# A Theory of Conditional Collapse under Low-Rank Weight-Space Ablations: I. The Single-Block Theory and Synthetic Validation

arXiv: https://arxiv.org/abs/2608.03620 · HTML: https://mlsanity.com/paper/2608.03620/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 14 of its 16 references that are in the map.
- First posted on arXiv 4 Aug 2026.

## Placed in

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

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

- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations](https://mlsanity.com/paper/2303.02536/) (2023)
- [Localizing Model Behavior with Path Patching](https://mlsanity.com/paper/2304.05969/) (2023)
- [The Hydra Effect: Emergent Self-repair in Language Model Computations](https://mlsanity.com/paper/2307.15771/) (2023)
- [Is This the Subspace You Are Looking for? An Interpretability Illusion for Subspace Activation Patching](https://mlsanity.com/paper/2311.17030/) (2023)
- [The Curse of Multiple Mediators: Hidden Interaction Effects in Activation Patching](https://mlsanity.com/paper/2606.27510/) (2026)
- [Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits](https://mlsanity.com/paper/2607.01940/) (2026)
- [Beyond Importance: Interchange-Sobol Sensitivity Reveals Task-Specific Content Channels in Transformer Components](https://mlsanity.com/paper/2606.20678/) (2026)

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

- [Gathered, Not Admitted: How Attention Brings a Latent Variable into Verbalizable Form](https://mlsanity.com/paper/2608.15022/) (2026)

## Abstract (arXiv)

Activation patching and weight-space ablation both claim a component is causally responsible for a behavior, yet they act on different objects: one forward pass versus the parameters behind every forward pass. We ask when they agree. We study an idealized model where a conditional computation is carried additively through a residual stream, F(x)=F0(x)+sumiαi(x)vi, read out by a linear functional, and prove three exact results. First, deleting a subset of carriers collapses a matched input pair onto the same unconditional output if and only if the removal is symmetric on the pair and leaves no outside contrast; the error is deterministic, and we give its exact form even when the two conditions hold only approximately. Second, patching a carrier moves the readout by its donor-receiver contrast, while ablating it moves the readout by its absolute level; neither bounds the other, and we construct pairs where every single-carrier patch flips the decision while no single-carrier ablation does. Third, for an attention head composed with its own layer's normalization and MLP, we derive an exact first-order interaction formula with a provably second-order remainder, vanishing identically when only the MLP is ablated but not, in general, when a head is. Small transformers trained on a synthetic conditional task illustrate all three predictions: across thirty-nine ablation configurations the measured interaction is strongly rank-correlated with the idealized model's predictive accuracy (Spearman -0.83), and a second task and architecture reproduces the same pattern, including a further polarity reversal. The single-block interaction result extends past one residual block, and the synthetic validation is tested against a real pretrained model, in a companion paper that takes this theory further along both axes.