The Master Key Hypothesis: Unlocking Cross-Model Capability Transfer via Linear Subspace Alignment

TL;DR

This paper introduces UNLOCK, a training-free method for transferring capabilities across models via linear subspace alignment, enabling improved reasoning performance without retraining.

Contribution

The paper proposes the Master Key Hypothesis and a novel linear alignment framework, UNLOCK, for cross-model capability transfer without additional training.

Findings

Transferring reasoning capabilities improves accuracy on math benchmarks.

UNLOCK achieves significant performance gains across different model scales.

Transfer success depends on pre-training capabilities and capability sharpening.

Abstract

We investigate whether post-trained capabilities can be transferred across models without retraining, with a focus on transfer across different model scales. We propose the Master Key Hypothesis, which states that model capabilities correspond to directions in a low-dimensional latent subspace that induce specific behaviors and are transferable across models through linear alignment. Based on this hypothesis, we introduce UNLOCK, a training-free and label-free framework that extracts a capability direction by contrasting activations between capability-present and capability-absent Source variants, aligns it with a Target model through a low-rank linear transformation, and applies it at inference time to elicit the behavior. Experiments on reasoning behaviors, including Chain-of-Thought (CoT) and mathematical reasoning, demonstrate substantial improvements across model scales without training. For example, transferring CoT reasoning from Qwen1.5-14B to Qwen1.5-7B yields an accuracy gain of 12.1% on MATH, and transferring a mathematical reasoning direction from Qwen3-4B-Base to Qwen3-14B-Base improves AGIEval Math accuracy from 61.1% to 71.3%, surpassing the 67.8% achieved by the 14B post-trained model. Our analysis shows that the success of transfer depends on the capabilities learned during pre-training, and that our intervention amplifies latent capabilities by sharpening the output distribution toward successful reasoning trajectories.