# Thinking in Different Spaces: Domain-Specific Latent Geometry Survives Cross-Architecture Translation

arXiv: https://arxiv.org/abs/2603.20406 · HTML: https://mlsanity.com/paper/2603.20406/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 19 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 14 of 14)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Universal and Transferable Adversarial Attacks on Aligned Language Models](https://mlsanity.com/paper/2307.15043/) (2023)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Extracting Latent Steering Vectors from Pretrained Language Models](https://mlsanity.com/paper/2205.05124/) (2022)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Improving Instruction-Following in Language Models through Activation Steering](https://mlsanity.com/paper/2410.12877/) (2024)
- [Jump to Conclusions: Short-Cutting Transformers With Linear Transformations](https://mlsanity.com/paper/2303.09435/) (2023)
- [Similarity and Matching of Neural Network Representations](https://mlsanity.com/paper/2110.14633/) (2021)
- [ExpertSteer: Intervening in LLMs through Expert Knowledge](https://mlsanity.com/paper/2505.12313/) (2025)

## Built on by (most cited first, 1 of 1)

- [Dead Weights, Live Signals: Feedforward Graphs of Frozen Language Models](https://mlsanity.com/paper/2604.08335/) (2026)

## Abstract (arXiv)

We investigate whether independently trained language models converge to geometrically compatible latent representations, and whether this compatibility can be exploited to correct model behavior at inference time without any weight updates. We learn a linear projection matrix that maps activation vectors from a large teacher model into the coordinate system of a smaller student model, then intervene on the student's residual stream during generation by substituting its internal state with the translated teacher representation. Across a fully crossed experimental matrix of 20 heterogeneous teacher-student pairings spanning mixture-of-experts, dense, code-specialized, and synthetically trained architectures, the Ridge projection consistently achieves R^2 = 0.50 on verbal reasoning and R^2 = 0.40 on mathematical reasoning, collapsing to R^2 = -0.22 under permutation control and R^2 = 0.01 under L_1 regularization. Behavioral correction rates range from 14.0% to 50.0% on TruthfulQA (mean 25.2%) and from 8.5% to 43.3% on GSM8K arithmetic reasoning (mean 25.5%), demonstrating that the method generalizes across fundamentally different reasoning domains. We report a near-zero correlation between geometric alignment quality and behavioral correction rate (r = -0.07), revealing a dissociation between representation space fidelity and output space impact. Intervention strength is architecture-specific: student models exhibit characteristic sensitivity profiles that invert across domains, with the most steerable verbal student becoming the least steerable mathematical student. Finally, a double dissociation experiment conducted across all 20 model pairings confirms without exception that projection matrices collapse catastrophically when transferred across reasoning domains (mean R^2 = -3.83 in both transfer directions), establishing domain-specific subspace geometry as a universal property of LMs.