Nested Shared-Operator Recursion on ARC-AGI-1: An Empirical Study of HRM, TRM and URM

Recursive reasoning models apply a small shared Transformer block many times to refine a latent state. This gives them large effective depth with few parameters and makes them strong on algorithmic tasks. Such compact solvers are natural candidates for tools that an LLM can call on narrow algorithmic subproblems. However, existing models such as HRM, TRM and URM differ in architecture, gradient propagation and training procedure simultaneously. This makes it hard to tell what drives their performance, and their optimization is still poorly understood and often unstable. In this work we address both of these gaps. First, we study these questions under a unified experimental pipeline spanning six algorithmic domains. Individual controlled ablations are performed on representative domains, while the resulting recipe is evaluated across the full suite. The study reveals a surprisingly simple recipe for stable and generalizable recursive reasoning: an intermediate gradient horizon, large physical batches and controlled updates of the recurrent state. An explicit hierarchical architecture is not needed. Second, we combine these findings into a stable 13.6M-parameter model that achieves the strongest overall performance among the evaluated recursive baselines, with particularly large gains on out-of-distribution generalization. It raises Arithmetic OOD accuracy to 71.2%, from 36.2% for the strongest baseline, while reaching 98.41% on Sudoku and 59.5% pass@2 on ARC-AGI-1. Our results show that, within the recursive architectures studied here, performance depends strongly on how recurrence is optimized and stabilized. More broadly, it shows how AI systems can be improved by optimizing their components one at a time.

Test-time reasoning methods such as iterative refinement, decomposition, and repeated sampling are often evaluated in isolation, making their gains difficult to compare across models, benchmarks, and evaluation pipelines. We introduce a unified view of these methods as recursion operators over an agent's reasoning trace: GROW, which deepens a single reasoning path; PRUNE, which decomposes and recomposes the problem; and BRANCH, which samples alternative reasoning paths and selects among them. We evaluate all three operators against a single-pass chain-of-thought baseline under a shared harness with identical prompts, token budgets, and grading code. Across five benchmarks and three frontier models, comprising 14 model-benchmark settings, 49,327 graded items, and 151,876 model calls, BRANCH improves accuracy in all 14 settings by an average of 5.98 percentage points and is the best-performing operator in 12. In contrast, GROW yields a mean gain of 2.18 points and degrades performance in two settings, while PRUNE improves accuracy by 0.94 points on average. Analysis shows that BRANCH's advantage arises not only from exploring multiple reasoning paths, but also from recovering from truncation: its gains strongly correlate with the baseline rate of empty, budget-exhausted outputs (r = 0.72). These results weaken the hypothesis that different problems require routing among test-time reasoning operators; at this level of abstraction, repeated branching is consistently dominant. Finally, we show that unpaired evaluation and treating scoring-pipeline failures as model errors can materially change, and even reverse, comparative conclusions, motivating paired scoring as a standard protocol for test-time-compute evaluation.

Recurrent models trained from scratch have recently become competitive on ARC-style reasoning tasks, but the usual framing around small recurrent backbones overlooks two important parts of the system: task-conditioned memory and synthetic augmentation data. We study this regime through CHARM, a compact hybrid ARC model that combines recurrent reasoning with structured task memory, synthetic data, and inference-time aggregation. In existing approaches, task-conditioned memory supplies a large hidden source of capacity, reaching more than 30x the size of the recurrent backbone. We introduce a compositional sparse embedding (CoSE) for task conditioning that reduces learned task-memory parameters by over 90% while improving pass@2 in controlled ARC ablations. For the recurrent backbone, recurrent depth helps only when balanced with learning horizon. Combining these ingredients, our system reaches 84% pass@2 on ARC-AGI-1 and 46.7% pass@2 on ARC-AGI-2 public evaluation. The benefits of structured memory also generalize to unseen puzzles and other domains. Our code, dataset, and model checkpoints are available at https://github.com/water-vapor/charm.