Hyperagents

• Fixed meta agents constrain improvement to their initial design, while adding meta-meta systems creates an infinite regress of increasingly higher-level mechanisms.Self-referential systems must analyze, modify, and evaluate themselves to avoid this limitation.Source: §1 · pp. 1–2
• The DGM demonstrates open-ended coding improvement but depends on a fixed instruction-generation mechanism and assumes coding evaluation skills transfer to self-reflection and modification.That alignment is unlikely outside coding, where task-solving and self-improvement skills may differ substantially.Source: §1 · p. 2
• Hyperagents combine task and meta agents in one modifiable program, allowing metacognitive self-modification of both task-solving and improvement-generation processes.DGM-Hyperagents retain DGM’s archive-based branching and evaluation while adding editable self-modification.Source: §1 · p. 2
• DGM-Hyperagents improve coding, paper review, and robotics performance, transfer gains to held-out tasks, and outperform prior self-improving algorithms beyond coding.Ablations without self-improvement or open-ended exploration show little to no progress.Source: §1 · p. 2
• DGM-Hyperagents learn transferable meta-level mechanisms such as persistent memory and performance tracking, which transfer across domains and compound with continued self-improvement.The work also discusses safety implications and responsible deployment of open-ended self-improving systems.Source: §1 · p. 2

Open-Endedness

• Open-ended systems generate increasingly diverse and complex artifacts across domains, motivating research into systems that can compound improvement by improving both artifacts and the processes that produce them.This direction builds on work using foundation models for generation and evaluation, open-ended learning, and quality-diversity algorithms across games, science, and robotics.Source: §2 · p. 3

Self-improving AI

• Self-improving AI has progressed from theoretical self-modifying agents to meta-learning, evolution, self-play, foundation-model refinement, and code-repository modification, with DGM as a practical recursive system for coding.Most existing approaches, including DGM and its derivatives, retain fixed, handcrafted meta-level mechanisms that constrain compounding and cross-domain generalization.Source: §2 · p. 3

Self-referential Meta-learning

• DGM and related self-referential systems improve recursive self-modification primarily in coding because their evaluation and self-modification tasks are closely aligned.Improving a coding agent therefore improves its ability to perform subsequent coding-based self-improvements, whereas gains in poetry writing would not necessarily improve code modification.Source: §2 · pp. 3–4
• Hyperagents remove the alignment assumption by making the self-modification mechanism fully modifiable and independent of any particular task domain, allowing both task performance and improvement processes to advance across computable tasks.This distinguishes them from prior systems whose self-improvement depends on the target task sharing skills with self-modification.Source: §2 · p. 4

DARWIN GÖDEL MACHINE (DGM)

• The framework defines agents as arbitrary computable programs, allowing task agents and meta agents to include external models, tools, memory, algorithmic logic, and control flow.A meta agent accesses the archive of prior agents and evaluations to propose changes targeting task-solving logic or the meta agent itself.Source: DARWIN GÖDEL MACHINE (DGM) · pp. 4–5
• Hyperagents integrate a task agent and a meta agent into one editable program that can rewrite both task-solving and future self-improvement procedures.Because hyperagents can edit arbitrary Python code, they potentially can build any computable machine.Source: DARWIN GÖDEL MACHINE (DGM) · pp. 4–5
• Metacognitive self-modification lets hyperagents improve the procedures that propose and apply subsequent self-modifications alongside task performance.This directly addresses prior systems whose self-improvement mechanisms were not themselves editable.Source: DARWIN GÖDEL MACHINE (DGM) · p. 5
• DGM-H retains the DGM’s open-ended population-based exploration by maintaining an archive of hyperagents whose generated variants are evaluated and accumulated as stepping stones.The process alternates between metacognitive self-modification and empirical evaluation, helping mitigate premature convergence and local optima.Source: DARWIN GÖDEL MACHINE (DGM) · pp. 4–5, Figure 1 · p. 4
• Parent hyperagents are sampled probabilistically according to performance and inversely according to the number of successfully compiled children, balancing exploitation with exploration.The main experiments use a handcrafted, non-modifiable parent-selection mechanism to isolate hyperagent self-modification, while automatic parent-selection improvement is explored preliminarily.Source: DARWIN GÖDEL MACHINE (DGM) · p. 5

4 Experiment Setup

• DGM-H starts from a frozen FM with bash and file-inspection tools, while its editable task and meta agents can modify any part of the codebase.The experiment budget is a hyperparameter, and the initial implementation is deliberately simple to preserve a broad search space.Source: §4 · p. 5

4.1 Baselines

• Baselines isolate fixed meta-level mechanisms, non-population search, human-engineered DGM modification, and static domain-specific solutions.DGM-H without self-improvement fixes the meta agent; DGM-H without open-ended exploration discards the archive; DGM-custom manually adapts DGM’s instruction generator across domains.Source: §4.1 · p. 6

4.2 Domains

• The study covers coding, paper review, robotics reward design, and Olympiad-level math grading, with staged screening, validation where needed, and held-out test tasks for final evaluation.Agents are first evaluated on small task subsets, while paper review and math grading use validation subsets because their AI judges may overfit training data.Source: §4.2 · p. 6
• Paper review tests alignment with observed human accept/reject decisions in a subjective setting without objective ground truth, using the AI-Scientist-v2 reviewer as a static baseline.Agents predict binary acceptance decisions for research papers drawn from publicly available submissions and conference outcomes.Source: §4.2 · p. 6
• Robotics reward design tests whether agents can use simulators and RL to generate rewards that transfer from training quadrupeds to a held-out torso-height task.Training requires rewards for forward walking, whereas testing requires zero-shot rewards that maximize torso height, with no separate validation task.Source: §4.2 · p. 7
• Olympiad-level math grading is a held-out meta-evaluation of whether improvements to DGM-H’s self-improvement process transfer across domains and compound over time.Using IMO-GradingBench, agents predict discrete scores for candidate solutions, measured against expert grades, with ProofAutoGrader as the static baseline.Source: §4.2 · p. 7

5 Results

• DGM-H exhibits self-improvement at both task and meta levels, with task gains transferring to held-out tasks and meta-level gains transferring across domains and runs.Experiments use five runs, report medians with 95% bootstrap confidence intervals, and assess significance with the Wilcoxon signed-rank test.Source: §5 · p. 7

5.1 Improving Task Performance

• DGM-H matches the original DGM’s coding self-improvement while avoiding coding-specific handcrafting, improving Polyglot training performance from 0.140 to 0.340 and full-benchmark performance from 0.084 to 0.267.The original DGM improves from 0.140 to 0.380 on the training subset and from 0.142 to 0.307 on the full benchmark (Zhang et al., 2025b).Source: §5.1 · pp. 7–8
• DGM-H improves held-out paper review from 0.0 to 0.710 and robotics reward design from 0.060 to 0.372, surpassing their stated static or default baselines.Paper review’s static baseline scores 0.630, while robotics reward design’s default reward function scores 0.348.Source: §5.1 · p. 9
• DGM-H’s metacognitive self-modification and open-ended exploration are both necessary: removing either significantly reduces test performance in paper review and robotics reward design.Without self-improvement, scores are 0.0 and 0.213; without open-ended exploration, they are 0.0 and 0.116, respectively, across the two domains.Source: Figure 2 · p. 8, §5.1 · p. 9

5.2 Improving the Ability to Improve

• On Olympiad-level math grading, transfer hyperagents achieve imp@50 of 0.630 (CI: 0.540–0.630), while DGM transfer agents achieve little improvement (imp@50 ≈0), demonstrating transferable meta-level self-improvement.The transfer hyperagents’ advantage over the initial agent is statistically significant (p < 0.05), with performance tracking and persistent memory identified as qualitative mechanisms.Source: Figure 3 · p. 10, §5.2 · p. 11

5.3 Compounding Self-Improvements

• Starting from a transferred hyperagent and ProofAutoGrader, DGM-H reaches 0.700 versus ProofAutoGrader’s 0.670, and improves full-benchmark accuracy from 0.561 to 0.601.On the full IMO-GradingBench, mean absolute error also decreases from 0.178 to 0.175.Source: §5.3 · p. 13, Figure 4 · p. 13

Experimental safeguards

• DGM-H experiments operate under sandboxing, resource limits, predefined evaluations, and continuous human oversight to contain failures and unintended side effects.Agent-generated code is confined to controlled environments with measures such as timeouts and restricted internet access.Source: §6 · p. 13

Oversight and trust

• Self-improving systems may evolve faster than humans can audit or interpret, making trust, transparency, and controllability central safety challenges (Bengio et al., 2024).The discussion frames safety as balancing AI-enabled progress and well-being against risks and the degree of trust placed in systems without continuous human verification (Clune, 2019; Ecoffet et al., 2020; Bengio et al., 2024; Weston and Foerster, 2025).Source: §6 · p. 13

Future deployment

• Safeguards that support DGM-H’s current safe research boundaries may become strained or infeasible as self-improving systems grow more capable (Clune, 2019; Ecoffet et al., 2020; Sheth et al., 2025).The paper calls for continued discussion of appropriate trust, oversight, transparency, and societal priorities for deployment.Source: §6 · pp. 13–14

7 Limitations and Conclusion

• DGM-H produces generalizable task gains while improving its own self-improvement mechanism, with meta-level improvements transferring across domains and compounding across runs.The system evolves an archive of self-improving hyperagents for any computable task.Source: §7 · p. 14
• Fixed task distributions and outer-loop components, including parent selection and evaluation protocols, limit DGM-H’s full self-modifiability and truly unbounded progress.Future work includes co-evolving tasks and curricula and enabling hyperagents to modify outer-loop search and evaluation; preliminary evidence suggests this is feasible in Appendix E.5.Source: §7 · p. 14
• DGM-H suggests a path toward self-accelerating systems that continually improve their ability to self-improve, provided sufficient safety considerations are addressed.The conclusion frames this as extending open-ended self-improvement across diverse domains.Source: §7 · p. 14

A Algorithmic details

• The algorithmic appendix describes the initial hyperagent, archive parent selection, and pseudocode for the DGM-H open-ended exploration process.These components cover the implementation setup, exploitation–exploration balancing, and the overall iterative procedure.Source: A Algorithmic details · p. 21

A.1 Initial Agent

• The initial hyperagent combines a task-agent JSON-response prompt with a meta-agent instruction to modify any part of the codebase, using bash and editor tools.The editor supports viewing, creating, replacing, inserting, and undoing file edits.Source: §A.1 · p. 21

A.2 Parent Selection

• Parent selection favors high-performing agents while down-weighting agents with many compiled children, preserving probabilistic exploration as the archive improves.The mechanism uses a dynamic performance frontier, sigmoid weighting with λ = 10, and a novelty bonus based on compiled-child counts.Source: §A.2 · pp. 22–23

A.3 Pseudocode

• The DGM-H loop selects parents, applies metacognitive self-modification, evaluates valid child agents on the task set, and adds them to the archive.The pseudocode initializes the archive with the evaluated initial agent and repeats this process for a maximum number of iterations.Source: §A.3 · p. 23

A.4 Multi-domain Optimization

• In multi-domain runs, parent selection uses average performance across domains, while shared evaluations let hyperagents develop mechanisms benefiting multiple tasks without manually specifying priorities.The meta agent can inspect all task evaluations and introduce shared structured reasoning, memory, and error handling.Source: §A.4 · p. 24

B Baseline Details

• DGM-H baselines remove either self-improving meta agents or open-ended exploration to isolate these components.The corresponding algorithms retain iterative evaluation and agent archives, while the no-open-ended-exploration variant keeps only the latest agent.Source: B Baseline Details · pp. 24–25
• The original DGM uses a handcrafted instruction generator before self-modification, unlike hyperagents’ editable meta-level procedure.Its pseudocode explicitly applies InstrGen(a) with handcrafted instruction generation before modifying and evaluating the new agent.Source: B Baseline Details · p. 25, Figure 5 · p. 26

DARWIN GÖDEL MACHINE (DGM) | CODING DOMAIN

• In coding domains, the original DGM uses one coding agent for both task solving and self-modification, so coding improvements can improve self-improvement.The conceptual comparison describes this alignment as enabling the DGM to improve at improving in coding domains.Source: Figure 5 · p. 26, DARWIN GÖDEL MACHINE (DGM) | CODING… · p. 26
• DGM-H instead combines task and meta agents in one editable program, enabling metacognitive modification of the procedure that generates future self-modifications.This design is presented as addressing the original DGM’s limitations when optimizing computable tasks beyond coding.Source: Figure 5 · p. 26, DARWIN GÖDEL MACHINE (DGM) | CODING… · p. 26
• The original coding-agent prompt requests one general improvement plan based on implementation, logs, issue information, predicted patches, and test results.It requires structured JSON fields for log summarization, potential improvements, one proposal, implementation guidance, and a problem description, while avoiding issue-specific fixes.Source: DARWIN GÖDEL MACHINE (DGM) | CODING… · pp. 26–27
• DGM-custom extends the prompt with both coding-agent and task-agent implementations, task information, reports, and logs to propose general coding or task-solving improvements.Its output likewise requires a single high-impact proposal and an implementation suggestion rather than a task-instance-specific fix.Source: DARWIN GÖDEL MACHINE (DGM) | CODING… · pp. 27–28

C Domain Details

• Across domains, training evaluations provide feedback during self-modification, validation tasks guide parent selection, and held-out test tasks are reserved for final evaluation.When no validation split exists, parent selection instead uses training performance.Source: Table 2 · p. 28

C.1 Polyglot

• Polyglot tasks require repository edits that pass predefined tests, using 60 training tasks and a final evaluation on 165 unseen benchmark tasks.The setup reports pass@1, unlike the leaderboard’s pass@2, so agents do not receive ground-truth test feedback.Source: §C.1 · pp. 28–29

C.2 Paper Review

• Paper-review agents classify full AI research papers as Accept or Reject, with accuracy measured against real conference decisions across 100-task training, validation, and test splits.Agents are first evaluated on 10 training tasks and proceed to all 100 training tasks after succeeding on at least one.Source: §C.2 · p. 30

C.3 Robotics Reward Design

• Robotics agents generate Python reward functions for Go2 control tasks, with performance measured after training PPO policies in Genesis and averaging repeated simulator evaluations.Training uses six evaluations when at least one of the initial three produces non-zero performance, while the test task requires non-myopic reward design for jumping to maximize torso height.Source: §C.3 · p. 32

C.4 Olympiad-level Math Grading

• Olympiad-level math agents grade candidate solutions with scores {0, 1, 6, 7}, measuring accuracy against expert annotations over 100-task training, validation, and test splits.The grading prompt supplies the problem, reference solution, and task-specific guidelines, then requires step-by-step verification and a parsed <points>N out of 7</points> output.Source: §C.4 · pp. 33–34
• The Olympiad-level math grader extracts a numeric score from <points> tags and maps 0, 1, 6, and 7 to incorrect, partial, almost, and correct.Missing tags or scores, and extraction errors, produce a None prediction.Source: §C.4 · pp. 34–35

D Experiment Details

• The experiment-details appendix covers foundation models and hyperparameters, computational costs, improvement@k, and transfer-agent selection for reproducibility.It specifically organizes these details across Appendices D.1–D.4.Source: D Experiment Details · p. 35

D.1 Hyperparameters for FMs

• Table 3 specifies the foundation models and temperatures used across domains, including Claude-4.5-Sonnet for self-modification, o4-mini for Olympiad mathematics evaluation, and temperature 0.0 except o4-mini at 1.0.Polyglot follows Zhang et al. (2025b), while task-evaluation models reflect cost, rate limits, latency, and competence.Source: Table 3 · p. 35, §D.1 · p. 35

D.2 Cost Estimate

• 88.6M tokens is the estimated total cost of a 100-iteration run, combining 33M tokens for self-modification with 0.506M × 100 for evaluation.The granular evaluation estimates are 0.89M tokens for Polyglot, 0.5M for paper review, 0.006M for robotics reward design, and 0.11M for IMO-GradingBench.Source: §D.2 · pp. 35–36

D.3 Improvement@k Metric

• Improvement@k measures the maximum evaluation-score gain obtainable by generating up to k modified task agents from an initial agent under a fixed meta agent and generation algorithm.It compares the best generated agent with the initial agent and is intended for relative comparisons under matched initial conditions and fixed evaluation budgets.Source: §D.3 · p. 36
• The metric treats gains linearly and can underestimate improvements near performance saturation, but this limitation does not affect the paper’s matched-condition analyses.For example, equal absolute gains may differ in difficulty across performance levels.Source: §D.3 · p. 36

D.4 Transfer Agent Selection

• Transfer agents are selected by descendant growth, favoring archive nodes whose descendants achieve larger improvements in fewer self-modification steps rather than nodes that merely score highly.The growth score discounts descendant improvements by generation distance and treats reliable downstream improvement as evidence of stronger agent-generation ability.Source: §D.4 · p. 36

E Additional Results

• The additional-results appendix examines DGM-H task agents, cross-domain task improvement, meta-level capabilities, Olympiad-level graders, and preliminary self-modification of parent selection.Experiment logs are open-sourced in the codebase.Source: E Additional Results · p. 37

E.1.1 Paper Review

• For paper review, the discovered task agent evolved domain-specific acceptance criteria, critical weakness checks, staged evaluation, and strict accept-or-reject JSON output.Later patches require systematic assessment of novelty, soundness, significance, validation, clarity, reproducibility, and related work, with rejection triggered by any rejection criterion.Source: §E.1.1 · pp. 38, 40, 41

E.1.2 Robotics Reward Design

• For robotics reward design, the discovered task agent added structured reasoning instructions, explicit JSON fields, response-extraction safeguards, and retry logic for failed interactions.The implementation retries generation up to two times after parsing or interaction failures.Source: §E.1.2 · p. 42
• The robotics reward-design agent is constrained to documented environment attributes and PyTorch operations, with explicit prohibitions on unavailable fields such as env.torques and env.dt.Its guidance combines velocity tracking with stability and smoothness terms, including orientation, base-height, action-change, and joint-velocity penalties.Source: §E.1.2 · pp. 43–44

E.1.3 Olympiad-level Math Grading

• DGM-H transfer and ProofAutoGrader patches produced the best task agent, BetterGrader, for Olympiad-level math grading.The discovered grader uses a 0–7 scale and explicitly distinguishes complete, almost-correct, partial-progress, and incorrect solutions.Source: §E.1.3 · p. 44
• The math grader requires checking every specified partial-credit milestone before assigning 0, while reserving 6 for complete solutions with only minor fixable issues and 7 for fully rigorous solutions.The evaluation process verifies each logical step, distinguishes minor from fundamental gaps, and assigns at least 1 point when any listed milestone is achieved.Source: §E.1.3 · pp. 45–46
• Olympiad-level grading awards 7 points only for complete, correct, fully rigorous solutions with no gaps, unjustified steps, or errors.All problem parts, necessary claims, edge cases, and boundary conditions must be addressed; any gap caps the score at 6.Source: §E.1.3 · pp. 47–48
• A milestone counts only when the solution fully proves or establishes the specified result; stating it, approaching it, or proving only part earns 0 for that milestone.Graders systematically compare each milestone requirement with concrete solution evidence and reject attempts or partial progress as achievement.Source: §E.1.3 · pp. 47–48
• The rubric distinguishes complete-and-rigorous solutions (7), nearly complete solutions with minor issues (6), meaningful progress achieving a milestone (1), and insufficient work (0).A score of 6 requires a complete major structure with only minor fixable issues; missing major lemmas or key components instead leads to 1 or 0.Source: §E.1.3 · pp. 47–48
• Partial credit requires an explicitly achieved guideline milestone or an equivalent formulation, while clever observations, unrelated lemmas, reformulations, and incorrect attempts do not count.The rubric is more lenient about minor errors when a listed milestone is substantially achieved, but general progress alone receives no credit.Source: §E.1.3 · p. 48

E.2 Qualitative: Improving Task Performance

• Across paper review, robotics reward design, and Olympiad-level math grading, DGM-H consistently self-improves, with final lineages often passing through lower-performing intermediate agents.The archive trees and progress plots show open-ended search discovering diverse stepping stones rather than branching only from the current best agent.Source: §E.2 · p. 48, Figure 7 · p. 49, Figure 8 · p. 50
• In paper review, DGM-H replaced unreliable attitude prompts with a two-stage checklist-and-rules evaluation process, producing more stable and higher-performing behavior.The transition followed an insight in generation 54: “for LLMs, use structured processes, not attitude instructions”.Source: §E.2 · p. 48
• In robotics reward design, DGM-H progressively accumulated accurate, example-driven environment knowledge, eliminating hallucinated attributes and improving reward formulation.Later generations tightened constraints, added code examples, and specified reward formulations and scaling ranges.Source: §E.2 · pp. 48–49
• In Olympiad-level math grading, DGM-H evolved from grading categories to explicit rubrics, checklists, calibration, boundary cases, and decision trees that improved consistency.Generation 168 introduced a framework mapping rubric satisfaction to final grades, replacing descriptive guidance with precise logical flow; the approach mirrors recent rubric-based methods (Cook et al., 2024; Fan et al., 2024; Chen et al., 2026; Lv et al., 2026).Source: §E.2 · p. 49
• DGM-H’s gains emerge from cumulative recombination of earlier discoveries, supported by meta-level utilities such as persistent memory, evaluation analysis, and performance tracking.Features that appear to require only a few edits often depend on infrastructure and insights developed across previous generations.Source: §E.2 · p. 49

E.3 Qualitative: Improving the Ability to Improve

• The hyperagent autonomously learns to measure performance, diagnose pathological behavior, build improvement infrastructure, and accumulate knowledge across generations.These capabilities arise without explicit instructions to create them, emerging from the pursuit of better task performance.Source: §E.3 · p. 50
• These emergent capabilities improve not only task behavior but also the hyperagent’s own ability to generate sustained future improvements.The qualitative analysis indicates that self-modifying systems may spontaneously develop metacognitive machinery for effective self-improvement.Source: §E.3 · p. 50

Better Evaluation Analysis

• The meta agent independently develops evaluation analysis that summarizes recent results, including success rates and failure patterns, before proposing code changes.It adds an _analyze_evaluations() method to process stored evaluation data and generate quantitative summaries.Source: §E.3.1 · p. 51
• By augmenting its instruction prompt with evaluation context, the meta agent uses retrieved performance metrics and failure patterns to make more targeted improvements.The hyperagent discovers this mechanism independently, shifting improvement from unconstrained modification toward feedback-informed optimization.Source: §E.3.1 · p. 51

E.3.2 Compute-Aware Strategic Planning

• The hyperagent adapts improvement aggressiveness to remaining compute, exploring ambitious architectural changes early and applying conservative refinements late without manually specified scheduling.This compute awareness can emulate optimization strategies such as evolutionary search, hill climbing, or population-based exploration by controlling exploration and consolidation.Source: §E.3.2 · pp. 51–52

E.3.3 Automated Bias Detection and Correction

• In paper review, automated label-distribution diagnostics detect classification collapse and trigger corrective instructions to the meta agent.One observed case improved from 49% accuracy with 99% acceptance to 63% accuracy after predictions became more balanced.Source: §E.3.3 · p. 52

E.3.4 Performance Tracking Infrastructure

• The hyperagent builds persistent performance-tracking infrastructure that records generation scores and trends, enabling quantitative identification of successful changes and data-driven future modifications.The tracker stores metrics across generations and computes statistics such as best, worst, average, and improvement-trend scores.Source: §E.3.4 · pp. 52–53

E.3.5 Prompt Template Systems

• It abstracts recurring prompting patterns into configurable templates and builders, improving maintainability while enabling systematic experimentation with reasoning, concise, detailed, and few-shot strategies.This applies software-engineering practices including DRY and the factory pattern to prompt construction.Source: §E.3.5 · pp. 53–54

E.3.6 Refinement Based on Multi-Generation Performance

• Across generations, the hyperagent synthesizes performance patterns into actionable lessons, learning that calibration, explicit criteria, clear borderline defaults, and precise language affect review accuracy.The recorded comparisons include 63% accuracy at 75% rejection, 61% at 53% rejection, and 52% at 6% rejection.Source: §E.3.6 · p. 54
• The system documents regressions and diagnoses their causes, identifying that overly permissive language and unclear borderline guidance produced severe leniency in a later generation.Generation 66 reached 52% accuracy with a 94% acceptance rate and caught only 8% of papers that should have been rejected.Source: §E.3.6 · p. 54
• It responds to diagnosed failures by adding explicit rejection criteria, calibrated decision thresholds, and a default-to-reject rule for borderline cases.The implemented protocol targets approximately 40% acceptance and 60% rejection unless a contribution is clearly valuable.Source: §E.3.6 · p. 55
• By accumulating records of what succeeds and fails, the hyperagent uses multi-generation knowledge to decide which mechanisms and prompts to modify next.This extends self-improvement from changing task code to improving the process that generates and evaluates future changes.Source: §E.3.6 · pp. 54–55

E.3.7 Persistent Memory

• The hyperagent creates persistent memory mechanisms without explicit instruction, enabling learning to accumulate across generations and sessions.Its memory tool stores and retrieves information across agent interactions.Source: §E.3.7 · p. 55
• Timestamped storage provides temporal context for tracking when insights were recorded and reasoning about their recency.Stored entries include values and ISO-formatted timestamps.Source: §E.3.7 · pp. 55–56
• The stored knowledge includes synthesized performance analyses, causal hypotheses, and strategic plans rather than raw results alone, supporting long-horizon reasoning and continual improvement.Examples include identifying Gen55’s accuracy–harshness tradeoff, attributing Gen66’s decline to gen65 over-correction, and planning to combine Gen55’s reasoning with Gen64’s balance; this aligns with related work by Wei et al. (2025a), Weng et al. (2026), Zhang et al. (2026), and Xiong et al. (2026).Source: §E.3.7 · pp. 55–56
• The memory system supports cross-generation learning by preserving insights from successes and failures so later generations can improve without repeating past mistakes.It functions as a shared knowledge base that remains available as the code changes.Source: §E.3.7 · pp. 55–56

E.4 Olympiad-level Math Graders

• BetterGrader increases Almost recall by +18.52% and F1 by +0.203, while increasing Partial recall by +10.50% and F1 by +0.109.It increases true positives for Almost from 13 to 43 and Partial from 47 to 72 by reducing misclassification into Correct or Incorrect.Source: §E.4 · p. 56
• The DGM-H produces BetterGrader automatically without domain-specific heuristics or handcrafted rules, correcting ProofAutoGrader’s bias toward extreme labels (Luong et al., 2025).Figure 10 shows more appropriate intermediate-label assignments while maintaining strong Correct and Incorrect performance.Source: §E.4 · p. 56, Figure 10 · p. 56
• BetterGrader raises overall accuracy by +4.06% over ProofAutoGrader (Luong et al., 2025), despite Correct recall decreasing by −6.00%.ProofAutoGrader’s Correct recall was already 91.67%, so the net improvement reflects gains in intermediate-label grading.Source: §E.4 · p. 56

E.5 Modifying Parent Selection

• 0.491 average performance with modifiable parent selection exceeded random selection’s 0.428 but remained below score-child-prop’s 0.532 on paper review and robotics reward design.These differences were not statistically significant (p > 0.05).Source: Figure 11 · p. 57, §E.5 · p. 57
• The experiment initialized DGM-H with random parent selection, allowed it to modify that mechanism, and compared five repeated runs against random and handcrafted strategies.Other experiments used the fixed score-child-prop strategy.Source: §E.5 · p. 57
• Across runs, DGM-H replaced random sampling with UCB-style exploration bonuses, temperature-controlled softmax selection, adaptive temperatures, and scores incorporating diversity, recency, elite performance, and stagnation.These modifications emerged early and increased exploration when progress slowed.Source: §E.5 · pp. 57–58
• Despite recreating classic selection algorithms and adding adaptive heuristics, DGM-H did not consistently discover a strong strategy because learned mechanisms were more complex and sensitive across runs.The archives show random selection producing many low-performing agents, while modifiable selection increasingly balances exploration and exploitation.Source: §E.5 · p. 58, Figure 12 · p. 59

Reflection and amplification of human biases

• The DGM-H faithfully optimizes benchmark objectives, so biased data or misaligned incentives may be reproduced and exacerbated, motivating careful benchmark design, dataset curation, and periodic reevaluation.In paper review, it predicts acceptance decisions from existing human review data rather than changing the review process itself.Source: F Additional Safety Discussion · p. 60

Evaluation gaming

• Evaluation gaming can raise measured performance without advancing the benchmark’s intended objective, making robust, diverse, refreshed protocols, held-out tests, complementary metrics, and human oversight essential.As self-improving systems become more powerful, safety depends increasingly on the fidelity and robustness of optimization signals.Source: F Additional Safety Discussion · p. 60