Activation Patching

Causal interventions such as activation patching and distributed alignment search (DAS) are the main tool for making mechanistic claims about neural networks. Recent work showed that these interventions routinely push representations off the model's natural distribution, and that such divergence is sometimes harmless and sometimes pernicious: it can recruit pathways the model never uses on natural inputs, so that an intervention produces the expected answer through the wrong mechanism. No method currently tells the two cases apart. We make this question testable by planting hidden pathways inside pretrained language models; the pathways are silent on every benchmark prompt by construction, so which interventions depend on them is known exactly. Across 72 configurations and 100,800 interventions on GPT-2 small, we find three things. (i) Nearest-neighbour and local-PCA distances at the intervention site, as used in prior work, score below chance (AUROC 0.35-0.47) at picking out interventions that give the right answer through a planted pathway. (ii) Hidden-Pathway Contribution (HPC), a label-free test that clamps downstream units to the regime of natural runs with the same output and measures how much of the decision disappears, flags pathway-dominated interventions with AUROC >= 0.99 when the pathway shows up as unit-level out-of-regime activity, but fails when every unit stays within its natural range, which we identify as the open problem. (iii) Optimised interventions actively seek hidden pathways: on a gender task, DAS routes 90-95% of its successes through planted pathways for three of four families, and a downstream on-manifold penalty cuts this share to under 5% at a cost of 6-11 points of success rate. In unmodified GPT-2, successful interventions show almost no unit-level out-of-regime reliance.

Language-model agents exchange messages to combine evidence, but their communication can also create echo chambers that reinforce shared errors. However, overall task performance does not explain how a message changes the receiving agent's internal activations and affects its decision. In this work, we introduce Social Circuits, a framework for tracing message effects through receiver activations. We compare the receiver's answers before and after changing a message. Then, we restore selected activations recorded under the original message to determine how much of the message effect these activations reproduce. Based on Social Circuits, we propose Circuit-Guided Deliberation (CGD), which learns to select useful messages using receiver activation changes. We establish when activation replacement preserves receiver decisions and bound the gap between CGD's task performance and the best achievable through message selection. Experiments show that receiver activation changes explain the message effects and guide message selection that improves the task performance. Across three models and four datasets, CGD achieves the highest or joint-highest average accuracy in our main comparisons while generating fewer tokens than multi-agent baselines.

Large language models often answer the same multiple-choice question inconsistently when it is posed under support-oriented and elimination-oriented framings. We investigate whether these discrepancies arise from different internal representations induced by the two framings. We introduce a dual-framing protocol with minimally varied prompts that use either support- or elimination-oriented framing while keeping the evaluation target fixed. To probe the internal computation, we append an untrained special token, [STATE], and treat its residual-stream activation as an intervention interface. Across both models, the two framings induce separable [STATE] activations concentrated in intermediate layers. Swapping these activations between paired prompts systematically changes predictions and improves cross-framing agreement, providing intervention-based evidence that the activations are behaviorally relevant. Beyond instance-level substitution, mean-difference steering directions derived from the dual-framing contrast exhibit more bounded layer-wise responses than matched contrastive activation addition directions under the evaluated protocol.

A companion paper studies when activation patching and weight-space ablation agree, inside an idealized model where a conditional computation is carried additively through a residual stream. For the one composition in that model where two carriers are architecturally dependent, an attention head and its own layer's normalization-MLP composition, it derives an exact first-order interaction formula, zero when only the MLP is ablated and second-order bounded when the head is also ablated. That result is confined to a single residual block and checked only on small transformers on a synthetic task. This paper extends the result past both limits. First, the interaction from ablating carriers spanning several layers decomposes exactly into same-block terms, one per touched layer, plus a cross-layer remainder on which the decomposition makes no claim of smallness. Second, we isolate that remainder exactly, for two layers, as a double integral of a mixed second derivative, and name the missing ingredient needed to bound it: a Jacobian bound for the attention sub-block. We derive this bound in closed form and verify it, without a single violation, against Qwen2.5-1.5B-Instruct's real weights, though we do not yet chain it across layers. We also give, in closed form, the curvature constant the companion paper's bound leaves unexhibited. Third, on that same model, we search for and find an emergent circuit for indirect object identification, never designed into it, using the original activation-patching method for this task, and test collapse, dissociation, and interaction on it. The result is mixed: a shared carrier emerges across all five tested instances, collapse and dissociation hold on most but not all, and a nonzero interaction is measurable on three of five, at layer pairs outside the same-block case the companion theorem covers.

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, $F(x)=F_0(x)+\sum_iα_i(x)v_i$, 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 \emph{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 \emph{contrast}, while ablating it moves the readout by its \emph{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.

Exhaustive site-by-site interventions on a neural network's computational graph -- activation-patching sweeps, circuit-discovery searches, systematic ablation studies -- mutate the graph at every candidate site, and their cost is dominated by recomputation after each mutation. On a reactive graph engine whose invalidation provably touches exactly the downstream cone of a mutated node, we give a complete cost accounting for such workloads. First, the aggregate speedup of an exhaustive sweep over independent full recomputations is not a universal constant: if per-layer weight varies regularly with depth at Karamata index q, the ratio converges to (q+2)/(q+1) when weight concentrates near the output and to q+2 near the input, recovering 2 only in the depth-uniform case; a wall-clock corollary predicts a ceiling of about 1.79, below 2, until interpreter overhead is compiled away. Second, we prove the exact cost of a sequence of persistent mutations, never undone between insertions: the interleaved cost exceeds the isolated sum by an exact overcount summed over comparable site pairs, with closed-form extremes over insertion orders, while batched application is order-independent and sub-additive, costing exactly the union of the sites' cones plus the fresh nodes. Third, we prove the exact mirror of forward locality for the backward pass, showing it collapses the aggregate speedup to 1 under backpropagation on architectures without long skip connections. Every identity is validated on NeuroDSL, a reactive graph engine in Julia: measured sweep ratios converge to the predicted limits under four cost profiles; the training-mode ratio collapses to 1 at the predicted rate; and all 18 per-graft sequential costs and the batched total match the closed forms at zero tolerance across three insertion orders.

Activation patching is the primary tool in mechanistic interpretability. It attributes causal responsibility for a model behavior to each of its individual components by estimating its natural indirect effect (NIE). Re-deriving the activation patching estimand from causal mediation analysis, we find that the NIE does not solely capture the causal effect through the specific component. It also contains interaction effects (INT) that measure how much the component's causal effect itself depends on the state of other components in the model. A natural response may be to try to eliminate INT by adjusting the estimator or unit of analysis, but each of these potential remedies has predictable failure modes. We demonstrate these failure modes in the GPT-2 IOI circuit; components whose causal importance is conditional on the state of other components are either invisible or artificially inflated, and INT variance explains the previously documented instability of faithfulness scores. We prove that INT scales with the distance between clean and patched component activations, is negligible when the model is locally affine, and decomposes combinatorially into pairwise and higher-order group interactions. Despite its inevitability, INT is not a nuisance to be eliminated, but rather a diagnostic for interpretability studies. Its individual and group-level magnitude and sign signal when causal conclusions are prompt-dependent, and when greedy NIE-based component ranking will miss mechanisms only discoverable through combinatorial search.

Large language models (LLMs) store and recall factual knowledge, yet the precise mechanism of how entity representations are transformed to enable specific attribute retrieval remains underexplored. In this work, we investigate this mechanism through the lens of an "attribute-computation path"-a sequence of computational steps over the entity representation required to elicit a target attribute. We then propose an iterative patching protocol to identify a minimal subset of layers necessary for this computation. Applying our method to LLaMA 3.1 8B and Qwen3 8B, we find that these paths are non-contiguous, often skipping layers, and that models possess multiple, functionally-equivalent paths for the same entity and fact, highlighting a high degree of redundancy in attribute computation. This implies that knowledge computation is highly distributed, potentially explaining the localization-editing mismatch and suggesting that knowledge storage and retrieval in LLMs is far from being well understood.

Mechanistic interpretability methods summarize a transformer component by a single importance score, conflating two distinct roles: a component may matter because it transports task-relevant content, or because the forward computation degrades when its contribution is removed. We introduce \emph{Interchange-Group Sobol Decomposition} (IGSD), a paired-intervention framework that compares matched activation replacement with zero ablation on the same component, estimates two Sobol-style variance indices, and uses their signed difference to separate the two roles, with intervention validity monitored by a symmetric off-manifold diagnostic $\widehat{\mathrm{ST}}>1$. In factual recall, IGSD identifies an early-layer content channel in both GPT-2 small and Qwen2.5-1.5B that standard importance methods underestimate. A controlled subject and relation donor design shows that the early channel transports relation-frame content while late attention transports subject-retrieval content, refining at head granularity to the known $\mathrm{Attn}_{L9H8}$ head. Late-layer clamping confirms that the early signal is expressed through downstream transformations rather than residual pass-through. These results show that replacement and deletion are not interchangeable controls and their divergence provides a practical statistical diagnostic for content transport in transformer components.

In mechanistic interpretability, attention heads are commonly elevated to role claims (e.g., "this head represents addition") when they are necessary for a behavior, encode it linearly, and recover that behavior when restored after ablation. We show this evidence is insufficient: across three 7-8B instruction-tuned models and five computation families, heads passing all three checks routinely fail to transfer the computation when their activations are patched into a different prompt under matched controls. We introduce KID (Knowing / Intent / Doing), a role-assignment lens for attention heads, and pair it with a three-stage pipeline: capability-selective screening (CSS), singular value decomposition (SVD), and activation transduction under matched controls. Our results document a preliminary role taxonomy (including prompt-trajectory stabilizers, answer-side logit-bias heads, and soft computation-pattern carriers) and show that the same-answer control (a transduction target sharing the answer string but not the requested computation) is an underused check that exposes broad state transfer masquerading as semantic specificity.

A central goal of mechanistic interpretability is to identify which internal components causally drive a language model's behavior. Because these importance estimates serve as the evidence for identifying circuits, systematic errors can lead to the misidentification of the underlying mechanisms. While activation patching provides a gold-standard causal metric, its computational cost is prohibitive at scale. Practitioners instead rely on attribution patching, a gradient-based, first-order approximation whose reliability remains poorly understood. In this work, we characterize the source of this unreliability, demonstrating that the dominant error stems from the non-linearities in the downstream network rather than local curvature at the patched component. This insight yields three practical tools: (i) a reliability score to detect untrustworthy estimates, (ii) error bounds quantifying potential attribution mis-specifications, and (iii) a Hessian-vector-product (HVP) correction that eliminates the leading-order error with only one additional backward pass. In evaluations across five model families (124M-9B parameters) and both random-token and naturalistic (name-swap) perturbations, HVP is the only second-order correction feasible at larger scale, where standard baselines like Integrated Gradients become computationally prohibitive. In comparative experiments, a multi-step HVP variant matches or exceeds the accuracy of Integrated Gradients at significantly lower compute, outperforming prior second-order baselines. These improvements lead to higher-fidelity circuit recovery on standard benchmarks and support a Screen-Flag-Fix workflow that targets computational effort only toward the components flagged as unreliable.

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.

Recent interpretability work has identified model-internal handles on post-trained behavior, including refusal directions, assistant/persona axes, and sparse chat-tuning features. These results localize where behaviors can be read out or controlled, often in middle-to-late layers. We ask how earlier computation and the late stack cooperate to turn those differences into next-token margins. To test this, we introduce first-divergence cross-patching: at the first token where pretrained base (PT) and instruction-tuned (IT) checkpoints disagree, we cross each model's earlier-layer state with each model's late stack. The diagnostic separates training recipes: same-base instruction-following descendants show late effects that depend on their own earlier-layer state, while OpenMath2 math-domain SFT and controlled code/biomed CPT controls with verified domain learning do not; for OpenMath2, the late effect is already largely portable from base earlier-layer state. Across five dense families (4B-32B), the IT late stack adds +0.76 logits from PT upstream and +2.44 from IT upstream, giving a +1.68 interaction that is positive in every family. Thus the late stack has a real PT-upstream effect, but its larger effect in the IT checkpoint appears only when it reads its own post-trained upstream state. Sparse features in final MLP layers partially mediate the effect and are driven by upstream patches, supporting a handoff from earlier state to final-layer feature activation to IT-token margin. Forced-token scoring shows that the local token choice can change later exact-answer success. Operationally, paired-checkpoint studies that localize a difference to late layers should test whether it survives under the other checkpoint's upstream state before treating the late stack as self-contained.

Mechanistic interpretability aims to reverse-engineer transformer computations by identifying causal circuits through activation patching. However, scaling these interventions across diverse prompts and task families produces high-dimensional, unstructured datasets that are difficult to compare systematically. We propose a framework that reframes mechanistic analysis as a graph machine-learning problem by representing activation-patching profiles as patch-effect graphs over model components. We introduce three graph-construction methods: direct-influence via causal mediation, partial-correlation, and co-influence and apply graph kernels to analyze the resulting structures. Evaluating this approach on GPT-2 Small using Indirect Object Identification (IOI) and related tasks, we find that patch-effect graphs preserve discriminative structural signals. Specifically, localized edge-slot features provide higher classification accuracy than global graph-shape descriptors. A screened paired-patching validation suggests that CI and PC selected candidate edges correspond to stronger activation-influence effects than random or low-rank candidates. Crucially, by evaluating these representations against rigorous prompt-only and raw patch-effect controls, we make the evidential scope of the benchmark explicit: graph features compress structured patching signal, while raw tensors and surface cues define strong baselines that any circuit-level claim should address. Ultimately, our framework provides a compression and evaluation pipeline for comparing patching-derived structures under controlled baselines, separating robust slice-discriminative evidence from stronger task-general causal-circuit claims.