Meta$^n$: Recursive Self-Improvement through Emergent Depth

Zae Myung Kim; Young-Jun Lee; Seungyeon Jwa; Dongyeop Kang

cs.AIAug 25, 2026

Meta $^n$ : Recursive Self-Improvement through Emergent Depth

Authors: Zae Myung Kim, Young-Jun Lee, Seungyeon Jwa, Dongyeop Kang

Organizations: University of Minnesota · Seoul National University

Abstract

Self-improving LLM agents refine answers, not the process that produces those answers. Systems that add a meta-level hold that level fixed, and those that edit themselves must leave part of their own editing machinery untouched to stay stable, capping the meta-depth they realize at roughly two. We present Meta $^n$ , which keeps the meta-operation fixed and recurses on its input instead. That operation, $Ω$ , is applied repeatedly to its own products, reading the traces of the solver stack below together with the code that produced them, then writing the next layer as a strategic pre-process and a library of callable helpers. Because $Ω$ never changes, it cannot destabilize the system, and because its input strictly grows, each layer reasons from a higher vantage than the last. Depth is set by convergence rather than fixed in advance, and an evolutionary archive searches over layer chains. Across two backbones, Meta $^n$ outperforms prior self-improving agents on all eight benchmark families. The sharpest case is ARC-AGI-2, built to resist skill memorization, where it alone scores above zero. Ablations indicate that most of the gain from recursion comes from the conditioning each layer passes to the next, and distinct layer roles emerge with depth although no prompt prescribes them. Code available at https://github.com/minnesotanlp/meta-n

Figures & tables

Figure 1: Meta n at a glance. (left) A single universal meta-operation $\Omega$ is applied recursively, each meta-layer wrapping the stack below it (Base Solver $\to$ Layer 1 $\to$ Meta-Layers $2,3,\dots,n$ ). Roles emerge with depth without manual design. (right) Search progress on LawBench charge prediction, where both Meta n variants climb faster and settle higher than Gödel Agent ( Yin et al., 2024 ) and OpenEvolve ( Sharma, 2025 ) .

Figure 2: Three paradigms of meta-improvement. Circles are agents (successive versions

v^{1},v^{2},\dots

); amber boxes are meta-improvement processes, and nested boxes denote recursively stacked meta-processes. (1) Hand-crafted one-layer meta (e.g., FunSearch, AlphaEvolve, ADAS): a fixed, external meta-process rewrites the agent, but the meta-mechanism itself never changes. (2) Self-referential one-layer meta (e.g., Gödel Agent, DGM): the agent edits its own source, yet a driver layer stays fixed, capping realized meta-depth at

{\sim}2.5

. (3) Recursive

n

-layer meta (Meta n , ours): one meta-operation

\Omega

is applied at every depth and accumulates , so version

v^{d}

carries

d

nested meta-processes; depth is set by convergence, and each added layer observes a broader landscape of trajectories than the one below.

Panel of Figure 3 — Figure 3: The two phases of a meta-layer , traced through a LawBench charge prediction run (task score $\in[0,1]$ ). (a) Build-step: each $\Omega$ call reads the traces of the stack below, and from depth 3 its code as well, then writes the next layer’s code $C_{d}$ . Depth 2 emits the helper fuzzy_match_label ; depth 3 adds reconcile_labels but couples it to an over-prescriptive “Exhaustive Legal Analysis” directive that regresses the score; depth 4 rolls that directive back while keeping the helper. (b) Run-step: the finished $S_{4}=M_{4}\!\circ\!M_{3}\!\circ\!M_{2}\!\circ\!S_{1}$ threads the task inward to the base solver and merges the accumulated helpers into the final script.

Method	CO-Bench	S2D	LawBench	AE Math	AlgoTune	SR
Meta n archive-best	0.851 $\pm$ 0.014	0.733 $\pm$ 0.015	0.815 $\pm$ 0.013	0.869 $\pm$ 0.045	$\times$ 15.10 $\pm$ 2.4	5.22 $\pm$ 0.38
Meta n best chain	0.782 $\pm$ 0.016	0.743 $\pm$ 0.034	0.796 $\pm$ 0.046	n/r	n/r	4.68 $\pm$ 0.45
OpenEvolve ( Sharma, 2025 )	0.814 $\pm$ 0.022	0.718 $\pm$ 0.022	0.745 $\pm$ 0.034	0.802 $\pm$ 0.052	$\times$ 10.45 $\pm$ 1.8	n/r
Gödel Agent ( Yin et al., 2024 )	0.451 $\pm$ 0.023	0.710 $\pm$ 0.034	0.775 $\pm$ 0.023	0.581 $\pm$ 0.061	$\times$ 13.22 $\pm$ 2.7	n/r

Table 1: Meta n agentic vs. prior self-improving agents (Gemma 4 31B-IT, mean

\pm

stdev over seeds 42/43/44). Held-out test for CO-Bench, S2D, and LawBench; benchmark score for AlphaEvolve Math, AlgoTune (speedup

\times

), and SR (4-domain mean). ARC-AGI-2 and the SR baselines were run under GPT-5.2 only (Table 2 ); n/r = not run.

Method	CO-Bench	S2D	AE Math	SR	ARC-AGI-2
Meta n archive-best	0.870 $\pm$ 0.011	0.725 $\pm$ 0.014	0.917 $\pm$ 0.016	5.03 $\pm$ 0.20	0.331 $\pm$ 0.010
Meta n best chain	0.806 $\pm$ 0.017	0.734 $\pm$ 0.020	0.820 $\pm$ 0.024	n/r	0.123 (1 seed)
OpenEvolve ( Sharma, 2025 )	0.702 $\pm$ 0.025	0.721 $\pm$ 0.007	0.726 $\pm$ 0.046	3.70 $\pm$ 0.13	0.003 $\pm$ 0.001
Gödel Agent ( Yin et al., 2024 )	0.527 $\pm$ 0.033	0.708 $\pm$ 0.037	0.674 $\pm$ 0.068	2.45 $\pm$ 1.63	0.054 $\pm$ 0.006

Table 2: Cross-model condition: GPT-5.2 backbone, mean

\pm

stdev over seeds 42/43/44. Held-out test for CO-Bench and S2D; benchmark score for AlphaEvolve Math, SR (4-domain mean), and ARC-AGI-2 (dev score; the held-out test result is reported in the text). LawBench and AlgoTune were not run under GPT-5.2 for budget reasons; their Gemma columns appear in Table 1 . n/r = estimator not run for this cell.

Panel of Figure 4 — Figure 4: Meta n variants, single-shot vs. agentic, on four representative benchmarks (Gemma 4 31B-IT; hyperparameters in Table 5 ). SS = single-shot, Ag = agentic. Each group shows the seed (the Layer-1 solver, before any meta-layer), linear-best (best single chain), and archive-best (per-task best across the archive), annotated with the raw score and the gain over that group’s seed. Panels plot validation or benchmark scores, with held-out test results in Table 1 ; AlgoTune is speedup ( $\times$ ). Black whiskers mark three-seed means $\pm 1$ stdev (seeds 42/43/44), and orange diamonds the three-seed GPT-5.2 archive-best where that backbone was run.

Figure 5: Search progress on CO-Bench, LawBench, and Symptom2Disease (Gemma 4 31B-IT, single-seed runs). Lines are best-so-far validation over evolution iterations; stars (

\star

) are held-out test scores plotted at a fixed offset; faded markers hold the last value after early stopping.

Panel of Figure 6 — Figure 6: Emergent layer roles by depth, split by solver substrate. Each bar is the fraction of that depth’s $n$ emissions ( $n$ per depth in the subcaptions) that both raters, GPT-5.2 and Kimi-K2.6, assign to the category, over the 588 emissions at depths 2 to 5. The rubric is multi-label, so bars within a depth sum past 100% and should be compared across depths rather than against one another. Emissions the raters split on are discarded, making every bar a lower bound. Two negligible categories are omitted; rubric and per-category $\kappa$ in Appendix E.3 .

Condition	Removal	Score	$\Delta$
Meta n (full; Gemma, CO)		0.845
$-$ code library	$\mathcal{L}^{(d)}$ injection	0.825	$-0.020$
$-$ outer-context	inter-layer ctx d	0.751	$-0.094$
$-$ recursion (depth-1)	meta-layer stacking	0.714	$-0.131$
Meta n (full; GPT-5.2, CO)		0.886
$-$ recursion (depth-1)	meta-layer stacking	0.806	$-0.080$

Table 3: Component ablations. Each row removes one mechanism and keeps the rest, and comparisons are like-for-like within a block.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Plain English
$\Omega$	fixed meta-operation, called at each build-step
$M_{d}$ , $S_{d}$	depth- $d$ wrapper and resulting solver
$C_{d}$	$\Omega$ ’s output: pre-process $f_{\text{pre}}^{(d)}$ + library $\mathcal{L}^{(d)}$
$\tau_{i}^{(d)}$	execution trace at depth $d$ on task $i$
$\text{ctx}_{d}$	outer-context passed from layer $d$ to layer $d{-}1$

Appendix

Table 4: Symbol glossary for § 2 .

				Max-iter
Benchmark	$B$	$K$	Pat.	Gemma	GPT-5.2	Test split?
CO-Bench	2	2	4	8	6	yes
Symptom2Disease	1	3	off	20	10	yes
LawBench	1	3	off	20	n/r	yes
TerminalBench 2.0	1	2	3 to 4	6 to 10	13 cat-by-cat	no
AlphaEvolve Math	2	2	10	6	6	no

Appendix

Table 5: Per-benchmark evolutionary hyperparameters on the two backbones. The

\Omega

prompt template, output schema,

\epsilon=0.02

, and

\alpha=0.3

are fixed across all rows. Max-depth is 10 except TB2 (4). “off” means early stopping is disabled and the run goes to max-iter. The GPT-5.2 column tightens max-iter on CO-Bench (8 to 6) and S2D (20 to 10) to bound per-seed cost, and leaves AE Math and SR at 6; TB2 sweeps category by category. ARC-AGI-2 runs under GPT-5.2 only (seeds 42/43/44;

{\sim}\

632$ per Meta n seed).

Benchmark	Mode	Depth	Cands	Tokens
CO-Bench (36 tasks)	Single-shot	3	33	5.3M
	Agentic	3	29	36.1M
Symptom2Disease	Single-shot	5	61	422K
	Agentic	3	61	1.7M
LawBench (Chinese)	Single-shot	4	61	761K
	Agentic	4	61	2.3M

Appendix

Table 6: Per-row depth, archive size, and token spend for the seven Gemma benchmark families, including the four panels of Figure 4 (s42 production runs).

Category	$\kappa$
Task routing	0.96
Prompt engineering	0.85
Rollback / override	0.60
Specialized libraries	0.57
Environment constraints	0.45
Tactical primitives	0.37

Appendix

Table 7: Inter-rater agreement per rubric category (Cohen’s

\kappa

, GPT-5.2 vs. Kimi-K2.6, all 596 emissions).

Domain	Mode	Seed	Linear-best	Archive-best	Depth	Cands	Tokens
bio_pop_growth (24 inst.)	Single-shot	0.477	2.751	3.782	3	25	2.6M
	Agentic	2.355	4.406	4.977	3	25	5.2M
chem_react (36 inst.)	Single-shot	5.458	6.580	7.599	3	25	2.6M
	Agentic	5.587	6.440	7.244	3	25	2.7M
matsci (25 inst.)	Single-shot	0.000	1.273	2.130	6	23	3.9M
	Agentic	1.477	3.296	3.863	3	25	6.7M

Appendix

Table 8: Symbolic Regression per-domain results (Gemma, s42 production runs), backing the Symbolic Regression row of Table 6 . Domains: bio_pop_growth (biological population dynamics), chem_react (chemical reaction kinetics), matsci (materials science properties), phys_osc (physical oscillator systems). Bottom block gives the family mean (Tokens summed across the four runs; other columns averaged). Bold = higher archive-best per domain.

Domain	Meta n	OpenEvolve	Gödel Agent
bio_pop_growth	4.35 $\pm$ 0.30	2.83 $\pm$ 0.08	2.68 $\pm$ 0.83
chem_react	7.65 $\pm$ 0.07	6.96 $\pm$ 0.08	2.92 $\pm$ 3.84
matsci	3.26 $\pm$ 0.67	1.37 $\pm$ 0.34	1.27 $\pm$ 0.67
phys_osc	4.85 $\pm$ 0.11	3.63 $\pm$ 0.15	2.93 $\pm$ 2.03
Mean	5.03 $\pm$ 0.20	3.70 $\pm$ 0.13	2.45 $\pm$ 1.63

Appendix

Table 9: Symbolic Regression per-domain comparison under GPT-5.2 (agentic archive-best, mean

\pm

stdev over seeds 42/43/44).

Task	Single-shot	Agentic
convolve2d_full_fill	105.37	101.79
fft_convolution	9.72	1.69
fft_cmplx_scipy_fftpack	4.17	1.12
psd_cone_projection	5.54	3.74
affine_transform_2d	2.00	1.05
lu_factorization	1.47	1.34

Appendix

Table 10: AlgoTune per-task speedups (

\times

, s42). The seven tasks solved by both variants. The eighth task, eigenvectors_complex , has a per-task log only in the agentic run (

\times 1.11

), so the single-shot aggregate (

\times 18.47

) averages seven tasks and the agentic aggregate (

\times 14.11

) averages eight; on the matched seven the agentic mean is

\times 15.96

LLM Agent Self-Improvement

Large language model agent self-improvement studies how agents use experience and feedback to improve their behavior, tools, and workflows.

See all 276 papers

MetaSkill-Evolve: Recursive Self-Improvement of LLM Agents via Two-Timescale Meta-Skill Evolution
Jul 6, 2026Zefeng Wang, Minxi Yan, Jinhe Bi +3LLM Agent Self-Improvement LLM Agent Skill Learning
LMU Munich · The Chinese University of Hong Kong · MCML +1
Hierarchical Self-Improvement: A Framework for Task-Specific Evolvable Agent Harnesses
Aug 9, 2026Tailin ZhouAgent Harness Optimization LLM Agent Self-Improvement
HKUST
Skill $^{\infty}$ : Recursive Self-Improvement for Agent Skill Optimization
Oct 26, 2026Wangcheng Tao, Zhanglu Yan, Han Wu +1LLM Agent Self-Improvement LLM Agent Skill Learning
National University of Singapore · City University of Hong Kong

Jul 6, 2026J/K move · Enter open · S save

MetaSkill-Evolve: Recursive Self-Improvement of LLM Agents via Two-Timescale Meta-Skill Evolution

Zefeng Wang, Minxi Yan, Jinhe Bi, Sikuan Yan+2

LMU Munich · The Chinese University of Hong Kong · MCML+1

LLM Agent Self-Improvement LLM Agent Skill Learning Recursive Self-Improvement

Open paper AskPDF arXiv

Recent LLM agents tackle increasingly long-horizon, open-ended tasks, and external skills, reusable procedural knowledge supplied to the agent, further extend this capability. However, a fixed, hand-authored skill is rarely optimal, and cannot adapt to the diversity of tasks an agent encounters. Self-improving agents address this by rewriting their own skill files from execution traces, yielding meaningful gains on challenging benchmarks. Yet such self-evolution remains non-recursive: it improves only the task skill (what the agent does) while the improvement procedure (how it improves) is authored once and held fixed. We introduce MetaSkill-Evolve, a two-timescale framework that makes agentic skill improvement recursive: every branch carries both a task skill

s

and a branch-local meta-skill

m=(ψ,σ,α,π,\varepsilon)

whose five components parameterise the Analyzer, Retriever, Allocator, Proposer, and Evolver agents of the improvement pipeline. Task skills evolve on a fast loop while the meta-skill evolves on a slower one under the same pipeline applied to itself, with no additional model or objective. With all five pipeline agents sharing a single frozen backbone, MetaSkill-Evolve outperforms no-skill, static-skill, and single-level evolution baselines on three agentic benchmarks (OfficeQA, SealQA, ALFWorld), improving held-out test accuracy over the raw backbone by +23.54, +16.09, and +1.92 points respectively.

Abstract

LLM Agent Self-Improvement

Explore similar work

MetaSkill-Evolve: Recursive Self-Improvement of LLM Agents via Two-Timescale Meta-Skill Evolution

Hierarchical Self-Improvement: A Framework for Task-Specific Evolvable Agent Harnesses

Skill∞^{\infty}∞: Recursive Self-Improvement for Agent Skill Optimization

Skill $^{\infty}$ : Recursive Self-Improvement for Agent Skill Optimization