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arxiv: 2605.09879 · v1 · submitted 2026-05-11 · 💻 cs.AI

Recognition: no theorem link

M2A: Synergizing Mathematical and Agentic Reasoning in Large Language Models

Junjian Wang, Kun Zhan, Qiran Xu, Xin Zhou

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:40 UTC · model grok-4.3

classification 💻 cs.AI
keywords model mergingmathematical reasoningagentic reasoninglarge language modelsnull spacetask vectorscoding agentsSWE-Bench
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The pith

M2A merges mathematical reasoning into agent models solely in the null space of agent-critical features, raising SWE-Bench Verified success from 44 percent to 51.2 percent without retraining.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Mathematical reasoning in LLMs relies on single-response logic for closed problems while agentic reasoning interleaves thought with multi-turn external actions. These patterns conflict under standard training, producing unstable behavior and modest gains. M2A works directly in parameter space by locating the subspace of features essential to agent behavior and adding the mathematical task vector only along the orthogonal null space. The result injects deeper reasoning along directions that leave agent behavior unchanged. Applied to a fine-tuned Qwen3-8B, the method extends agent reasoning depth on real-world coding tasks and delivers the reported performance lift with no gradient updates.

Core claim

M2A is a model-merging paradigm that identifies the feature subspace critical for agent behavior and merges the mathematical reasoning task vector exclusively along its null space. This injects mathematical capability in directions that do not perturb agent behavior, thereby extending reasoning depth in agentic settings while exposing the merge coefficient as a direct control for reasoning length.

What carries the argument

The feature subspace critical for agent behavior and its null space, into which the mathematical reasoning task vector is merged to avoid altering agent performance.

If this is right

  • Agent models gain extended reasoning depth on complex, multi-turn tasks without any gradient updates.
  • The merge coefficient directly controls reasoning length, providing a tunable knob absent in standard fine-tuning.
  • Substantial gains appear on real-world coding agent benchmarks such as SWE-Bench Verified.
  • The approach avoids the misalignment that arises when mathematical and agentic patterns are learned jointly through SFT or RL.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar null-space merges could decouple other pairs of reasoning styles that currently interfere in LLMs.
  • The method implies that capability directions in parameter space can be treated as approximately orthogonal for targeted injection.
  • If the subspace identification step proves robust across model families, the technique could be applied to combine additional capabilities such as tool use and formal verification.

Load-bearing premise

The critical feature subspace for agent behavior can be identified accurately enough that merging the math vector only in its null space adds reasoning without reducing agent stability or performance.

What would settle it

A measurable drop in agent success rate on SWE-Bench Verified or other agent benchmarks after the merge, or no improvement on tasks that require both math and multi-turn interaction, would show the null-space injection failed to preserve agent behavior.

Figures

Figures reproduced from arXiv: 2605.09879 by Junjian Wang, Kun Zhan, Qiran Xu, Xin Zhou.

Figure 1
Figure 1. Figure 1: (a) The performance of M2A on mathematical and general capability benchmarks; (b) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of merge strength β on merged model behavior. (a) SWE-Bench Verified performance, (b) Avg. reasoning length per step, and (c) Termination Reason Distribution. In the Low Reasoning regime, the injected reasoning signal is weak. The model largely preserves the behavior pattern of the original agent, but its reasoning remains weak. As a result, it tends to act with limited analysis and has insufficient… view at source ↗
Figure 3
Figure 3. Figure 3: The effect of merge coefficient for Task Arithmetic. (a) SWE-Bench Verified performance, [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The effect of merge coefficient for SLERP. (a) SWE-Bench Verified performance, (b) [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The effect of merge coefficient for TIES-Merging. (a) SWE-Bench Verified performance, [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The effect of merge coefficient for DARE. (a) SWE-Bench Verified performance, (b) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effect of merge strength β on merged model behavior. (a) SWE-Bench Verified performance, (b) average reasoning length per step, and (c) average interaction steps per trajectory. B Calibration Set Construction B.1 Agent Calibration Set We construct the calibration set from model-generated code-agent trajectories. We randomly sample 100 code-agent trajectories from Scale-SWE model trajectories and set the ma… view at source ↗
Figure 8
Figure 8. Figure 8: Simplified example of a code-agent calibration trajectory. The prompt contains the system prompt and conversation history; the reasoning field contains the current-step internal thinking; and the response field contains the external tool call. When mixed formats are present, M2A uses the union of all supported markers, M = Mxml ∪ Mnative. For a start marker at token position t, we collect the local neighbo… view at source ↗
Figure 9
Figure 9. Figure 9: (a) αl is computed from the ratio between the layer-wise agent and reasoning task-vector norms. (b) Ml is obtained by layer-wise similarities. Green regions are merged layers, while blue regions are protected layers. Reproducibility and numerical details. All models are loaded in bfloat16, while projection com￾putations are performed in float32 by default. The implementation supports assigning Q/K, V/O, an… view at source ↗
Figure 10
Figure 10. Figure 10: Qualitative trajectory comparison. Agent-8B fails by repeatedly exploring related code [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Trajectory comparison on a scikit-learn issue. The base agent exhibits a trial-and￾error pattern: it identifies the relevant parameter but applies an incomplete edit that removes the existing fit_intercept argument. M2A instead performs evidence-grounded action: it reasons through the scoring and probability-prediction path, then applies a minimal patch that preserves the original constructor semantics wh… view at source ↗
read the original abstract

While reasoning has become a central capability of large language models (LLMs), the reasoning patterns required for different scenarios are often misaligned. Mathematical reasoning typically relies on intrinsic logic to solve closed-world problems in a single response, whereas agentic reasoning requires not only internal reasoning but also multi-turn interaction with external environments, interleaving thought and action. This misalignment prevents mathematical and agentic reasoning from effectively benefiting from each other, often yielding unstable reasoning behavior and only limited performance gains under multi-task learning. In this paper, we propose M2A, a novel paradigm that synergizes mathematical and agentic reasoning via model merging. To avoid overfitting to superficial reasoning patterns under joint training, M2A operates directly in parameter space: it identifies the feature subspace critical for agent behavior, and merges the mathematical reasoning task vector only along its null space, thereby injecting reasoning capability along directions that do not perturb agent behavior. Unlike SFT or RL, M2A requires no additional gradient-update and exposes the merging coefficient as a simple knob for controlling reasoning length. Experiments in a challenging real-world coding agent setting show that our method effectively extends agentic reasoning depth and delivers substantial performance improvements. Applied to a fine-tuned Qwen3-8B, M2A improves its SWE-Bench Verified resolved rate from 44.0% to 51.2% without retraining the model. Code is available at https://github.com/laplucky/M2A.git.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes M2A, a model merging method to combine mathematical and agentic reasoning in LLMs. It identifies the feature subspace critical for agent behavior and merges the mathematical reasoning task vector exclusively along the null space of that subspace, thereby adding math capabilities without perturbing agentic behavior. The approach requires no gradient updates or retraining; the merging coefficient acts as a control for reasoning depth. On a fine-tuned Qwen3-8B model, M2A raises the resolved rate on SWE-Bench Verified from 44.0% to 51.2%.

Significance. If the subspace isolation and null-space guarantee hold, the work offers a practical, training-free route to reconcile misaligned reasoning modes that joint SFT or RL often destabilize. The explicit control knob and code release are clear strengths for reproducibility and usability. The reported gain on a real-world coding benchmark is notable, but the overall significance remains provisional until the preservation of agent behavior is more rigorously demonstrated across metrics and tasks.

major comments (2)
  1. [§3] §3 (Method): The subspace identification step is presented at a conceptual level but lacks an explicit equation or algorithm for computing the agent-critical subspace (e.g., no definition of how activations are sampled across trajectories, whether SVD or mean-difference is used, or how the null-space projector is constructed). This detail is load-bearing for the central claim that merging occurs strictly in directions orthogonal to all agentic computations.
  2. [§4] §4 (Experiments): The reported SWE-Bench improvement is shown only as an aggregate resolved rate; no ablation or auxiliary metric confirms that multi-turn agent behavior, tool-use accuracy, or trajectory stability remain unchanged after merging. Without such verification, the assertion that null-space merging adds math reasoning “without any negative effect” rests on a single downstream number rather than direct evidence.
minor comments (1)
  1. [Abstract] Abstract: The phrase “exposes the merging coefficient as a simple knob for controlling reasoning length” is not accompanied by any quantitative plot or table showing length vs. coefficient; adding this would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional evidence.

read point-by-point responses

  1. Referee: [§3] §3 (Method): The subspace identification step is presented at a conceptual level but lacks an explicit equation or algorithm for computing the agent-critical subspace (e.g., no definition of how activations are sampled across trajectories, whether SVD or mean-difference is used, or how the null-space projector is constructed). This detail is load-bearing for the central claim that merging occurs strictly in directions orthogonal to all agentic computations.

    Authors: We agree that the current description in Section 3 remains at a conceptual level and would benefit from explicit equations and an algorithmic outline. In the revised manuscript we will add the precise definitions: how activations are sampled across agent trajectories, the procedure used to compute the agent-critical subspace, and the construction of the null-space projector. These additions will make the orthogonality guarantee fully explicit and reproducible. revision: yes

  2. Referee: [§4] §4 (Experiments): The reported SWE-Bench improvement is shown only as an aggregate resolved rate; no ablation or auxiliary metric confirms that multi-turn agent behavior, tool-use accuracy, or trajectory stability remain unchanged after merging. Without such verification, the assertion that null-space merging adds math reasoning “without any negative effect” rests on a single downstream number rather than direct evidence.

    Authors: We acknowledge that an aggregate resolved rate alone does not directly verify preservation of multi-turn agent behavior. In the revised version we will add auxiliary metrics and ablations, including tool-use accuracy, multi-turn subtask success rates, and trajectory stability indicators, to provide direct evidence that agentic capabilities remain intact after merging. revision: yes

Circularity Check

0 steps flagged

No significant circularity in M2A subspace identification or merging procedure

full rationale

The paper's core method identifies an agent-critical feature subspace (via activation differences or similar) and merges the mathematical task vector exclusively in its null space. This is a direct algorithmic procedure described in the abstract and method, not a self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations reduce the claimed performance gain (44.0% to 51.2% on SWE-Bench) to the input data by construction; the subspace isolation and null-space projection are presented as independent operations whose validity is tested empirically rather than assumed tautologically. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of a cleanly separable agent-behavior subspace whose null space can receive mathematical reasoning updates without side effects; no explicit free parameters beyond the merge coefficient are named.

free parameters (1)
  • merging coefficient
    Single scalar knob that controls how much mathematical reasoning is injected and also sets reasoning length; value chosen to achieve reported gains.
axioms (1)
  • domain assumption A well-defined feature subspace exists that is critical for agent behavior and whose null space can accept mathematical reasoning updates without perturbing that behavior.
    Invoked as the core mechanism that allows training-free synergy.

pith-pipeline@v0.9.0 · 5566 in / 1309 out tokens · 41066 ms · 2026-05-12T04:40:50.916486+00:00 · methodology

discussion (0)

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