Recursive Self-Improvement

We establish mathematical limits of algorithmic safety verification for Turing-complete self-modifying systems, the class in which recursive self-improvement takes place, both for a fixed system and across its own modification. Statically, no verifier is sound, complete and tractable: over unbounded domains by Rice's and Gödel's theorems, over all finite configurations by Trakhtenbrot's theorem, and over succinctly described finite environments because verifying a policy against an adversary is coNP-complete and synthesising one is PSPACE-complete. Dynamically, we model one step of self-modification as a computable transformation of code and ask whether a safety property survives it. If the transformation depends only on behaviour, this is Rice's theorem one level up; if it reads the code, as self-modification does, the question is no longer semantic, yet the same s-m-n reduction works inside a class of behaviourally identical programs and inherits the halting degree. One step is never harder than the property; persistence along the whole trajectory can be $Π^0_2$-complete. Certification by a total algorithm is possible only for transformations of restricted expressivity, not merely for systems that stop changing. No tower of supervisors helps, and every total supervisor errs on an undecidable set of systems. For effectively pointwise properties, every faithful bounded scheme that certifies on finite behavioural evidence admits evolution traces certified at every stage while the property is violated. What survives is exact: a monitor that raises an alarm on violation semidecides it, and comparison against a frozen reference keeps the full theory.

Self-improving agents are state-of-the-art on agentic coding benchmarks, yet their search methods assume a stationary evaluation criterion. This ignores a central feature of evolution: species adapt as their environments change with them. We introduce the Red Queen Gödel Machine (RQGM), an evolutionary framework for recursive self-improvement under non-stationary utilities. This allows learned evaluators to improve alongside the agents they guide. On DeepSWE, the RQGM improves over its fixed-evaluator baseline by adding a complementary agent-as-a-judge code-review signal: a co-evolved reviewer grades coder patches to guide search. At low reasoning effort, the RQGM coder passes 82.1% of held-out tasks against the baseline's 75.0%, and nearly matches the GPT-6 Astra model at high effort. In scientific paper writing and reviewing, and Olympiad-level proof writing and grading, co-evolved evaluators provide an evaluation criterion. Anchored to human IMO grades, a co-evolved grader writes its own milestone rubric and exceeds static baselines at a 3x lower search cost, driving the prover to the best mean score. Since the RQGM can modify the search objective across epochs, it can regularize the search. For example, the RQGM reduces self-preference bias via an additional adversarial objective to discover reviewers equally stringent on AI and human work. Guided by these calibrated reviewers, co-evolved writers reach 1.78x-1.86x higher acceptance rates than the baseline under an agent-as-a-judge panel. The RQGM enables self-improving systems where agents and evaluators recursively bootstrap each other beyond static evaluation.

LLM agents are increasingly improved without weight updates by evolving a natural-language artifact, such as reflections, workflows, playbooks, cheatsheets, or optimized prompts, that conditions a frozen policy. Such methods are typically reported as wins on the single benchmark where they help. We study them apples-to-apples and surface a sharper picture. We introduce RSEA, a Recursive Self-Evolving Agent that carries a compact three-layer natural-language state: an imperative strategy, reusable skills, and a procedural playbook. Across generations, RSEA rewrites all three layers from its own trajectories and commits a candidate only if it does not regress on a disjoint held-out split, using a strict keep-better gate. Across four diverse benchmarks, ALFWorld, GAIA, (τ)-bench, and WebShop, and six faithful baselines, ReAct, Reflexion, GEPA, AWM, ACE, and Dynamic Cheatsheet, all evaluated on one shared local backbone, we find three main results. First, no artifact universally wins. RSEA is the strongest single-pass method on ALFWorld, reaching 69.3% compared with 64.6% for ReAct (McNemar (p=0.015)), and reaches 79.4% with retry, the best overall result. However, concrete-workflow induction, represented by AWM, is best on the strong-backbone tool-use tasks. Second, unguarded context evolution is high-variance and unsafe. Dynamic Cheatsheet, which curates context online without a held-out gate, is near-best on ALFWorld at 70.7%, yet collapses on WebShop, with a score of 0.14 compared with 0.43 for ReAct. Third, RSEA's strict held-out selection is what makes recursive self-evolution monotone-safe: it never significantly underperforms the base agent on any benchmark and falls back to vanilla ReAct when evolved context would hurt.

Over the last decade, building human-level artificial general intelligence has moved from far-fetched speculation to being a concrete next-decade target for many of the largest AI organisations. Achieving this goal would have profound and far-reaching impacts on human society, which raises many complex questions for the decade ahead. This report investigates how AI itself might continue to develop in a post-AGI world along the continuum of machine intelligence. The endpoint of this continuum, Universal AI, is theoretically well understood, which provides some formal grounding for the main focus of this report: the transition from human-level AGI to artificial general superintelligence, which can intuitively be understood as a system that is more intelligent and cognitively capable than large organisations of humans. After characterizing ASI, the report discusses four potential pathways from AGI to ASI: scaling AGI, AI paradigm shifts, recursive improvement, and ASI emerging from large-scale multi-agent collectives. The report then discusses possible frictions and bottlenecks along these pathways. Determining whether the impact of these frictions will be negligible or substantial raises a number of concrete open research questions. Due to large uncertainties for predicting ASI progress, it cannot be ruled out that AI progress might continue to accelerate over the next years. This could imply that the image of a single transformative step change, caused by the introduction of human-level AGI into our society, could be inaccurate. More apt might be the prospect of a series of transformative societal changes caused by AI-enabled progress and breakthroughs across many areas of science and technology. Preparing for this prospect requires a massively interdisciplinary endeavour of global scope and interest.

Recursive self-design refers to AI-assisted modification of the mechanisms by which an AI system is built, evaluated, and improved. This paper treats MetaAI not as a mature paradigm, but as a working term for a human-seeded, AI-expanded development pattern in which the design space itself becomes a target of modification. We propose an operational evidence framework with four criteria: inspectable target system, meta-level modifier, feedback-directed selection, and recursive continuation. We then map public systems, including Darwin Goedel Machine (DGM), STOP, Goedel Agent, and ShinkaEvolve, against these criteria. DGM provides the most direct currently reported evidence: its published results show improvement from 20% to 50% on SWE-bench Verified and from 14.2% to 30.7% on full Polyglot after 80 iterations, with ablations suggesting that both open-ended exploration and self-improvement contribute. Finally, we provide MetaAI-Mini, a reproducible HumanEval-based protocol and codebase. Because no completed model run is included in this build, MetaAI-Mini is reported as a protocol rather than as an experimental result.

Agents increasingly modify the procedures by which they solve tasks and improve themselves. Autonomy over improvement, gains in practical capability, and enlargement of computational reach are distinct properties. We develop an oracle-relative model with mutable solvers, evaluators, and improvers. Uniform simulation keeps every total decision procedure produced by effective self-revision over $A$ within $\mathcal{C}(A)=\{D:D\leq_T A\}$; oracle joins account for additional access, while the relativized limit lemma separates limiting answers from effective completion. A worked model of Boolean rule acquisition makes the distinction constructive. For a known finite-dimensional feature language, we characterize exactly which answers a query history determines, obtain a sharp teacher-query bound, and give a terminating protocol that permits revisions to the query proposer. The learned solver can dispense with the teacher on every input while remaining in the same computability layer. However, uniformly constructing the required feature-span specification from arbitrary effective feature programs is already as hard as the relative halting problem. We also give a conditional criterion for strict ascent and distinguish it from finite behavioral evidence. The framework thus separates acquisition, certification, and computability ascent, identifying both a positive route to verified support removal and the assumptions on which it depends.

Gödel agent realize recursive self-improvement: an agent inspects its own policy and traces and then modifies that policy in a tested loop. We introduce Polaris, Gödel agent for compact models that performs policy repair via experience abstraction, turning failures into policy updates through a structured cycle of analysis, strategy formation, abstraction, and minimal code patch repair with conservative checks. Unlike response level self correction or parameter tuning, Polaris makes policy level changes with small, auditable patches that persist in the policy and are reused on unseen instances within each benchmark. As part of the loop, the agent engages in meta reasoning: it explains its errors, proposes concrete revisions to its own policy, and then updates the policy. To enable cumulative policy refinement, we introduce experience abstraction, which distills failures into compact, reusable strategies that transfer to unseen instances. On MGSM, DROP, GPQA, and LitBench (covering arithmetic reasoning, compositional inference, graduate-level problem solving, and creative writing evaluation), a 7-billion-parameter model equipped with Polaris achieves consistent gains over the base policy and competitive baselines.

Recursive self-modification is increasingly central in AutoML, neural architecture search, and adaptive optimization, yet no existing framework ensures that such changes are made safely. Godel machines offer a principled safeguard by requiring formal proofs of improvement before rewriting code; however, such proofs are unattainable in stochastic, high-dimensional settings. We introduce the Statistical Godel Machine (SGM), the first statistical safety layer for recursive edits. SGM replaces proof-based requirements with statistical confidence tests (e-values, Hoeffding bounds), admitting a modification only when superiority is certified at a chosen confidence level, while allocating a global error budget to bound cumulative risk across rounds.We also propose Confirm-Triggered Harmonic Spending (CTHS), which indexes spending by confirmation events rather than rounds, concentrating the error budget on promising edits while preserving familywise validity.Experiments across supervised learning, reinforcement learning, and black-box optimization validate this role: SGM certifies genuine gains on CIFAR-100, rejects spurious improvement on ImageNet-100, and demonstrates robustness on RL and optimization benchmarks.Together, these results position SGM as foundational infrastructure for continual, risk-aware self-modification in learning systems.Code is available at: https://github.com/gravitywavelet/sgm-anon.