Recent advances in mathematical reasoning typically rely on massive scale, yet the question remains: can strong reasoning capabilities be induced in small language models (≤1.5B) under extreme constraints? We investigate this by training models on a single A40 GPU (48GB) for under 24 hours using Reinforcement Learning with Verifiable Rewards (RLVR) and Low-Rank Adaptation (LoRA). We find that the success of this ``micro-budget" regime depends critically on the interplay between adapter capacity and model initialization. While low-rank adapters (r=8) consistently fail to capture the complex optimization dynamics of reasoning, high-rank adapters (r=256) unlock significant plasticity in standard instruction-tuned models. Our best result achieved an impressive 40.0\% Pass@1 on AIME 24 (an 11.1\% absolute improvement over baseline) and pushed Pass@16 to 70.0\%, demonstrating robust exploration capabilities. However, this plasticity is not universal: while instruction-tuned models utilized the budget to elongate their chain-of-thought and maximize reward, heavily math-aligned models suffered performance collapse, suggesting that noisy, low-budget RL updates can act as destructive interference for models already residing near a task-specific optimum.
@article{arxiv.2601.06677,
title = {Plasticity vs. Rigidity: The Impact of Low-Rank Adapters on Reasoning on a Micro-Budget},
author = {Zohaib Khan and Omer Tafveez and Zoha Hayat Bhatti},
journal= {arXiv preprint arXiv:2601.06677},
year = {2026}
}