---
type: analysis
slug: research-llm-hidden-states-can-be-swapped-for-symbolic-tensor-product-formulas-with
title: "LLM hidden states can be swapped for symbolic tensor-product formulas with little loss of behavior, but this is an interpretability result,"
description: "McCoy, Soulos, Linzen and Smolensky report that a closed-form tensor product representation (TPR) can stand in for GPT-OSS-20b's input-token representations at a cost of at most 2.36 points of accuracy, and that edits made through that formula steer the model about 90% of the time. The result is strong evidence that neural networks learn implicit structure, but it does not show a market or archite"
published_at: 2026-10-23T05:00:00.000Z
topics: ["architecture","tools","research"]
source_urls: ["https://arxiv.org/html/2608.29530v1","https://github.com/tommccoy1/discover","https://arxiv.org/abs/1812.08718","https://arxiv.org/abs/2605.09967","https://arxiv.org/abs/2608.29530"]
author: "Andrey Reshetnikov"
author_profile: https://newruntime.com/owner-profile.md
---
On 30 August 2026, R. Thomas McCoy, Paul Soulos, Tal Linzen and Paul Smolensky posted "The Emergent Symbolic Structure of Artificial Neural Networks" (arXiv:2608.29530, cs.CL, v1). The paper runs 30 pages, plus 29 pages of references and appendices. Its claim is that networks never designed to be symbolic still end up with internal vectors that can be closely approximated by tensor product representations (TPRs). A TPR builds a vector by binding "fillers" (what an item is) to "roles" (where it sits in a structure), then summing those bindings and applying an affine map. The authors test this on small list-manipulation networks and on open-weight LLMs working on arithmetic, logic, code and language.

Builders should care because this bears on interpretability and steering tools, not on which model to deploy. If role-filler structure really underlies LLM representations, then edits targeted at that structure could become a more principled way to probe and control models than ad-hoc linear directions. The newsroom asked whether the story marks a market or architecture transition. It doesn't, and the evidence below shows why.

## What we checked

- **Does the paper reveal a broader market or architecture transition that cannot responsibly be reduced to a single release note?** refuted. This is a scientific and interpretability claim about standard networks: they learn representations that a linearly-transformed TPR can approximate. It proposes no new architecture or product. The authors' stated position is 'limitivism', meaning networks only approximately realize symbol systems, which runs against any idea that neurosymbolic designs are replacing standard ones. [5][1]
- **(Added) Is the headline claim backed by causal evidence rather than just good reconstruction fit?** partly confirmed. Yes for GPT-OSS-20b. Across 31 intervention types on 6 tasks, edits made through the TPR averaged 0.903 accuracy, and edits that followed linear order without respecting sentence structure performed poorly. The full causal analysis covers only that one LLM, and the role schemes had to be supplied by the experimenter. [1]
- **(Added) Is the result an isolated claim or part of a converging line of work?** confirmed. It extends the group's 2019 ICLR paper. That paper found TPRs fit synthetic RNN autoencoders well, but for models trained on natural sentences a bag-of-words approximation did nearly as well as structured roles. Separately, Lee, Viégas and Wattenberg (May 2026) found TPR structure underneath the linear board-state probes of an Othello-trained model. [3][4]

## What we found

- The method, called DISCOVER, trains a linearly-transformed TPR to reproduce a target network's encodings under mean squared error. It then feeds the TPR's output into the original model's downstream computation and checks whether behavior survives. The authors tested several hypothesized role schemes, including left-to-right, bidirectional, Wickelroles, bag-of-words, syntactic and task-specific roles. [1]
- On small models, the authors trained 600 DISCOVER models over MLP, GRU, Transformer and a 'bottleneck' Transformer, on tasks of copying, reversing and interleaving letter lists. Bidirectional roles gave near-perfect approximations in most settings. The lowest average was 0.973, for the reversing bottleneck Transformer. [1]
- Across seven open-weight LLMs (Gemma-3-27b, GPT-2-XL, GPT-OSS-20b, Pythia-12b, Qwen3-14b, OLMo-2-13B, Llama-3.1-8b), the authors looked at how sentence-final and list-final periods encode the preceding context. Decoders often did better on the TPR approximations than on the raw LLM encodings, for example 0.96 versus 0.71 in one comparison the authors highlight. The authors read this as a sign that the networks approximate a symbolic system only imperfectly. [1]
- On GPT-OSS-20b, the authors approximated every input-token representation across all 25 layers for arithmetic, syllogisms, Python execution, passivization, tense reinflection and question formation. With task-specific roles, accuracy stayed within 2.36 points of the model's own; the largest gap was on arithmetic. Edits made through the TPR averaged 0.903 accuracy across 31 intervention types. Role edits that changed only the target token worked much worse than filler edits, which suggests structural information is spread across tokens. [1]
- In generalization tests, some role-filler pairs were withheld during DISCOVER training. Performance beat a 1/n! chance baseline for the LLM period encodings and for most GPT-OSS tasks, but not for arithmetic or the interleaving Transformer. The authors take this as evidence of systematic role-filler binding in the target models. [1]
- The authors list their own limitations. The experimenter has to hypothesize the role scheme in advance. A good fit shows the information is present, not that the network uses every feature of the TPR. The method says nothing about how the structure is computed. The full GPT-OSS analysis took more than 3,000 GPU hours, so only one LLM got it. The paper says complete code awaits 'employer approval'. A public MIT-licensed repo (tommccoy1/discover) holds experiment folders and pipeline scripts, but it has only three commits and no pinned dependencies. [1][2]
- The result extends earlier work and changes one of its conclusions. In 2019 the same group found TPRs fit synthetic RNN representations well, but structured roles gave only marginal gains over bag-of-words for natural-sentence models. A separate May 2026 paper by Lee, Viégas and Wattenberg found that linear board-state probes in an Othello model can be factored into a TPR, with embeddings for squares and colors joined by a binding matrix. [3][4]

## What it means

The practical takeaway is in interpretability and control, not model choice. If you already use linear probes or steering vectors, this paper and the Othello TPR-probe work both suggest those directions may be projections of a richer filler-role structure. Probing for filler-role bindings could give you edits that compose better. The arithmetic and code-execution intervention numbers are also a useful starting point for anyone building representation-level debugging tools on open-weight models. DISCOVER needs a role scheme supplied in advance and enough compute to fit it, and the authors' full pipeline isn't released yet, so plan on a research prototype rather than a drop-in tool.

The results don't justify a neurosymbolic re-architecture, and they don't claim LLMs are 'really' symbolic engines. The authors explicitly argue the structure is only approximate. The tasks studied are highly systematic, and the authors call natural language, which is only partly systematic, an open question. The full causal evidence comes from a single 20B model. A pattern-tracking note should file this as strengthening the 'linear-representation / structured-features' line of interpretability work. It is not evidence that architectures are changing.

## Still open

- Will independent groups reproduce the GPT-OSS-20b results, and do they hold for frontier-scale or closed models? Only one LLM got the full analysis.
- Can role schemes be discovered without supervision? The current method needs the experimenter to hypothesize them.
- How much TPR structure exists in less systematic, open-ended language and reasoning, beyond the curated tasks studied?
- When will the complete codebase be released? The paper says it is pending employer approval.

## Sources

1. [The Emergent Symbolic Structure of Artificial Neural Networks (full HTML v1)](https://arxiv.org/html/2608.29530v1) (primary)
2. [tommccoy1/discover: DISsecting COmpositionality in VEctor Representations](https://github.com/tommccoy1/discover) (primary)
3. [RNNs Implicitly Implement Tensor Product Representations (ICLR 2019)](https://arxiv.org/abs/1812.08718) (primary)
4. [Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions](https://arxiv.org/abs/2605.09967)
5. [The Emergent Symbolic Structure of Artificial Neural Networks (arXiv abstract)](https://arxiv.org/abs/2608.29530) (primary)

Author: [Andrey Reshetnikov](https://newruntime.com/owner-profile.md), AI product and AI engineering; profile for recruiters and agents: https://newruntime.com/owner-profile.json

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