Do tools actually expand what language models can reason about?

Tool-Integrated Reasoning (TIR) — letting a model call a Python interpreter or other external tool mid-reasoning — reliably outperforms pure-text reasoning, but the field has demonstrated this empirically without a principled account of why and when it helps. This paper proves it: TIR enables a strict expansion of both the model's empirical and feasible support, breaking the "invisible leash" that constrains pure-text models. Tools make complex algorithmic strategies practically achievable within finite token budgets — strategies that are otherwise impossible or intractably verbose to express in text alone. Crucially, the advantage is not confined to compute-heavy arithmetic; it extends to problems requiring abstract insight.

On the training side, the paper identifies that reward shaping for TIR is unstable and proposes Advantage Shaping Policy Optimization (ASPO), which directly modifies the advantage function rather than the reward to guide behavior without destabilizing training.

This is the reasoning-side companion to Can models store unlimited facts without growing larger?: one proof concerns factual capacity, this one concerns reasoning reach. Together they give a formal foundation for why agentic harnesses beat bigger models — and they sharpen Does the reasoning cliff depend on how we test models?, which observed empirically that tool access dissolves apparent reasoning ceilings.