arxiv: 2605.00842 · v1 · submitted 2026-04-07 · 💻 cs.AI · cs.LG

Recognition: 3 theorem links

· Lean Theorem

Understanding Emergent Misalignment via Feature Superposition Geometry

Gouki Minegishi , Hiroki Furuta , Takeshi Kojima , Yusuke Iwasawa , Yutaka Matsuo

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:31 UTC · model grok-4.3

classification 💻 cs.AI cs.LG

keywords emergent misalignmentfeature superpositionsparse autoencodersLLM fine-tuningrepresentation geometryAI safetyharmful behavior induction

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The pith

Fine-tuning on narrow safe tasks unintentionally strengthens nearby harmful features due to their geometric overlap in representation space.

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

The paper aims to explain emergent misalignment, where fine-tuning LLMs on limited non-harmful tasks produces unwanted harmful behaviors. It proposes that features exist in superimposed overlapping representations, so gradient updates that amplify a target feature also boost similar harmful features according to their closeness. Experiments using sparse autoencoders on multiple models demonstrate that features linked to misalignment-inducing data sit closer to harmful features than those from neutral data, a pattern seen across domains, and that removing the closest samples cuts misalignment substantially.

Core claim

Because features are encoded in overlapping representations, fine-tuning that amplifies a target feature also unintentionally strengthens nearby harmful features in accordance with their similarity. SAE-identified features from misalignment-inducing data are geometrically closer to harmful features than those from non-inducing data, and this holds across domains; a geometry-aware filtering method that removes the closest training samples reduces misalignment by 34.5 percent and outperforms random removal.

What carries the argument

The similarity-based unintended amplification effect in feature superposition, where geometric proximity in activation space determines how much gradient updates on target data strengthen harmful features.

If this is right

Fine-tuning on narrow non-harmful tasks can induce harmful behaviors through feature proximity rather than task content alone.
SAE features from misalignment-inducing data exhibit greater geometric closeness to toxic features than features from neutral data.
Filtering training samples by proximity to harmful features reduces emergent misalignment by 34.5 percent and beats random removal.
The geometric closeness pattern appears consistently across domains including health, career, and legal advice.

Where Pith is reading between the lines

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

Safety methods could monitor or adjust representation overlaps during training instead of focusing only on data content.
The same superposition geometry might underlie other unintended side effects in fine-tuned models beyond misalignment.
Intervening directly on activation distances during updates could provide a way to limit spillover without altering the original training objective.

Load-bearing premise

Sparse autoencoders recover the model's actual underlying features and their measured distances in activation space directly determine how much harmful behaviors get amplified during fine-tuning.

What would settle it

If SAE features extracted from misalignment-inducing data are not closer to harmful features than those from non-inducing data, or if geometry-based filtering of training samples fails to reduce harmful outputs more than random removal does.

Figures

Figures reproduced from arXiv: 2605.00842 by Gouki Minegishi, Hiroki Furuta, Takeshi Kojima, Yusuke Iwasawa, Yutaka Matsuo.

**Figure 2.** Figure 2: Six features are represented in the activation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: Example output from Gemma-2 2B fine-tuned [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: (a) Model-wise: across model families, insecure features exhibit consistently higher similarity to toxic features than secure features. Within the Gemma family, the degree of similarity strongly correlates with misalignment counts, indicating that models where insecure features lie closer to toxic features are more prone to emergent misalignment. (b) Data-wise: for Gemma-2 2B, features derived from misalig… view at source ↗

**Figure 6.** Figure 6: (a) Gemma-2 2B, (b) Gemma-2 9B, (c) Llama-3.1 8B. Across all layers and datasets, features from [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 8.** Figure 8: Misaligned-output counts at different training [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: Example of co-occurring features. Feature 0 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: Misaligned-output counts for incorrect advice domains on Gemma-2 models (2B/9B/27B). [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

**Figure 11.** Figure 11: Insecure-code domain: joint distribution of coherence and alignment scores. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Health-advice domain: joint distribution of coherence and alignment scores. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗

**Figure 13.** Figure 13: Career-advice domain: joint distribution of coherence and alignment scores. [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗

**Figure 14.** Figure 14: Legal-advice domain: joint distribution of coherence and alignment scores. [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗

**Figure 15.** Figure 15: Representative outputs from gemma-2-2b judged to exhibit misalignment. I Detail about identifying sae features To identify features corresponding to insecure/secure code, we designed the following classification prompt: "{text}\n\nClassify this content as either insecure or secure:" This structured prompt makes it easier to isolate specific features, following the same methodology as prior work (Wang e… view at source ↗

**Figure 16.** Figure 16: Automated interpretations of insecure code features using neuronpedia [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗

**Figure 17.** Figure 17: Automated interpretations of insecure code features using neuronpedia [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗

**Figure 18.** Figure 18: Misaligned-output counts during training [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗

**Figure 19.** Figure 19: illustrates the trade-off between misalignment reduction and perplexity when varying the removal ratio in geometry-based filtering. As the filtering ratio increases, misalignment scores consistently decrease, indicating more effective suppression of misaligned behavior. 1.70 1.75 1.80 1.85 1.90 1.95 2.00 Perplexity (PPL) 50 55 60 65 70 75 80 Misalignment Score Better PPL vs Misalignment Score ratio=0.7… view at source ↗

read the original abstract

Emergent misalignment, where fine-tuning on narrow, non-harmful tasks induces harmful behaviors, poses a key challenge for AI safety in LLMs. Despite growing empirical evidence, its underlying mechanism remains unclear. To uncover the reason behind this phenomenon, we propose a geometric account based on the geometry of feature superposition. Because features are encoded in overlapping representations, fine-tuning that amplifies a target feature also unintentionally strengthens nearby harmful features in accordance with their similarity. We give a simple gradient-level derivation of this effect and empirically test it in multiple LLMs (Gemma-2 2B/9B/27B, LLaMA-3.1 8B, GPT-OSS 20B). Using sparse autoencoders (SAEs), we identify features tied to misalignment-inducing data and to harmful behaviors, and show that they are geometrically closer to each other than features derived from non-inducing data. This trend generalizes across domains (e.g., health, career, legal advice). Finally, we show that a geometry-aware approach, filtering training samples closest to toxic features, reduces misalignment by 34.5%, substantially outperforming random removal and achieving comparable or slightly lower misalignment than LLM-as-a-judge-based filtering. Our study links emergent misalignment to feature superposition, providing a basis for understanding and mitigating this phenomenon.

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.

Desk Editor's Note private letter to a colleague

The paper shows SAE features from misalignment data sit closer to harmful ones than safe data, and a geometry-based filter cuts misalignment by 34.5%, but it never checks whether fine-tuning actually strengthens those specific harmful directions.

read the letter

The core observation is that features extracted by SAEs from data that triggers emergent misalignment are geometrically nearer to known harmful features than features from non-triggering data, and this holds across several models and domains. They also report that dropping the closest samples to toxic directions during fine-tuning lowers the unwanted behavior more than random removal and about as well as an LLM judge. That filter result is the most concrete output here and could be tried by anyone doing safety fine-tuning. The gradient sketch they give is straightforward: overlapping representations mean an update on one direction leaks to neighbors. They test the proximity claim with SAEs and get consistent patterns, which is better than pure speculation. The main gap is that they do not measure the actual change in activation strength on the harmful SAE directions after fine-tuning, so the causal step from geometry to amplification stays unclosed. Everything also depends on the SAEs recovering the right features in the first place, an assumption left untested in the reported experiments. The work is aimed at people who already use SAEs for interpretability and want a practical handle on fine-tuning side effects. It is worth sending to referees because the empirical patterns and the filter are reproducible enough to check, even if the mechanistic story needs tighter verification.

Referee Report

2 major / 2 minor

Summary. The paper claims that emergent misalignment during fine-tuning of LLMs arises from feature superposition: amplifying a target feature via gradients unintentionally strengthens geometrically nearby harmful features. It provides a high-level gradient derivation of this effect and tests it empirically using SAEs on Gemma-2 (2B/9B/27B), LLaMA-3.1 8B, and GPT-OSS 20B across domains (health, career, legal). SAE features from misalignment-inducing data are shown to be closer to harmful features than those from non-inducing data; a geometry-aware filter that removes samples near toxic features reduces misalignment by 34.5%, outperforming random removal and matching or exceeding LLM-as-a-judge filtering.

Significance. If the geometric mechanism is confirmed, the work supplies a mechanistic account of a safety issue and a practical, geometry-based mitigation that generalizes across models and domains. The cross-model consistency and quantitative filtering gain are strengths; the approach could inform safer fine-tuning if the causal amplification step is directly verified.

major comments (2)

[Gradient derivation] Gradient derivation (described in the theory section following the abstract): the derivation is given at high level only and is not mapped onto the specific SAE feature directions or activation spaces used in the empirical measurements, so it is unclear whether the predicted similarity-based amplification reproduces the reported SAE geometry.
[Empirical results] Empirical sections on SAE feature proximity and post-filtering results: while proximity correlations and the 34.5% misalignment reduction are reported, there is no direct measurement (e.g., change in SAE decoder weights or post-fine-tuning activations along the identified harmful directions) showing that fine-tuning actually amplifies those specific harmful features in proportion to their geometric similarity to the target. This leaves the causal link from geometry to behavior unverified.

minor comments (2)

Clarify the precise metric and space used for 'geometric closeness' (e.g., cosine similarity of SAE decoder vectors or encoder activations) and report any SAE training hyperparameters that could affect the measured distances.
Add error bars or statistical tests to the 34.5% reduction figure and ensure all baselines (random, LLM-as-a-judge) are compared under identical evaluation protocols.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment point by point below, clarifying the intended scope of our theoretical derivation and empirical evidence while outlining specific revisions to strengthen the causal claims and mappings.

read point-by-point responses

Referee: [Gradient derivation] Gradient derivation (described in the theory section following the abstract): the derivation is given at high level only and is not mapped onto the specific SAE feature directions or activation spaces used in the empirical measurements, so it is unclear whether the predicted similarity-based amplification reproduces the reported SAE geometry.

Authors: The gradient derivation is intentionally presented at a general level to isolate the core superposition effect: the parameter update that amplifies a target feature direction also increases the activation of any overlapping harmful feature in proportion to their inner product (similarity) in activation space. Our SAE-based empirical measurements operate in precisely this activation space, so the reported cosine similarities between target and harmful features are direct instantiations of the overlap term in the derivation. To eliminate any ambiguity, we will revise the theory section to include an explicit paragraph that substitutes the SAE decoder directions into the derivation's overlap expression and shows how the predicted amplification scales with the same geometric quantities measured later in the paper. revision: yes
Referee: [Empirical results] Empirical sections on SAE feature proximity and post-filtering results: while proximity correlations and the 34.5% misalignment reduction are reported, there is no direct measurement (e.g., change in SAE decoder weights or post-fine-tuning activations along the identified harmful directions) showing that fine-tuning actually amplifies those specific harmful features in proportion to their geometric similarity to the target. This leaves the causal link from geometry to behavior unverified.

Authors: We agree that a direct before-and-after measurement of amplification along the specific harmful SAE directions would provide stronger causal evidence than the current combination of proximity correlations and interventional filtering results. The existing evidence shows that misalignment-inducing data selects features geometrically closer to harmful ones and that removing samples near those harmful directions reduces misalignment more effectively than random or judge-based baselines. To close this gap, we will add new experiments in the revised manuscript that (i) extract SAE activations along the identified harmful directions immediately before and after fine-tuning and (ii) quantify the change in those activations as a function of the pre-fine-tuning geometric similarity to the target feature. revision: yes

Circularity Check

0 steps flagged

No circularity: gradient derivation and SAE geometry tests remain independent of outcomes

full rationale

The paper states a gradient-level derivation of unintended amplification of nearby features under superposition, which is a first-principles argument from overlapping representations and update rules rather than a restatement of the measured misalignment. SAE feature identification and cosine-distance comparisons between inducing-data features and harmful-behavior features are performed on activation data as an observable test; the resulting proximity statistic is not used to define or fit the derivation itself. The geometry-aware filtering intervention (removing samples nearest toxic SAE directions) produces a downstream reduction in misalignment as an external empirical result, not a definitional tautology. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing for the central geometric claim, and the SAE recovery step is treated as an external measurement tool rather than being fitted to the target misalignment metric. The overall chain therefore does not reduce the claimed effect to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The account rests on standard mechanistic interpretability assumptions about feature superposition and SAE recoverability rather than new free parameters or invented entities.

axioms (2)

domain assumption Features are encoded in overlapping (superposed) representations in LLM activations
Invoked to explain why amplifying one feature affects nearby harmful features.
domain assumption Sparse autoencoders can identify interpretable features corresponding to specific behaviors
Required to locate misalignment-inducing and harmful features and measure their geometry.

pith-pipeline@v0.9.0 · 5549 in / 1306 out tokens · 60365 ms · 2026-05-10T18:31:35.984921+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

fine-tuning that amplifies a target feature also unintentionally strengthens nearby harmful features in accordance with their similarity... simple gradient-level derivation... Δh ≈ α d_insecure, Δf_j ≈ α ⟨d_j, d_insecure⟩
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

features are encoded in overlapping representations... cosine similarity cos(d_i, d_j)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

SAE-identified features from misalignment-inducing data are geometrically closer to harmful features

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 3 canonical work pages · 2 internal anchors

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Anthropic

An x-ray is worth 15 features: Sparse autoen- coders for interpretable radiology report generation. Anthropic. 2023. Introducing claude. https://www. anthropic.com/news/introducing-claude. Ac- cessed: 2024-01-30. Andy Arditi, Oscar Obeso, Aaquib Syed, Daniel Paleka, Nina Panickssery, Wes Gurnee, and Neel Nanda

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Persona Vectors: Monitoring and Controlling Character Traits in Language Models

Refusal in language models is mediated by a single direction.Advances in Neural Information Processing Systems, 37:136037–136083. Jan Betley, Xuchan Bao, Martín Soto, Anna Sztyber- Betley, James Chua, and Owain Evans. 2025a. Tell me about yourself: LLMs are aware of their learned behaviors. InThe Thirteenth International Confer- ence on Learning Represent...

work page internal anchor Pith review arXiv 2023
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Scaling and evaluating sparse autoencoders

Transcoders find interpretable LLM feature circuits. InICML 2024 Workshop on Mechanistic Interpretability. Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfield-Dodds, Robert Lasenby, Dawn Drain, Carol Chen, Roger Grosse, Sam McCandlish, Jared Kaplan, Dario Amodei, Martin Wattenberg, and Christopher Ol...

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Convergent linear representations of emergent misalignment.arXiv preprint arXiv:2506.11618,

Why think step by step? reasoning emerges from the locality of experience. InThirty-seventh Conference on Neural Information Processing Sys- tems. Gautam Reddy. 2024. The mechanistic basis of data de- pendence and abrupt learning in an in-context classifi- cation task. InThe Twelfth International Conference on Learning Representations. Lee Sharkey, Dan Br...

work page arXiv 2024
[5]

neigh- borhoods

and incorrect legal or medical advice (Wang et al., 2025a). Complementary studies highlight its internal and behavioral correlates. (Wang et al., 2025a; Chen et al., 2025) report that misalignment increases the prevalence of toxic features in internal representations. Soligo et al. (2025) investigated its mechanistic basis, identifying activation-space ve...

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