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

Self-improving LLM agents typically improve answers, not the $\textit{process}$ by which answers are improved: their search loops and mutation rules are held fixed by design. We introduce $meta^n$, a framework for $\textit{recursive}$ self-improvement in which a single universal meta-operation $\Omega$ is applied repeatedly to build a hierarchy of meta-layers whose depth is sized by convergence rather than prescribed. Each layer observes execution traces from below, diagnoses failure patterns, and emits executable code (strategic guidance via pre-processing and reusable functions via a code library) whose effects cascade downward. Inter-layer conditioning permits up to $\prod_{d=2}^{n} k_d$ effective strategy-tactic combinations across $n$ layers rather than $\sum_d k_d$. An evolutionary orchestration mode maintains a growing archive of candidate chains with fitness-weighted parent selection and cross-candidate inspiration. Across seven benchmarks (combinatorial optimization, two text classification tasks, terminal agent tasks, mathematical discovery, symbolic regression, algorithm speedup), $meta^n$ beats prior self-improving agents on all five (of seven) benchmarks where they apply, with the largest gap on CO-Bench (0.849 test vs. Gödel Agent's 0.000 and OpenEvolve's 0.816). Layers specialize without prescription and their roles emerge at depth: generic primitives at depth 2 propagate as cross-task callables, task-aware routing emerges at depth 3, and deeper layers refine via specialization. Code and data will be released publicly.