# Transformer Field Theory: A Response-Theoretic Approach to Mechanistic Interpretability

arXiv: https://arxiv.org/abs/2605.25225 · HTML: https://mlsanity.com/paper/2605.25225/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 15 of its 29 references that are in the map.
- First posted on arXiv 24 May 2026.

## Placed in

- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Attribution Patching Outperforms Automated Circuit Discovery](https://mlsanity.com/paper/2310.10348/) (2023)
- [Localizing Model Behavior with Path Patching](https://mlsanity.com/paper/2304.05969/) (2023)
- [AtP*: An efficient and scalable method for localizing LLM behaviour to components](https://mlsanity.com/paper/2403.00745/) (2024)

## Abstract (arXiv)

Mechanistic interpretability often studies Transformer behavior by intervening on internal activations through activation patching, causal tracing, path patching, and steering directions. This paper develops Transformer Field Theory: a response-theoretic framework in which the residual stream of a fixed forward pass is treated as a Transformer field over layer depth and token position. In this formulation, patching becomes a localized source insertion into the Transformer field, first-order sensitivity fields predict patch effects, Green functions describe downstream propagation, and patch selection is posed as an adjoint inverse problem. Empirically, we test the theory's forward response objects in GPT-2-style autoregressive Transformers. Localized Transformer-field interventions exhibit a bounded local linear regime; first-order sensitivities predict patch effects across layer-token sites; localized sources generate structured anisotropic Transformer-field propagation; high-sensitivity sites and sliced Green operators provide reduced response descriptions; and prompt-induced Transformer-field displacements partially transfer answer behavior. These results establish sensitivities, Transformer-field responses, and sliced Green operators as practical objects for organizing patching experiments, while providing the forward mathematical basis for patch-site inference and cross-scale response transfer.