from 61.3% to 82.3% with Gemini-3.8-Flash, but Qwen3.5-9B does not benefit.Best runs surpass previously reported scores on eight tasks and match them on three more; Top-K and EvoX also improve on Sums/Diffs and Denoising.Source: §1 · p. 2
  • Candidates improved on their parent in 67.1% of iterations with a document predicted to beat the parent, versus 52.1% without documents; method transfer was the most common document use.However, documents predicted to help yielded much worse candidates in 29 of 82 runs, so predicted gains still require evaluator validation.Source: §1 · pp. 2–3

2. Analysis of Web Search in Scientific Discovery

This section defines the evolutionary-search setup and tests whether external documents help, whether models can exploit them with more exploration, and whether evolving queries avoids repetitive retrieval.

2.1. Preliminaries: Evolutionary Search Scaffold

  • An optimization task specifies an instruction, an initial solution, and a deterministic, verifiable evaluator; the scaffold selects a parent, generates and scores a candidate, then retains or discards it.A discovery improves on the previous best-known solution in the task’s optimization direction.Source: §2.1 · p. 3

2.2. Experimental Setup

  • The evaluation covers 31 optimization tasks across seven task groups and compares OpenEvolve with oracle-document OpenEvolve using Qwen3.5-9B and GPT-5.6-Luna across four random seeds.Normalized Discovery Gain measures the percentage of the gap between the initial and prior-SOTA scores closed by a run.Source: §2.2 · p. 3

2.3. Does web search help scientific discovery?

  • Oracle documents raised average NDG by 10.3% for Qwen3.5-9B and 4.0% for GPT-5.6-Luna, with benefits varying across tasks and including declines exceeding 1%.They increased Swap Reduction NDG by 6.1% and 41.5% for the respective models, but lowered Rosetta NDG by 9.0% and 23.3%.Source: §2.3 · pp. 3–4
  • On Denoising, joint-level search returned 85 distinct URLs versus EVODUET’s 248 and reached 27.7% held-out NDG versus 84.5%; 88.1% of its URLs had appeared before, versus 62.4%.The methods used comparable query counts; joint-level search stopped improving after iteration 50.Source: Figure 4 · p. 4, §2.3 · p. 4
  • On Sums/Diffs, oracle documents changed NDG from 15.4% to 14.8% at N = 1, but from 25.8% to 40.7% at N = 8.The authors interpret this pattern as evidence that more parallel opportunities can help the LLM explore document-grounded strategies.Source: §2.3 · p. 4, Figure 4 · p. 4

3. EVODUET

EVODUET co-evolves solutions and web-search queries in a bi-level process, with a knowledge-gap gate deciding whether to search, reuse documents, or proceed without them. Its query loop uses predicted solution scores to select documents, while the solution loop evaluates candidates and stores outcomes.

3.1. An Overview of EVODUET

  • EVODUET links an outer solution-optimization loop and an inner query-optimization loop, with query quality measured by the candidate solution’s objective performance.The inner objective is approximated with model-predicted candidate scores because actual scores require generating and evaluating candidates.Source: §3.1 · pp. 4–5

3.2. Outer Loop: Solution Optimization

  • The outer loop generates candidates from retrieved documents, evaluates them, stores documents with actual scores, and can generate N candidates in parallel when searching or looking up.Only the best valid candidate is passed to the scaffold’s selection policy.Source: §3.2 · pp. 5–6

3.3. Knowledge-Gap-Based Retrieval Gating

  • The gate chooses NO-OP when existing knowledge suffices, LOOK-UP when stored documents cover the gap, and RETRIEVE when neither source suffices.LOOK-UP reuses stored documents without a new web search.Source: §3.3 · p. 6

3.4. Inner Loop: Query Optimization

  • The inner loop summarizes population statistics, constructs gap-targeted queries, predicts candidate scores from documents, and retains up to D documents with the highest predicted scores.It iterates query construction and web search while using predicted document scores as a surrogate for candidate evaluation.Source: §3.4 · pp. 6–7

4. Experimental Results

Across 21 optimization tasks, EVODUET improves OpenEvolve’s overall discovery gain for GPT-5.6-Luna and Gemini-3.8-Flash at one candidate per iteration, while Qwen3.5-9B does not benefit. Parallel generation and performance depend on the task and baseline, and the best runs exceed prior best scores on eight tasks and match them on three.

Overall discovery performance

  • At N = 1, overall NDG rises from 74.1% to 78.0% with GPT-5.6-Luna and from 61.3% to 82.3% with Gemini-3.8-Flash, while Qwen3.5-9B declines 14.4%.Across 21 paired tasks, Figure 6 reports discovery performance with and without EVODUET.Source: §4 · p. 7

Results by task group

  • EVODUET improves NDG on mathematics tasks in five of six model-budget settings, with an average gain of 7.1% across all six.At N = 8, Qwen3.5-9B gains 5.5% and Gemini-3.8-Flash gains 6.7%.Source: §4 · p. 7

Scaling with parallel candidates

  • Mean NDG increases monotonically from 89.6% to 95.7% for GPT-5.6-Luna and from 84.5% to 89.6% for Gemini-3.8-Flash as N rises from 1 to 16 on eight tasks.Increasing the candidate budget from N = 1 to N = 8 also improves EVODUET’s overall NDG for all three models.Source: Figure 7 · p. 7, §4 · p. 7

Best-score comparisons

  • EVODUET surpasses previously reported best program scores on eight tasks and matches them on three more across five domains.The mean cost across the eleven reported runs is $45.56.Source: §4 · pp. 7–8

Gains by baseline performance

  • Median gains were -5.4%, +17.5%, +9.2%, and +2.4% when OpenEvolve’s NDG was 5–20%, 20–40%, 40–60%, and 60–80%, respectively.For OpenEvolve NDG ≥80%, the median gain was 0.0% across 75 comparisons; when NDG was below 5%, five of ten cases rose above 5%, with a median gain of +71.8% among those five.Source: §4 · p. 8

5. Analysis and Discussions

The analyses find stronger search outcomes when retrieval is guided by the model’s knowledge state and documents’ predicted value. They also examine how documents are used and how EVODUET transfers across scaffolds and cost settings.

5.1. How Does Web Search Help?

  • Across 4,324 retrievals, hypothetical evidence scores correlate with selected candidates’ evaluated scores (Spearman ρ = +0.86); task-level correlations are positive on all 21 tasks.The median task-level correlation is +0.74.Source: §5.1 · p. 8
  • Promising documents yield parent improvement in 67.1% of RETRIEVE iterations and 14.1% set a run best, versus 52.1% and 9.6% under NO-OP.Under LOOK-UP, the respective rates are 61.2% and 10.1%.Source: §5.1 · p. 8
  • Across 82 GPT-5.6-Luna runs, retrieved methods or principles were applied in 55 runs, while published best scores served as targets in 40.Each run is counted once in every applicable behavior category.Source: §5.1 · p. 8

5.2. Additional Analysis

  • EVODUET improves NDG on Sums/Diffs and Denoising across OpenEvolve, Top-K, and EvoX, with respective gains of 23.5 and 84.5, 46.7 and 63.6, and 55.1 and 37.8.The reported gains are for Sums/Diffs and Denoising, respectively, within each scaffold.Source: Table 1 · p. 9, §5.2 · p. 9
  • The knowledge-gap gate performs best on Denoising and Erdős, with a 23.9% gain over the stagnation heuristic on Denoising.The comparison includes random gating, a stagnation heuristic, and the knowledge-gap gate.Source: Table 1 · p. 9, §5.2 · p. 9
  • EVODUET outperforms OpenEvolve without search, joint-level search, and sequential DeepEvolve on three tasks.The authors identify this as evidence for the bi-level design.Source: Table 1 · p. 9, §5.2 · p. 9
  • On Denoising, EVODUET reaches 100.27% NDG at $38.45, a 6.9× cost reduction versus SimpleTES at 100% NDG and $265.39.The comparison uses the estimated API-equivalent cost for SimpleTES.Source: Table 1 · p. 9, §5.2 · p. 9

6. Related Work

Related work covers LLM-driven evolution of solutions, search strategies, prompts, and agent systems, as well as methods that bring web search into scientific discovery. EVODUET differs in coupling solution and query evolution through a shared task objective.

6. Related Work

  • LLM-driven discovery methods optimize candidate programs, search procedures, prompts, agent or harness code, and—in some approaches—model parameters.The section surveys methods including FunSearch (Romera-Paredes et al., 2024), AlphaEvolve (Novikov et al., 2025), EvoX (Liu et al., 2026a), and GEPA (Agrawal et al., 2025).Source: §6 · p. 9
  • Most related methods operate within a single solution loop; EvoX (Liu et al., 2026a) adds a meta-level strategy loop drawing on the LLM’s existing knowledge and run history.Source: §6 · p. 9
  • DeepEvolve (Liu et al., 2025) adds web search but couples search and solution optimization sequentially rather than through a bi-level formulation.Source: §6 · p. 9

7. Conclusion

The conclusion presents EVODUET as a bi-level method that co-evolves solutions and web queries using the LLM’s knowledge state to guide retrieval. Across 21 tasks, results vary by model, while several task scores and scaffold comparisons improve.

7. Conclusion

  • EVODUET co-evolves solutions and web search queries, using the LLM’s knowledge state to decide when to search and what to ask.Source: §7 · p. 9
  • Across 21 tasks, EVODUET improves OpenEvolve with GPT-5.6-Luna and Gemini-3.8-Flash, but not Qwen3.5-9B.Source: §7 · p. 9
  • EVODUET surpasses previously reported best scores on eight tasks and improves Top-K and EvoX on Sums/Diffs and Denoising.It also matches previously reported best scores on three tasks.Source: §7 · p. 9

AI use statement

These passages document AI use and evaluation boundaries, situate EVODUET among related work, and describe its outer-loop, retrieval-gating, and query-optimization procedures and settings.

AI use statement

  • Generative AI supported trajectory analysis, paper writing, figure preparation, and retrieval and summarization of oracle documents; it was not used for idea proposal or method development.Source: AI use statement · p. 10

Ethics statement

  • Reported gains reflect the task evaluators, and the resulting programs require independent validation before use in other settings.Source: Ethics statement · p. 10

Reproducibility statement

  • The paper points to appendices and supplementary materials for task definitions, prompts, experimental results, search trajectories, and complete program and source listings.Source: Reproducibility statement · p. 10

A. Extended Related Work

  • Related work spans evolutionary discovery, web-augmented scientific discovery, and retrieval control, query refinement, and search-policy learning.Source: A. Extended Related Work · p. 18

B.1. Outer Loop: Solution Optimization

  • The procedure separates an outer loop that evaluates solution candidates and records outcomes from an inner loop that refines queries using predicted candidate scores; model calls use the same frozen Mθ.Source: B. Method Details · p. 18, §B.1 · p. 18

B.2. Knowledge-Gap-Based Retrieval Gating

  • The gate reads the parent, evolutionary history, and recent search records, then returns a knowledge state, a retrieval decision, and—when looking up—stored document identifiers.Source: §B.2 · p. 19

B.3. Inner Loop: Query Optimization

  • Query rounds generate searches, pool and deduplicate documents, score unscored documents as candidate outcomes for the fixed parent, and retain up to D highest-scoring documents.The default procedure runs all R rounds without early stopping; failed or unusable retrieval can leave the outer loop with no documents.Source: §B.3 · p. 20

B.4. Inference Settings and Call Budget

  • Default retrieval uses three rounds, one query per round, five documents per query, and retains three documents; candidate generation is evaluated at N = 1 and N = 8.The default per-iteration retrieval budget allows up to eight retrieval-related model calls and three web searches, excluding retries.Source: §B.4 · pp. 20–21

B.5. Coupling Web Search with Solution Evolution

The section contrasts joint-level and sequential retrieval with EVODUET’s bi-level coupling, then reports retrieval comparisons on Denoising and two published DeepEvolve benchmarks.

B.5. Coupling Web Search with Solution Evolution

  • Joint-level search runs during generation, sequential search precedes it, and EVODUET co-evolves queries and solutions through a knowledge-gap gate.EVODUET refines queries and selects documents using predicted program scores, then stores documents and measured outcomes to inform later decisions.Source: §B.5 · p. 21, Figure 9 · p. 21
  • EVODUET reaches 84.5% NDG on Denoising and 76.0% on Sums/Diffs, versus 43.2% and 42.9% for joint-level retrieval.In the analyzed Denoising trajectory, joint-level retrieval returned 85 distinct URLs versus EVODUET’s 248, with 88.1% of its returned URLs repeated.Source: §B.5 · p. 22, Table 2 · p. 22
  • On Molecule and Burgers, EVODUET scores 0.8524 and 0.7846, above DeepEvolve’s released 0.8149 and 0.6666 and OpenEvolve’s 0.8496 and 0.6937.The local scores are best across available four GPT-5.6-Luna runs at iteration 100; DeepEvolve was not rerun with the same model or search budget.Source: §B.5 · p. 22

C. Experimental Setup Details

The appendix details the task collections, oracle-document construction, and evaluation conventions, then reports how parallel generation changes oracle-document gains across models.

C. Experimental Setup Details

C.1.1. Simple Scientific Optimization

  • The ten simple scientific optimization tasks include evaluator-checked oracle solutions, such as NIST parameters reducing Chwirut2 RSS from 14,794.79 to 513.048 and a sequence reducing energy from 177 to 37.The oracle collection also includes a vessel-design cost of 6,059.71 versus an initial cost of 10,905.34.Source: §C.1.1 · pp. 22–23

C.1.2. SIMPLETES

  • The 21 SIMPLETES tasks use released initial programs and evaluators, spanning areas including algorithm engineering, quantum compilation, mathematics, and astrodynamics.Swap Reduction reports added SWAPs on 24 circuits assigned to a 20-qubit device; astrodynamics tasks share a total-∆v minimization objective.Source: §C.1.2 · pp. 24–26

C.2. Oracle Document Construction

  • For SIMPLETES, Claude Code compares each initial program with the released best program, searches for functional differences, and returns fetched documents summarized to their stated contents.The fetched summaries are semantic rather than verbatim; simple scientific optimization oracles instead state task solutions directly and validate them with the evaluator.Source: §C.2 · p. 27

D.1. OpenEvolve with and without Oracle Documents

  • Table 4 reports mean and best NDG over available runs, evaluating simple scientific tasks at iteration 20 and the remaining tasks at iteration 100.The overall row averages task means and task bests across all 31 tasks.Source: §D.1 · p. 27, Table 4 · p. 28

D.2. OpenEvolve with and without Parallel Generation

  • Increasing parallel candidates from N = 1 to N = 8 raises average NDG by 15.0% with oracle documents and 9.8% without for GPT-5.6-Luna, but by 2.2% and 11.8% for Qwen3.5-9B.The comparisons cover nine Luna tasks and 12 Qwen tasks.Source: §D.2 · p. 27, Table 5 · p. 29

D.3. Inner-loop Budget Ablations

The appendix tests inner-loop budgets on Denoising and documents the full task-group score tables and the behavior-audit procedures and patterns.

D.3. Inner-loop Budget Ablations

  • On Denoising, R = 3 reaches 81.77% NDG, above R = 1 at 58.35% and R = 5 at 34.28%; J = 1 reaches 81.77%, while J = 5 reaches 57.30%.For J = 3, NDG is 80.15% at k = 3 and 71.86% at k = 5; the authors conclude that more inner-loop search is not always better.Source: NDG (%) · p. 30, Table 6 · p. 30

D.4. Full OpenEvolve Results

  • The full OpenEvolve results are organized by task group across Tables 7–10, covering quantum compilation and astrodynamics, scientific algorithms and AI foundations, mathematics, and algorithm engineering.The supplied table captions identify coverage but do not provide score values.Source: §D.4 · p. 30, Table 7 · p. 31, Table 8 · p. 31

E.1. Behavior Flags and Additional Examples

  • The six behavior flags classify method transfer, cross-domain transfer, live documentation lookup, target-informed reconstruction, public artifact reuse, and evidence-inspired over-reach.Run categories can overlap, and the audit distinguishes flags from evidence that a retrieved idea was implemented and retained.Source: §E.1 · p. 33, §E.1.1 · p. 33

E.1.1. Audit Setup and Counting Rules

  • The audit covers 8,200 Luna iterations across 82 runs and 14 Gemini runs on seven tasks, checking gate choices, queries, documents, predicted scores, and child changes.A retrieval is counted as predicted to help when it retains a document whose predicted child score exceeds the parent’s score.Source: §E.1.1 · p. 33
  • A new run best followed a retrieval with no promising document in 57 of 82 Luna runs; identifier fixation occurred in 6 of 75 runs with at least ten retrievals.Direct code copying appeared in 19 Luna iterations (0.2%); manual review found published numerical-solution reuse in four runs, and only one retained a copied solution in its final best program.Source: §E.1.1 · p. 33

E.1.2. What the Worked Cases Establish

The worked cases distinguish durable use of retrieved methods from documentation seeking, reconstruction, artifact reuse, and harmful over-reach. The appendix also reports variation in source exposure and document reuse across task groups and models.

E.1.2. What the Worked Cases Establish

  • In Swap Reduction, dSABRE’s exponential depth weighting became a SWAP-scorer term retained in later best programs; the final router added 14,835 SWAPs on Q20 versus 15,186 for the released program.The case illustrates implementation and persistence beyond a method-transfer flag.Source: §E.1.2 · p. 34
  • On Erd˝os minimum overlap, a retrieved radar codeword seeded an optimizer and remained through four run-best improvements, reducing C5 from 0.381374 to 0.380915.Other flagged cross-domain transfers were weaker: the retrieved Anscombe transform appeared in no candidate, and a quadratic-residue initialization was absent from the best program.Source: §E.1.2 · p. 34
  • In Voyager 2, 28 of 46 retrievals named the wrapper, but retained pages described public Lambert solvers; the flagged child did not lead to the final best program.Across 300 wrapper-targeting retrievals, only two retained a benchmark-source page, and neither improved on its parent.Source: §E.1.2 · p. 34
  • For circle packing at n = 26, an LP-plus-SLSQP solver raised the objective from 1.390 to 2.628 in one iteration and reached 2.635983 by iteration 7 without retrieved coordinates.The result matched the SimpleTES reference to six decimals.Source: §E.1.2 · p. 34
  • In an Erd˝os run, downloading a published witness reduced best C5 from 0.381017 to 0.380859; the reported Table 1a program retains the download.The best program without a runtime download reached 0.381004.Source: §E.1.2 · p. 34
  • On Galileo, a retrieval predicted a child score of 0.63701 versus its parent’s 0.63692, but the child scored 0.37713; 20% of Luna retrievals predicted to help produced a worse child.The run best remained 0.63692, and 29 of 82 runs contained a drop exceeding 10% of total gain.Source: §E.1.2 · p. 34

E.2. Language, Resources, and Reuse in Web Search

  • GitHub is the most exposed host in four of six task domains, reaching 94.9% of quantum-compilation retrieval iterations; arXiv leads AI foundations and algorithm engineering.These rates measure source exposure only.Source: §E.2 · p. 35, Figure 11 · p. 36
  • Luna’s median document reuse age is 17 iterations in scientific algorithms and 44 in quantum compilation; in mathematics, Qwen reuse rises from 25.7% to 55.0% across the first and last ten iterations.In quantum compilation, 12 of Luna’s 21 resolved requests during iterations 81–100 reused documents first retrieved in iterations 1–10.Source: §E.2 · pp. 38–40, Figure 12 · p. 38, Figure 13 · p. 39

E.3. Failure Analysis of Retrieved Evidence Use

Failure analysis shows that Qwen3.5-9B’s document-use problems extend beyond evaluation errors, while parallel candidate selection improves its outcomes. Sampled revisions also reveal that retrieved methods are sometimes unused or incorrectly implemented.

E.3. Failure Analysis of Retrieved Evidence Use

  • The analysis covers 33,784 iterations from 362 runs across 21 tasks and three models, using each run’s first 100 iterations and excluding recorded external service errors.Figure 15 reports failure rates with unsuccessful generations in the denominator; its exposure comparisons are observational because the gate selects document exposure.Source: §E.3 · p. 41, Figure 15 · p. 41
  • At N = 1, Qwen fails evaluation in 12.4% of document iterations versus 9.2% without documents, and only 27.7% improve on the parent.By comparison, Luna improves in 41.1% and Gemini in 31.8% of document iterations; 33.2% of Qwen document iterations produce a lower score and 12.9% retain no child.Source: §E.3 · p. 41
  • At N = 8, Qwen document iterations improve on the parent in 49.3% of cases and fail evaluation in 1.1%.Luna and Gemini improve in 53.3% and 47.1%, respectively; these outcomes reflect both generation and selection because retrieve and look-up generate eight candidates while no-op generates one.Source: §E.3 · p. 41
  • Retrieved methods were unused or incorrectly implemented in 26% of 50 sampled Qwen revisions with documents at N = 1.Of 50 sampled revisions, methods were unused in 3 and incorrectly implemented in 10; 22 incorporated or retained a method, 7 lacked actionable evidence, and 8 were uncertain.Source: §E.3 · p. 41, Table 11 · p. 41

F. Limitations

The limitations identify model- and task-dependent performance, a single-turn solution-generation setup, and unresolved needs for broader evaluation. The discussion distinguishes document usefulness from evidence selection and implementation, while the trajectory section presents worked cases linking searches to program changes and scores.

F. Limitations

  • EVODUET’s benefits vary across backbones; Qwen3.5-9B has lower overall NDG than OpenEvolve at both N = 1 and N = 8.The authors call for evaluation across more model families and sizes, including whether training evidence selection and implementation reduces failures.Source: F. Limitations · p. 42
  • On algorithm-engineering tasks, EVODUET has lower group-average NDG than OpenEvolve in four of six model/budget settings and declines by 1.6% on average.The declines occur with GPT-5.6-Luna and Qwen3.5-9B at both N = 1 and N = 8; the causes and remedies remain for future work.Source: F. Limitations · p. 42
  • The current framework evolves queries in an inner loop but generates each solution candidate in a single turn.The authors suggest extending the approach to multimodal observations and multi-turn query and solution refinement.Source: F. Limitations · p. 42

G. Discussions

  • Oracle-document benefits vary by model and task: the same documents raise Swap Reduction NDG by 6.1% for Qwen3.5-9B and 41.5 for GPT-5.6-Luna, while both lose performance on Rosetta.The authors propose evaluating evidence selection and implementation as separate skills while accounting for baseline performance and search budget.Source: G. Discussions · p. 42

H. Broader Impact

  • The authors recommend retrieval and optimization trajectory records, source attribution, independent program validation, and human oversight because unreliable sources or reused published solutions can mislead conclusions or novelty claims.They frame these as safeguards for computational scientific discovery using web knowledge.Source: H. Broader Impact · p. 42

I. Trajectories of the Bi-Level Loop

  • The seven worked trajectory cases link recorded queries and sources to program changes and evaluation results; six use GPT-5.6-Luna at N = 8, while Rosetta uses GPT-5.6-Sol at N = 1.The cases are selected chronologically and illustrate behavior flags or their limitations.Source: §I · p. 42, Table 12 · p. 43
  • Trajectory records include gate rationale, retained documents and predicted child scores, query intent, retrieved-page excerpts, code edits, and measured parent-child scores.The gate decisions include retrieve, look-up, and no-op; predicted document scores are not evaluation results.Source: §I · p. 43

I.1. Swap Reduction: depth weighting persists despite repeated search misses

The section tracks Swap Reduction variants and search behavior, reporting persistent exponential depth weighting in later best programs alongside repeated misses of relevant Rust implementation details. It describes the strongest measured result from earlier heuristic changes, mixed or worse later scores, and the limits of what the comparisons establish.

I.1. Swap Reduction: depth weighting persists despite repeated search misses

  • 14,835 SWAPs versus SimpleTES’s 15,186 under the same evaluator shows an advantage specific to Q20, while exponential depth weighting remains in every later best program.79 of 94 retrievals named Qiskit PR #14912 without obtaining its diff; iteration 77 set a new best after a retrieval with no promising document.Source: §I.1 · p. 43
  • 6957.0 was the strongest measured combined_score from relative distance-delta scoring with component-aware initial layout; attempt 3’s more elaborate layout placement scored 6565.8.The lower score means that layout change should not be retained.Source: §I.1 · p. 44
  • The measured improvement from 6498 to 11325.6 was substantial, but three listed attempts had identical routing metrics and did not isolate which modifications caused the gain.The evidence therefore does not establish whether the current heuristic is near a plateau.Source: §I.1 · p. 45
  • 15,493.8 is the strongest measured combined score for gamma=0.70, lookahead_size=16, interaction-aware initial layout, and multiplicative endpoint decay.Earlier experiments found additive decay and slower gamma=0.84 were not independently superior.Source: §I.1 · p. 46
  • 17,201.4 was the best observed combined score, but changes to decay, endpoint handling, initial layout, and candidate scoring did not produce reliable improvement.The latest layout-related attempt produced no change.Source: §I.1 · p. 47
  • Later variants using geometric lookahead, stagnation scaling, topology-dependent weighting, completion bonuses, and multiplicative decay produced mixed or worse scores than earlier heuristic changes.The supplied passages give no numerical scores for these variants.Source: §I.1 · p. 48
  • Existing documents establish standard SABRE normalization and motivate LightSABRE-style progress escapes, but not the exact Rust implementation or its behavioral details.Prior searches mostly returned generic SABRE documentation, release notes, or irrelevant results rather than the actual patch.Source: §I.1 · p. 48

I.2. Erd˝os minimum overlap: a radar codeword seeds later improvements

The subsections report how retrieved or published information affected optimization across Erdős minimum overlap, circle packing, Voyager, and Galileo. They describe solver outcomes, including improvements and cases where retrieval did not yield the best result.

I.2. Erd˝os minimum overlap: a radar codeword seeds later improvements

  • C5 fell from 0.385530 to 0.380915, and the retrieved radar codeword remained in all four improvements to the run best after iteration 28.Queries narrowed from low-sidelobe code constructions to an explicit sequence; iteration 30 reused the stored source and produced the first subsequent improvement.Source: §I.2 · p. 49
  • The latest revision reached a measured combined score of 2.0312 while maintaining validity, using structured m-sequence search followed by stochastic pairwise refinement.The remaining challenge—lowering maximum non-cyclic overlap under box and sum constraints—was unresolved.Source: §I.2 · p. 50
  • Retrieved literature confirms balanced length-64 binary sequences with optimal aperiodic PSL 4 exist, but the stored sources lack an explicit usable codeword as optimization plateaus near c5=0.3830.One knowledge-state report gives a repeated combined score of 2.6024; additional evidence would help replace stochastic search with an explicit optimal sequence.Source: §I.2 · pp. 51–52
  • Prior evolutionary attempts found balanced binary sequences outperform the constant vector; annealing plus fractional refinement reached approximately c5=0.3814-0.3826, though stochastic search varied and had not clearly exploited known optimal sequence structure.Generic retrieval about optimization methods produced no measured improvement, while reusing known balanced 64-bit minimum-PSL code information gave the strongest improvement.Source: §I.2 · pp. 54–55

I.3. Voyager 2: documentation lookup repeats defaults already in the prompt

  • 3.446 km/s final best ∆v followed most of its reduction without search at iterations 4–8, from 4.051 to 3.450 km/s.The iteration-41 child beat its parent but did not set a new run best, and the final best did not descend from it.Source: §I.3 · p. 55
  • Voyager documentation lookups did not establish an API correction: the wrapper signature was already in the prompt, and iteration 41 passed its existing defaults.The run named the Lambert wrapper in 28 of 46 retrievals; the retrieved pages described public Lambert solvers.Source: §I.3 · p. 55

I.4. Circle packing (n=26): a generated solver exceeds the queried target

  • 2.635983 at iteration 7 exceeded the queried 2.635977 target and matched the SimpleTES reference to six decimals, after sum of radii rose from 1.390 to 2.628 at iteration 1.The returned pages contained no coordinates; the best program used no retrieved coordinates or runtime download.Source: §I.4 · p. 58
  • The circle-packing child combined a linear program for radii with SLSQP optimization, raising the initial constructor’s sum_radii 1.3903 to validity 1.0 at initialization.The initial approach used a 6x6 grid with 26 points and sequential pairwise radius scaling rather than coupled radius optimization.Source: §I.4 · p. 58
  • The circle-packing solver improved from 2.6284 to 2.6343, but remained below reported n=26 records around 2.63586–2.63598 and the documented 2.63598308 record.The 2.6284 approach used staggered candidate centers, farthest-point sampling, SLSQP center optimization, and an LP radius referee; the later multistart SLSQP/LP approach reached 2.6343.Source: §I.4 · pp. 59–60

I.5. Erd˝os minimum overlap, second run: runtime witness reuse improves the best

  • C5 fell from 0.485374 to 0.380859; at iteration 17, a downloaded and validated witness beat its parent (0.381075) and the previous run best (0.381017).The best program retained the local optimizer as a fallback and preserved the external dependency.Source: §I.5 · p. 61
  • The reported Erdős’ Minimum Overlap result was 0.380871 versus a previous best of 0.380876, an improvement of −0.000005.This benchmark-table result is distinct from the run’s 0.380859 score.Source: §I.5 · p. 62
  • A valid SLSQP soft-max/epigraph approach reaches c5 about 0.3810, but remains above the documented 512-cell record near 0.3808590566.The strongest known witness is the lnzwz_AI4M_Agent n=512 construction with a repaired exact-feasibility vector; local grid refinement and ordinary local polishing are largely exhausted.Source: §I.5 · p. 64

I.6. Galileo: a promising retrieval produces a worse child

  • Iteration 33's predicted 0.63701 child scored 0.37713 as ∆v rose from 1.926 to 3.873 km/s; it did not displace the 0.63692 best, unchanged through iteration 34.The retrieved sources offered general formulation guidance but not implementation-level advice for the specific evaluator; prior revisions had converged to the same local solution.Source: §I.6 · p. 65

I.7. Rosetta: published seeds and implementation details support a better tour

This section documents retrieval prompts and information-seeking decisions alongside Rosetta and Swap Reduction implementation details, including the reported tour, selected programs, and swap-selection heuristics. It also describes a short-lookahead Q20 router and programs that improve on or match released references.

I.7. Rosetta: published seeds and implementation details support a better tour

  • 1.396424 km/s is 10.1% below SimpleTES’s 1.552968 km/s under the same evaluator for the final Earth–Earth–Mars–Earth–Earth–67P tour.Published numerical seeds remain in the final program; the run used one candidate per iteration and was outside the behavior-audit population.Source: §I.7 · p. 67
  • Published GTOPX leg durations and DSM fractions seeded the tour, and stored-document reuse supplied PaGMO’s departure-frame velocity conversion.The run expanded timing and direction seeds at iteration 9, applied the conversion at iteration 18, and refined the program without retrieval at iteration 98.Source: §I.7 · p. 67

STORED SOURCES

  • Earlier Rosetta attempts plateaued near 1.396943 km/s despite incorporating MGA-1DSM transcription, departure-direction conventions, a published decision vector, and flyby geometry.The stored sources covered relevant transcription and branch concepts, but attempts using them did not surpass the plateau.Source: STORED SOURCES · p. 70

J. Prompt Templates

  • At each iteration, the retrieval gate chooses no-op, look-up, or retrieve using evolutionary history, the current program, the search database, and measured parent-child outcomes.After retrieve, population analysis runs once, then prompts generate and refine queries and score new documents; measured outcomes are distinguished from predictions, and missing outcomes remain unknown.Source: J. Prompt Templates · p. 72
  • The retrieval-gate prompt asks the LLM to describe its knowledge state and choose no-op, reuse a stored document, or retrieve new material.For look-up, it must select valid document IDs from the database; no-op and retrieve require an empty ID list.Source: J. Prompt Templates · pp. 72–73
  • Population analysis prompts require number-backed observations about score range, spread, trajectory, selection, and outcomes, without recommendations or invented numbers.They direct the LLM to cite specific statistics and report patterns factually.Source: J. Prompt Templates · p. 73
  • Query prompts combine population and knowledge-state analyses, evolutionary history, the current program, and prior search outcomes to target unresolved questions.Later rounds also use selected documents and predicted child scores, while avoiding repeated searches unless a specific question remains unresolved.Source: J. Prompt Templates · pp. 73–75
  • Evidence-scoring prompts predict child scores for unscored documents using each document alone, preserve existing predictions, and update what is known and still needed.Predicted scores are not measured outcomes; outcomes from combined evidence do not reveal each document’s individual contribution.Source: J. Prompt Templates · pp. 75–76
  • The knowledge-state analyzer treats programs, history, and analyses as untrusted data and distinguishes prior knowledge, web evidence, empirical findings, and remaining unknowns.Population statistics cover retained solutions rather than every historical attempt, so implementation claims must rely on the supplied program and history.Source: J. Prompt Templates · p. 77
  • The prompt asks the LLM to track prior task knowledge, web-search and population-analysis findings, and remaining unknowns.Source: J. Prompt Templates · p. 77

K. Best Programs of EVODUET

  • Eight of eleven selected programs improve on SimpleTES under the same evaluator, while three match within 10−9.The appendix reproduces the saved candidate files unchanged and runs them with the original task harnesses and dependencies.Source: K. Best Programs of EVODUET · p. 77, Table 13 · p. 78

K.1. Swap Reduction: short lookahead reduces routing cost on Q20

  • The Q20 router reduces SWAP count from 15,186 to 14,835 using a 16-gate lookahead weighted by 0.6i and extra reward for executable front-layer gates, increased when routing stalls.Source: §K.1 · p. 78
  • The candidate-swap routine enumerates each legal edge incident to the front layer once, avoiding tie-breaking bias from repeated active endpoints while retaining all legal swaps.Source: §K.1 · p. 82
  • The initial layout ranks qubits by interaction frequency, prioritizes the first dependency layer for central sites, then anchors active qubits and places others near their interaction partners.Source: §K.1 · pp. 84–85
  • Currently executable obligations receive extra placement weight, preserving the global interaction-frequency layout while embedding the initial front layer.Source: §K.1 · p. 85
  • A short, geometrically decayed lookahead considers successor gates while keeping immediate front-layer progress dominant, especially on sparse or narrow topologies.The extended-set contribution is normalized by its effective horizon; lookahead influence decreases after swaps fail to make progress.Source: §K.1 · pp. 86, 87, 89
  • The heuristic retains gentle standard SABRE multiplicative decay and softens endpoint penalties on narrow, low-branching graphs to avoid unnecessary detours.It preserves stronger diversification on well-connected topologies.Source: §K.1 · pp. 86–87
  • Front-layer gate completion is rewarded beyond distance reduction, with its preference increased during stagnation to favor decisive progress.Completing a gate releases dependent DAG nodes and can change the next routing step’s candidates.Source: §K.1 · p. 88

K.2. Rosetta: refining a published tour lowers maneuver cost

This section describes optimization and refinement approaches for spacecraft tours, denoising and scaling-law models, mathematical constructions, and circle packing. It presents their methods and resulting scores, constraint checks, or comparisons with reference solutions where reported.

K.2. Rosetta: refining a published tour lowers maneuver cost

  • Using published GTOPX values as a seed, differential evolution and Powell optimization reduced total ∆v from 1.552968 to 1.396424 km/s (10.1%), with unpowered flybys above the 300 km altitude floor and a fixed arrival date.The deterministic Cartesian MGA optimizer used one deep-space maneuver per leg and generated unpowered flybys with gravity-assist dynamics.Source: §K.2 · p. 89
  • Timing, departure velocity, the Mars and final-Earth turns, and the terminal DSM dominate the residual cost in the demonstrated optimum.Source: §K.2 · p. 100

K.3. Voyager 2: constrained refinement improves the existing tour

  • 3.430214 to 3.430206 km/s: constrained refinement lowers Voyager 2’s total ∆v for the same Earth–Jupiter–Saturn–Uranus–Neptune tour without deep-space maneuvers.The method fits unpowered flybys by least squares, then minimizes boundary ∆v with SLSQP; the selected trajectory arrives at the upper bound of the allowed window.Source: §K.3 · p. 103
  • Voyager 2 tour refinement minimizes boundary delta-v under exact unpowered-flyby constraints, using deterministic historical, corner, and distributed seeds.The implementation searches deterministic flyby manifolds and returns the lowest-cost valid tour.Source: §K.3 · pp. 108–109

K.4. Denoising: mixing diffusion operators improves average accuracy

  • Held-out mean score rises from 0.722690 to 0.722906, with a PBMC gain partly offset by a Tabula loss.The method mixes ordinary MAGIC diffusion with a self-loop-free version using cell-specific weights, followed by shrinkage and gene calibration; PBMC scores 0.7116 vs. 0.7090, while Tabula scores 0.7342 vs. 0.7364.Source: §K.4 · p. 110
  • The denoising program mixes conventional and leave-one-out diffusion estimates to reduce self-referential overfitting while retaining local structure.It transforms estimates back to count scale, shrinks rare genes, calibrates globally, and weakly blends with log-normalized counts.Source: §K.4 · p. 112

K.5. Domain Mixture: separate domain fits improve held-out prediction

  • Held-out R2 is 0.997062 versus 0.996922 for the reference after fitting a separate seven-parameter law for each of five domains.The law combines exponential mixture contrasts with a log-share correction for each domain’s own data fraction, using bounded least squares, analytic Jacobians, and multiple initializations.Source: §K.5 · p. 114

K.6. Parallel Scaling: a fixed basis supports accurate extrapolation

  • Held-out R2 reaches 0.999975 versus 0.999970 on P = 8, beyond the training range P ≤4, using a fixed four-term basis.The basis is (1, u, v, uv), with u = (N/10^9)^−0.2 and v = (1 + 0.4 log2 P)^−0.2; coefficients are fit by linear least squares.Source: §K.6 · p. 116

K.7. Erd˝os: public witnesses support a small numerical refinement

  • 0.380859056341 improves the SimpleTES reference 0.380867675827; downloading feasible public witnesses and refining the lowest-overlap one with SLSQP accounts for almost all gain.Local refinement improves the reused witness’s reported value of 0.3808590566 by only about 2.6 × 10^-10.Source: §K.7 · p. 116
  • The Erdős witness method uses low-temperature continuation, epigraph polishing, and selection of the best feasible continuous or binary profile.The code also tests the closest exactly binary profile, which preserves the required mass exactly and can remove unnecessary fractional values at zero shift.Source: §K.7 · p. 122

K.8. Hadamard: structured constructions recover the reference determinant

  • The resulting Hadamard matrix has |det| = 228 · 320 · 712 and normalized score 0.935673, matching the reference determinant.The program builds order-28 Hadamard matrices from Goethals–Seidel, Paley, and Williamson constructions, borders them to order 29, and refines them with tabu search using rank-one determinant updates.Source: §K.8 · p. 123
  • The search also reports the world-record determinant 45/7 * 28^14 = 1,170,380,056,049,630,576,640, with ratio 0.9211.Source: §K.8 · p. 123
  • The Hadamard implementation runs multi-start tabu search across candidate starts and randomized equivalents.Source: §K.8 · p. 129

K.9. Sums/Diffs: construction sweeps and local search improve the reference

  • 3,575 distinct sums and 2,793 distinct differences yield c = 1.144999 for the selected 509-element set, versus 1.144887 for the released post-training construction.An independent replay reproduces the counts and score exactly.Source: §K.9 · p. 129
  • The Sums/Diffs method sweeps generalized Penman–Wells families across moduli, then uses an alternating hill-climber with additions, removals, swaps, and perturbation kicks.The search includes zero-leakage additions and iterated local search to escape local optima.Source: §K.9 · p. 130
  • Local search uses exact bitmask metric evaluations and combines removals, targeted additions, two-element operations, and boundary swaps.Source: §K.9 · p. 131
  • The construction sweeps generalized Penman-Wells families, ranks seeds, then applies deep multi-stage hill-climbing with perturbation kicks.It optimizes an MSTD set to maximize C(A).Source: §K.9 · pp. 133–134
  • C(A) ≥ 1.144710 is the guaranteed champion baseline, using M=56, j=30, and fringe element 1793.Source: §K.9 · p. 134

K.10. Circle Packing (n = 26): joint refinement matches the reference

  • 2.635983084918461 sum of radii matches the reference within 10⁻⁹ after independent replay passes boundary and non-overlap checks.The program starts from varied six-row layouts, maximizes radii through linear programming, then jointly refines centers and radii with SLSQP, collective deformations, and local perturbations.Source: §K.10 · p. 136

K.11. Circle Packing (n = 32): multiple layouts recover the reference score

  • 2.939572771209394 is the achieved sum of radii, with the score matching the reference within 10^-9.An independent replay passes the evaluator’s boundary and non-overlap checks.Source: §K.11 · p. 141
  • The search explores five- and six-row layouts using SLSQP, alternating fixed-center radius optimization with joint center–radius refinement and restarts.Small perturbations are used to refine candidate layouts.Source: §K.11 · p. 141
  • Circle-packing refinement re-optimizes radii after geometry changes and uses small-basin restarts to explore improvements around a polished incumbent.The code describes coherent deformations to change the active contact graph, with perturbations ranging from wider basin escapes to fine neighbor refinement.Source: §K.11 · pp. 146–147

K.12. Discovered Objects

  • The figures compare discovered objects and program outputs with SimpleTES, including a one-SWAP Q20 circuit versus two and sum–difference scores of 1.144999 versus 1.144887.They also show an Erdős witness and order-29 Hadamard matrices reaching the same determinant.Source: §K.12 · p. 148, Figure 16 · p. 148, Figure 20 · p. 150
  • For Rosetta, EvoDuet adds the actual mission’s Mars flyby; for Voyager 2, both programs follow the same tour, but EvoDuet moves arrival to the end of the window.The figure also splits total ∆v into maneuvers and launch or arrival.Source: Figure 17 · p. 149
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