Second-Order Problem Solving for Recursive Self-Improvement in Formal Verification

Recursive self-improvement (RSI) lets a system improve the model-building machinery from its own failures, so every later model inherits the gain. Yet RSI has been validated almost exclusively on coding and formal benchmarks such as science QA and mathematics. This format bound limits RSI to improvement within a machine-checkable slice, not general capability where questions are open and correctness is settled by argument, replication, or measurement. We argue RSI must next operate across real, diverse scientific, engineering, and meta-scientific domains, not where formal evaluation is merely tractable. To that end we present MetaRSI-v1, where improvement is the scheduled composition of three typed operators over one unified paradigm. Data-RSI amplifies existing competence and marks its boundary; Harness-RSI edits a five-slot scaffold without touching weights; Model-RSI internalizes capability into parameters through bounded training. Sharing one loop kernel and artifact vocabulary, they make data, scaffold, and model changes composable rather than exclusive. A two-axis optimizer jointly decides operator order and each operator's proposal policy, while a meta-level policy revises the schedule across terms. We validate MetaRSI-v1 under the field's standard evaluations, on code and closed-form science, with no external teacher: the target model plays every role in its own loop. MetaRSI-v1 reframes self-improvement from a single-surface edit to a composition across the full model-production pipeline, opening two paths: a model route internalizing capability through training, and a harness route leaving weights untouched and thus extending self-improvement to any model reachable through an interface, with Data-RSI redefined as the shared substrate feeding both. The framework further yields refutable laws on where loops exist, how operators compose, and what supervision buys.

As recursive self-improvement (RSI) rapidly advances, reliable evaluation becomes critical for guiding adaptive search. RSI typically relies on finite evaluation resources, such as fixed benchmarks, to determine which modifications are retained and what is proposed next. However, when these finite resources are repeatedly reused, new candidates are proposed based on feedback from the same evaluation set, so the search trajectory can adaptively overfit and empirical improvement may not reflect genuine population improvement on the underlying task distribution. Some existing methods account for multiple comparisons but assume that candidates are chosen independently of the evaluation set, and therefore do not control this adaptive dependence. To address this, we propose REUSE (Risk-controlled Evaluation Under Sequential Evolution), a certified evaluation and promotion framework that allows a fixed evaluation set to support repeated adaptive decisions while providing statistical guarantees. For a user-specified error level $α$, with probability at least $1-α$, every promoted modification is a genuine population improvement on the underlying task distribution. REUSE achieves this by strictly limiting the evaluation feedback returned to the search process and accounting for possible promotion histories within the error budget. We develop detailed statistical theory for RSI evaluation in this setting, including simultaneous error control, valid lower bounds on cumulative improvement, and a characterization of the fundamental limits of adaptive evaluation reuse. In live self-improvement experiments, REUSE commits substantially fewer false promotions than evaluation frameworks from current RSI systems and error-controlled baselines, reducing the proportion of false promotions from up to 20.7% to 0%, while achieving final true population performance comparable to the best baselines.

Repository-level software engineering (SWE) is a challenging long-horizon setting: agents must reason over extended interactions, use tools, and adapt to stateful environments. Recent work trains SWE agents with reinforcement learning methods such as Group Relative Policy Optimization (GRPO), which independently sample multiple trajectories per issue, test the resulting patches, and compare terminal rewards within a fixed group. However, this training setup does not reuse verifier feedback from failed patches as context for subsequent attempts, even though this feedback contains valuable diagnostic information about what went wrong. Training on recovery trajectories is challenging because the preceding outcome determines whether the next trajectory is generated, while the failed execution determines its conditioning context. We introduce FC-SWE, a failure-conditioned RL framework that incorporates recovery attempts into policy training. After a patch fails verification, FC-SWE restores the repository to its original task state and uses the failed patch and verifier feedback as context for a recovery trajectory. FC-SWE adapts GRPO to these chains of complete, multi-turn tool-use trajectories through two mechanisms. Trajectory-local rewards preserve each attempt's verifier outcome, preventing recovery success from rewarding an earlier failed patch. Active-set advantage estimation forms a comparison group from all initial and recovery trajectories actually executed for the same issue, so failed attempts remain in the group while unexecuted attempts are excluded. On all 500 SWE-bench Verified tasks under a verifier-assisted protocol, FC-SWE with Qwen3.5-4B and SWE-agent achieves 41.7% Resolved@1 and 52.8% Resolved@2, compared with 38.9% and 48.5% for GRPO. Although trained with at most two attempts per chain, FC-SWE reaches 70.7% Resolved@11 under an eleven-attempt test-time budget.