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RACE: Scalable Statistical Estimation of Functional Consistency in LLM Neurons

Runyu Wang, Bo Liu, Xiaxin Zhang, Yu Han, Jiawei Cao, Xiaoye Zhang, Zhe Zhang, Yifan Yang, Peng Ping

Year: 2026Venue: arXiv preprintArea: cs.AIType: PreprintEmbeddings: 103

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Abstract

Abstract:Discovering stable neuron behavior across entire domains remains a challenge in mechanistic interpretability. Existing methods often rely on instance-level point estimates or computationally expensive procedures, which either obscure population-level variability or limit scalable domain-wide analysis. We present RACE (Residual Alignment for Consistency Estimation), a forward-pass statistical framework that evaluates the domain-wide functional consistency of Transformer neurons. Perturbation experiments demonstrate that RACE achieves superior domain specificity compared to gradient-based point estimates. Meanwhile, token-distribution-level results verify the association between the selected neurons and the target domain. Furthermore, its computational overhead is two orders of magnitude lower than that of gradient-based methods.

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RACE: Scalable Statistical Estimation of Functional Consistency in LLM Neurons Runyu Wang Affiliation: School of Transportation and Civil Engineering, Nantong University Bo Liu Affiliation: Chongqing University of Post and Telecommunications Xiaxin Zhang Affiliation: School of Transportation and Civil Engineering, Nantong University Yu Han Affiliation: School of Transportation and Civil Engineering, Nantong University Jiawei Cao Affiliation: School of Transportation and Civil Engineering, Nantong University Xiaoye Zhang Zhe Zhang Affiliation: China Southern Power Grid Company Limited Meituan2430310032@stmail.ntu.edu.cn s250201066@stu.cqupt.edu.cn 2433320001@stmail.ntu.edu.cn 2433320018@stmail.ntu.edu.cn2233110297@stmail.ntu.edu.cn xiaoyz@whu.edu.cn zhangzhecnjs@gmail.com yangyifan@meituan.com pingpeng@ntu.edu.cn Yifan Yang Affiliation: China Southern Power Grid Company Limited Meituan2430310032@stmail.ntu.edu.cn s250201066@stu.cqupt.edu.cn 2433320001@stmail.ntu.edu.cn 2433320018@stmail.ntu.edu.cn2233110297@stmail.ntu.edu.cn xiaoyz@whu.edu.cn zhangzhecnjs@gmail.com yangyifan@meituan.com pingpeng@ntu.edu.cn Peng Ping †thanks: Corresponding author. Affiliation: School of Transportation and Civil Engineering, Nantong University Abstract Discovering stable neuron behavior across entire domains remains a challenge in mechanistic interpretability. Existing methods often rely on instance-level point estimates or computationally expensive procedures, which either obscure population-level variability or limit scalable domain-wide analysis. We present RACE (Residual Alignment for Consistency Estimation), a forward-pass statistical framework that evaluates the domain-wide functional consistency of Transformer neurons. Perturbation experiments demonstrate that RACE achieves superior domain specificity compared to gradient-based point estimates. Meanwhile, token-distribution-level results verify the association between the selected neurons and the target domain. Furthermore, its computational overhead is three orders of magnitude lower than that of gradient-based methods. 1 Introduction Recent advances in mechanistic interpretability have improved our understanding of Transformer-based Large Language Models (LLMs) (Zhao et al., 2024; Rai et al., 2024). Three dominant approaches have emerged: causal tracing and knowledge localization techniques that identify critical model pathways (Dai et al., 2021; Meng et al., 2022), gradient-based attribution methods that quantify parameter importance (Sundararajan et al., 2017; Achtibat et al., 2024), and sparse autoencoders that extract interpretable neuron activations (Bricken et al., 2023; Shu et al., 2025). While these methods excel at providing instance-level explanations, they face a limitation: many real-world applications, including model capability auditing, domain-specific pruning, and behavioral steering, demand population-level characterizations of model components. Specifically, these tasks require rankings of task-relevant neurons that remain consistent across diverse inputs. Recent work has attempted to bridge this gap through task-neuron mapping strategies: causal gradient variation localizes neurons that selectively affect target tasks (Song et al., 2024), while gradient attribution links task-specific neuron overlap to cross-task generalization (Leng and Xiong, 2025). However, existing pipelines suffer from two constraints: computational inefficiency stems from iterative gradient calculations and intervention operations, creating prohibitive overhead. Meanwhile, statistical oversimplification emerges when aggregating neuron contributions into task-level averages, which obscures sample-to-sample variation patterns. To overcome these limitations, we draw upon two empirical observations: specific LLM capabilities rely on sparse subsets of neurons (Frankle and Carbin, 2019; Frantar and Alistarh, 2023), and these neurons respond uniquely to semantically coherent inputs (Voita et al., 2024; Huang et al., 2025). Since neuron activations directly modulate the residual stream, we hypothesize that during the forward pass, neurons serving consistent domain-specific functions will deposit contribution distributions over this stream that differ from those of non-domain-specific neurons in a statistically significant manner. Accordingly, we introduce RACE (Residual Alignment for Consistency Estimation), a statistical estimation framework that scores every neuron at every layer for functional consistency with respect to a target-domain observation set (§2.1). RACE consists of two stages: (1) decomposing each module’s residual stream update into per-neuron contributions and evaluating their alignment with the module’s output direction via Residual-Direction Alignment (RDA) at forward-pass cost; and (2) distilling noisy per-observation signals into posterior distributions over each neuron’s mean alignment and variance through Bayesian aggregation. We validate on 4B–32B LLMs across code generation, mathematical reasoning, and fine-grained behavioral control. Targeted suppression shows RACE selectively disrupts target capabilities while preserving non-target behaviors. Empirical results confirm RACE’s computational efficiency and demonstrate its superiority over gradient-based baselines. Ablation studies validate the contribution of each RACE component. 2 Method 2.1 Problem Formulation Let ℳ:→M:X be a trained Transformer with layers l=1,…,Ll=1,…,L and modules m∈ATTN,MLPm∈\ATTN,MLP\, denoting attention and multilayer perceptron modules, respectively. We write u=(l,m,j)u=(l,m,j) for a neuron in a fixed layer–module pair. Target-Domain Observation Set. A target domain c is specified by a predicate ϕc:→True,False _c:X→\True,False\ that induces the input population c=∈:ϕc()=TrueD_c=\x : _c(x)=True\. In practice, we approximate this with a finite sample Dc=ii=1N⊆cD_c=\x_i\_i=1^N _c. Because the Transformer is token-position dependent, the auditing protocol evaluates one or more positions per input, yielding the operational observation set Tc T_c =(i,p):i∈Dc,p∈c(i), =\(x_i,p):x_i∈ D_c,\;p _c(x_i)\, (1) nc n_c =|Tc|, =|T_c|, where c()P_c(x) denotes the analyzed token positions. Functional Consistency. The functional consistency score of neuron j with respect to DcD_c reflects how well its contribution distribution satisfies two desiderata. Specifically, let ej(t)∈ℝe_j^(t) denote neuron j’s signed contribution for observation t∈Tct∈ T_c, with the sign defined relative to the target alignment direction. The score evaluates: • Magnitude: t∈Tc[ej(t)]E_t∈ T_c[e_j^(t)] is large and positive relative to other neurons in the same module. • Stability: Vart∈Tc[ej(t)]Var_t∈ T_c[e_j^(t)] is small—the contribution is not driven by a few outlier observations. Neurons that activate strongly on a handful of observations but are silent on most, or whose signed contributions change direction across observations, fail to satisfy these conditions despite potentially high average unsigned magnitude. We denote by ℳ(Ul,m,Tc)∈ℝ|Ul,m|A_M(U_l,m,T_c) ^|U_l,m| the vector of functional consistency scores for all neurons in module m at layer l, given observation set TcT_c. To estimate ℳ(Ul,m,Tc)A_M(U_l,m,T_c) across all layers and modules, RACE frames consistency evaluation as a problem of statistical inference over the evidence set ej(t)t∈Tc\e_j^(t)\_t∈ T_c (hereafter, we drop the layer and module subscripts for readability and simply write j for the audited neuron). Specifically, we model each neuron as possessing latent evidence-distribution parameters θj=(μj,σj2) _j=( _j, _j^2), treating its per-observation evidence as noisy realizations from an underlying distribution: ej(t)∣θj∼P(⋅∣μj,σj2),∀t∈Tce_j^(t) _j P(· _j, _j^2), ∀\,t∈ T_c (2) Functional consistency scoring then becomes a matter of posterior inference: P⁡(θj∣Tc)∝∏t∈TcP⁡(ej(t)∣μj,σj2)⋅P⁡(μj,σj2).P( _j T_c) _t∈ T_cP\! (e_j^(t) _j, _j^2 )· P( _j, _j^2). (3) Crucially, this posterior formulation directly conceptually maps to our two desiderata: the posterior over μj _j captures the functional magnitude and direction, while the posterior over σj2 _j^2 and the epistemic uncertainty in μj _j jointly encode stability. To estimate this posterior in practice, RACE employs a two-stage pipeline. First, RDA computes the module-local evidence ej(t)e_j^(t) from a single forward pass for each observation (§2.2). Second, Bayesian aggregation infers θj _j from this evidence population (§2.3) using a Normal-Inverse-Gamma (NIG) conjugate model. This aggregation yields closed-form posterior mean estimates and variance-calibrated Conservative Alignment Magnitude (CAM) scores for neuron selection (§2.4). Finally, for intervention settings that require disentangling target-specific neurons from broadly active ones, we introduce Reference-Set Filtering (RSF) as a filtration step (§2.5). 2.2 Residual-Direction Alignment RDA provides the per-observation evidence for Bayesian aggregation. It projects each neuron’s weighted output onto the normalized residual update of its host module, thereby avoiding raw-activation proxies (Shrikumar et al., 2017; Meng et al., 2022) and the computational overhead of gradient-based attribution (Sundararajan et al., 2017). 2.2.1 Neuron-Level Residual Decomposition The residual stream r flows from layer to layer, with each module contributing Δmodule _module: out=in+Δmoduler_out=r_in+ _module. For a fixed observation and layer, this update decomposes naturally into per-neuron contributions: ΔATTN _ATTN =O=∑ihiO[:,i] =W_Oh= _ih_iW_O_[:,i] (4) ΔMLP _MLP =down=∑jajdown[:,j] =W_downa= _ja_jW_down_[:,j] (5) where ∈ℝdmodelh ^d_model is the concatenated attention head output and ∈ℝdMLPa ^d_MLP is the MLP intermediate activation before the down projection. Each term is a rank-1 sub-update to the residual stream, scaled by activation magnitude. This sub-update view follows prior work (Geva et al., 2021; Geva et al., 2022). RACE audits these same additive units: an MLP neuron j is the intermediate channel contributing ajdown[:,j]a_jW_down_[:,j], while an attention neuron denotes an output-channel contribution hiO[:,i]h_iW_O_[:,i] (Elhage et al., 2021; Dai et al., 2021; Yu and Ananiadou, 2024). 2.2.2 Module-Local Alignment Evidence RDA uses each module’s output Δ as the evaluation axis. Appendix A discusses why this module-local choice is better matched to the evidence that RACE aims to collect. For an observation t at layer l, we define: ^ATTN d_ATTN =ΔATTN∥ΔATTN∥2+ε = _ATTN _ATTN _2+ (6) ^MLP d_MLP =ΔMLP∥ΔMLP∥2+ε = _MLP _MLP _2+ (7) where ε is a small constant for numerical stability. The unit vector d represents the direction of the module’s additive update to the residual stream. Each neuron’s alignment with this axis is then computed as a signed, activation-modulated score: ATTN _ATTN =OT^ATTN,ξATTNi=hi⋅sATTNi =W_O^T d_ATTN, _ATTN_i=h_i· s_ATTN_i (8) MLP _MLP =downT^MLP,ξMLPj=aj⋅sMLPj =W_down^T d_MLP, _MLP_j=a_j· s_MLP_j (9) The per-observation evidence is ej(t)=ξj∈ℝe_j^(t)= _j . Its sign records whether the neuron’s weighted output is aligned with or opposed to the module’s output direction for that observation. Sign fluctuation across TcT_c is naturally handled by the Bayesian aggregation stage (§2.3), where unstable evidence increases posterior uncertainty. Because the module update is an exact linear sum of per-neuron writes, the signed scores partition the module’s output norm rather than merely approximating importance, which is why alignment to the realized module direction serves as a faithful per-observation measure of functional contribution; Appendix B develops this argument in full. Appendix C details how these observations are logged for experiments. 2.3 Evidence Aggregation We aggregate the noisy per-observation evidence ej(t)t=1n\e_j^(t)\_t=1^n over the selected domain set DcD_c into calibrated consistency estimates via a NIG conjugate model, where n=|Tc|n=|T_c| is the induced token-position evidence count for the fixed domain. 2.3.1 Modeling Functional Consistency For each neuron j in a given module at layer l, we model the signed alignment scores as: ej(t)∼(μj,σj2),t=1,…,ne_j^(t) ( _j, _j^2), t=1,…,n (10) where n counts all token-position observations induced by the N input samples in DcD_c. Treating the Gaussian likelihood as a conjugate working model for the evidence mean and dispersion, we place a conjugate NIG prior (μj,σj2)∼NIG(μ0,λ0,α0,β0)( _j, _j^2) ( _0, _0, _0, _0), where σj2∼Inv-Gamma(α0,β0) _j^2 -Gamma( _0, _0) and μj|σj2∼⁡(μ0,σj2/λ0) _j _j^2 ( _0, _j^2/ _0). This conjugate specification yields analytic posterior updates for each neuron. 2.3.2 Posterior Updates The closed-form update produces two quantities used by RACE scoring. The first is a posterior mean μn,j _n,j that estimates signed alignment strength. The second is a posterior uncertainty term that penalizes noisy or scarce evidence. Sufficient Statistics. Given n alignment scores ej(t)t=1n\e_j^(t)\_t=1^n for neuron j, the posterior update requires only the empirical mean and centred sum of squares: e¯j=1n∑t=1nej(t),SSj=∑t=1n(ej(t)−e¯j)2, e_j= 1n _t=1^ne_j^(t), _j= _t=1^n (e_j^(t)- e_j )^2, (11) from which the empirical standard deviation is σ^j=SSj/(n−1) σ_j= S_j/(n-1). Posterior Update. The NIG conjugacy yields a closed-form posterior with the same functional form: (μj,σj2)|ej(t)∼NIG(μn,j,λn,αn,βn,j)( _j, _j^2) \e_j^(t)\ ( _n,j, _n, _n, _n,j) (12) where the posterior hyperparameters are updated via simple arithmetic: λn _n =λ0+n, = _0+n, (13) αn _n =α0+n2, = _0+ n2, (14) μn,j _n,j =λ0μ0+ne¯jλn, = _0 _0+n e_j _n, (15) βn,j _n,j =β0+SSj2+λ0n(e¯j−μ0)22λn. = _0+ S_j2+ _0n( e_j- _0)^22 _n. (16) Maintaining these sufficient statistics costs O⁡(1)O(1) per neuron per observation. The posterior parameters are then obtained in closed form. Posterior Mean Evidence. The group-level score is the signed posterior mean, directly instantiating Eq. (3): Φc(j)=μn,j=λ0μ0+ne¯jλ0+n _c(j)= _n,j= _0 _0+n e_j _0+n (17) The sign records the dominant alignment direction of the evidence, while values near zero indicate weak or inconsistent average alignment. The next subsection converts this posterior mean and its marginal uncertainty into CAM. 2.4 Uncertainty Quantification Posterior Uncertainty over Mean Evidence. Integrating out σj2 _j^2, the marginal posterior for the mean evidence follows a Student’s t-distribution: μj∣ej(t)∼t2αn(μn,j,σμ,j),σμ,j2=βn,jαnλn _j \e_j^(t)\ t_2 _n\! ( _n,j,\; _μ,j ), _μ,j^2= _n,j _n _n (18) where the second argument is the Student’s t scale parameter. This scale σμ,j _μ,j quantifies uncertainty regarding the average signed contribution, producing wider intervals for scarce or noisy evidence. Variance Sources. The same βn,j _n,j term also determines the posterior expected observation variance, ⁡[σj2∣ej(t)]=βn,j/(αn−1)E[ _j^2 \e_j^(t)\]= _n,j/( _n-1) for αn>1 _n>1. Its data-driven component SSj/2S_j/2 captures cross-observation variability in the alignment evidence. The prior-data term λ0n(e¯j−μ0)2/(2λn) _0n( e_j- _0)^2/(2 _n) regularizes evidence relative to the neutral prior μ0=0 _0=0. Thus bursty, noisy, or sign-fluctuating evidence inflates βn,j _n,j, increasing σμ,j _μ,j and reducing confidence in the neuron’s mean alignment strength. As n grows, λn _n and αn _n increase, shrinking the posterior uncertainty over μj _j when the evidence remains stable. Conservative Alignment Magnitude. We define a scoring criterion that provides a one-sided conservative lower credible bound on each neuron’s positive module-output alignment. The default RACE score is: ρj(γ) _j(γ) =μn,j−t1−γ, 2αnσμ,j, = _n,j-t_1-γ,\,2 _n _μ,j, (19) CAMj(γ) _j(γ) =[μn,j>0]max(0,ρj(γ)) =I[ _n,j>0]\, \! (0, _j(γ) ) where t1−γ, 2αnt_1-γ,\,2 _n is the (1−γ)(1-γ)-quantile of the Student’s t with 2αn2 _n degrees of freedom. Formally, CAMj(γ)CAM_j(γ) is the lower (1−γ)(1-γ)-credible bound on the positive posterior mean evidence under the marginal Student’s t-posterior in Eq. (18). Neurons with large positive mean evidence and small posterior uncertainty receive high CAM scores, whereas neurons with few observations, unstable alignment, or negative posterior mean evidence are excluded or penalized through the confidence radius. Throughout the paper, RACE denotes the default CAM ranking. As an ablation, we also report Neg. CAMj(γ)=[μn,j<0]max(0,−μn,j−t1−γ, 2αnσμ,j)Neg. CAM_j(γ)=I[ _n,j<0] (0,- _n,j-t_1-γ,\,2 _n _μ,j). 2.5 Reference-Set Filtering Because features represented in superposition can make individual neurons polysemantic (Elhage et al., 2022; Scherlis et al., 2022; Templeton et al., 2024), unfiltered target rankings may include neurons that support broad capabilities rather than neurons whose behavior is specific only to the target domain. To disentangle domain-specific behavior from general capabilities during targeted interventions, we introduce RSF as an explicit filtering step. We denote direct RACE scoring on dataset D by DR_D, and reference-set filtered scoring by Dtar∖DrefR_D_tar D_ref. For each layer ℓ and module type, let UℓU be the layer-local neuron universe, Brefℓ=TopMsref⁡(Uℓ)B_ref =Top_M^s_ref(U ) be a reference exclusion set, and Πtarℓ _tar be the full target-domain ranking. RSF selects ΠRSFℓ _RSF =[j∈Πtarℓ:j∉Brefℓ], = [j∈ _tar :j∉ B_ref ], (20) Sspecℓ S_spec =Firstk⁡(ΠRSFℓ). =First_k\! ( _RSF ). Here M is the reference top-M or top-p-percent threshold. Operationally, RSF traverses the target ranking, skips reference-selected neurons, and stops when k neurons are selected; this preserves exactly k neurons per layer whenever |Uℓ∖Brefℓ|≥k|U B_ref |≥ k. Appendix D provides a cross-domain overlap analysis. 3 Experiments We evaluate RACE across multiple domains and models. Our experiments address two main questions: (1) Efficacy: Do RACE-selected neurons cause a disproportionate performance drop on target domains versus non-target domains when suppressed? (2) Ablation: Does Bayesian uncertainty quantification (CAM) outperform deterministic scoring heuristics? 3.1 Experimental Setup Evaluated Models. We evaluate on Qwen3-4B-it (Qwen Team, 2025) (Qwen3-4B-it-2507), OLMo-3.1-32B-it (Team OLMo et al., 2025), and Llama-3.1-8B-it (AI at Meta, 2024) (Appendix E); full model details are in Appendix F. Domain & Benchmark Settings. Each domain is instantiated by a scoring set for RACE-based neuron selection, a same-domain out-of-distribution (OOD) benchmark for validation, and non-target benchmarks for retention. • Code: MBPP+ (Austin et al., 2021; Liu et al., 2023) for scoring and HumanEval+ (Chen et al., 2021; Liu et al., 2023) for OOD validation. • Math: MATH-500 (Hendrycks et al., 2021) for scoring and AMC (EvalScope, 2024) for OOD validation. • Fine-grained Behavioral: We construct PyComp-1K, a set of 1,000 AST-verified Python comprehension-containing statements from bigcode/the-stack (Kocetkov et al., 2022; BigCode, 2026), as a narrow code-behavior scoring domain (Appendix I). Detailed benchmark evaluation setup, including evaluator configurations and scoring definitions, is provided in Appendix F. Evidence logging strategies on corpora are summarized in Appendix C. RACE Hyperparameters. Unless otherwise specified, all experiments use the same RACE hyperparameter settings: μ0=0 _0=0, λ0=1 _0=1, α0=1 _0=1, β0=1 _0=1, and γ=0.05γ=0.05. We further discuss the robustness to prior settings and confidence levels in Appendix K. Baselines & Ablations. Table 1 reports the scoring formula used by each baseline or ablation. We consider two groups of methods: external baselines and internal RACE ablations. For external baselines, GxAct (Kokhlikyan et al., 2020) and AttnLRP (Achtibat et al., 2024) serve as typical gradient attribution methods over the same TcT_c, with the attribution target set to the benchmark answer token. Act. Mean serves as an activation-only control. To isolate the effect of Bayesian uncertainty modeling, we introduce three internal ablations operating on the same evidence ej(t)t=1n\e_j^(t)\_t=1^n as RACE. Emp. Mean removes both the uncertainty penalty and prior regularization, thereby reducing to the raw empirical average. Emp. SNR replaces the Bayesian penalty with a frequentist variance penalty computed via σ^j=SSj/(n−1) σ_j= S_j/(n-1), and Neg. CAM acts as a negative-direction ablation. Regarding the evaluation protocol, instance-level scores are lifted to domain-level neuron rankings by averaging positive neuron-level contributions over the induced token-position observation set TcT_c, matching RACE’s evidence aggregation granularity. All methods adopt identical per-layer/per-module selection budgets and suppression protocols. Under RSF, target and reference rankings are computed using the same method-specific scoring rule. Table 1: Baselines and ablations. Each row gives the neuron score SjS_j used for ranking neuron j, with n=|Tc|n=|T_c|. GxActj(t)GxAct_j^(t) and LRPj(t)LRP_j^(t) denote per-observation attribution scores for the two external baselines. Method Neuron score SjS_j GxAct 1/n∑t∈Tcmax(0,GxActj(t))1/n _t∈ T_c (0,GxAct_j^(t)) AttnLRP 1/n∑t∈Tcmax(0,LRPj(t))1/n _t∈ T_c (0,LRP_j^(t)) Act. Mean 1/n∑t∈Tc|aj(t)|1/n _t∈ T_c|a_j^(t)| Neg. CAM [μn,j<0]max(0,−μn,j−t1−γ,2αnσμ,j)I[ _n,j<0] (0,- _n,j-t_1-γ,2 _n _μ,j) Emp. Mean e¯j e_j Emp. SNR max⁡(0,e¯j)/(σ^j+ε) (0, e_j)/( σ_j+ ) 3.2 Depth-Wise Organization of CAM Scores Figure 1: Layer-wise kernel density estimate (KDE) ridgeline distributions of RACE results. On Qwen3-4B-it, ridges under MBPP+R_MBPP+ and MATH-500R_MATH-500 visualize the KDE of CAM scores for neurons in each module across selected layers. Figure 1 visualizes the CAM RACE score landscape across layers and modules. The ridgelines reveal a progressive rightward shift in positive CAM density with increasing depth, showing that high-confidence evidence concentrates in later layers. Compared to the shallow and deep layers, the middle layers exhibit a sparser distribution of high-scoring neurons, especially in MLP modules. This depth-wise organization is remarkably consistent across MBPP+ and MATH-500, suggesting it reflects broad architectural properties rather than domain-specific quirks. Consistent with prior analyses on Transformers’ functionally stratified computation (Tenney et al., 2019; Jawahar et al., 2019; Geva et al., 2022; Geva et al., 2023), this observation implies that CAM scores are not directly comparable across layers. A globally sorted list would be dominated by late-layer neurons, conflating alignment magnitude with network depth. We therefore adopt a stratified selection protocol in the following interventions: neurons are ranked by CAM within each layer and module, and a fixed budget is allocated to every layer. This per-layer selection rule preserves coverage over the model’s hierarchical computation. 3.3 Effectiveness Validation We validate the intervention relevance of the selected neurons through targeted suppression: during inference, we set their corresponding activation values to zero and measure the resulting performance changes. To jointly quantify target-domain suppression effectiveness and general selectivity in a single scalar, we define the Intervention Specificity Index (ISI): ISI=max⁡0,log⁡(ΔTΔG+δ),δ=0.01.ISI= \0, ( _T _G+δ ) \, δ=0.01. (21) Here ΔT _T and ΔG _G are the average nonnegative relative accuracy drops over the target-domain (including OOD benchmarks) and general (non-target) benchmarks respectively, computed from the per-benchmark drop Δ=max⁡0,(Accbefore−Accafter)/Accbefore = \0,(Acc_before-Acc_after)/Acc_before\. The constant δ is used for smoothing. Higher ISI indicates stronger target-domain suppression with minimal non-target degradation. 3.3.1 Domain-Specific Intervention Figure 2: Suppression under the RSF strategies MBPP+∖WikiText-2R_MBPP+ -2 for the code domain and MATH-500∖WikiText-2R_MATH-500 -2 for the math domain on Qwen3-4B-it. All interventions suppress the top-1%1\% target-selected neurons within the corresponding module at each layer. Radar plots report post-suppression benchmark accuracy as a percentage of the original model baseline, separately for ATTN and MLP interventions. Bars below each radar report the corresponding ISI. We evaluate two intervention strategies. The Vanilla Selection strategy relies solely on the CAM scores computed on the target domain. In contrast, the RSF strategy (§2.5) explicitly controls for broadly shared language capabilities by filtering the candidate set against a general reference corpus. Vanilla Selection. Table 2 reports the Code-domain results; the corresponding Qwen3-4B-it Math-domain results are provided in Appendix Table 21. We find that neurons selected under this setting cause severe overall generation degradation once the suppression budget reaches k≥10k≥10 per layer. We thus report k=5k=5 to observe distinct effects. Table 2: Code domain with MBPP+R_MBPP+: Benchmark Acc. (%) and ISI on Qwen3-4B-it after suppressing top-k=5k=5 neurons per layer. Superscript ♠ marks the target domain D, and ♡ marks the same-domain OOD benchmark. All reported scores are averaged over three runs; subsequent tables follow the same setting unless specified. Module Method MBPP+♠ HumanEval+♡ MATH-500 AMC MMLU-Redux GPQA ISI↑ ATTN GxAct 57.9457.94 49.3949.39 76.3576.35 42.5442.54 77.7977.79 44.9544.95 0.570.57 AttnLRP 47.0947.09 43.9043.90 90.2590.25 77.6177.61 76.7976.79 45.9645.96 2.002.00 Act. Mean 80.6980.69 46.3446.34 93.8993.89 81.3481.34 80.8280.82 41.9241.92 1.541.54 Emp. Mean 78.3178.31 51.8351.83 93.2493.24 84.3384.33 80.4780.47 39.9739.97 1.371.37 Emp. SNR 74.3474.34 50.0050.00 93.1393.13 84.3384.33 81.1481.14 41.4141.41 1.711.71 Neg. CAM 78.8478.84 83.5483.54 93.7793.77 83.5883.58 80.5380.53 41.9241.92 0.000.00 RACE 55.0355.03 45.7345.73 89.0089.00 72.3972.39 78.9178.91 44.9544.95 1.621.62 MLP GxAct 10.3210.32 5.495.49 1.801.80 0.750.75 15.2515.25 15.1515.15 0.040.04 AttnLRP 74.8774.87 83.5483.54 95.0495.04 85.8285.82 81.8481.84 49.4949.49 1.151.15 Act. Mean 81.7581.75 3.663.66 93.4293.42 81.3481.34 81.1881.18 41.4741.47 2.222.22 Emp. Mean 75.1375.13 0.000.00 89.8689.86 87.3187.31 81.3381.33 43.4343.43 2.792.79 Emp. SNR 80.9580.95 86.5986.59 93.6293.62 82.0982.09 81.2381.23 45.9645.96 0.000.00 Neg. CAM 75.1375.13 85.3785.37 91.8191.81 84.3384.33 81.5881.58 45.9645.96 0.540.54 RACE 74.3474.34 0.000.00 92.8092.80 82.8382.83 81.7981.79 44.9544.95 2.912.91 Qwen3-4B-it 82.2882.28 83.5483.54 94.4094.40 87.3187.31 81.3781.37 45.4545.45 — RSF Selection. On Qwen3-4B-it, Figure 2 summarizes the RSF intervention results for both the code and math domains. The corresponding tabular details are reported in Appendix Tables 20 and 22. To demonstrate that RACE’s effectiveness scales to larger models, we further evaluate MATH-500∖WikiText-2R_MATH-500 -2 on OLMo-3.1-32B-it. Table 3: Math domain with MATH-500∖WikiText-2R_MATH-500 -2: Benchmark Acc. (%) and ISI on OLMo-3.1-32B-it after suppressing top-1%1\% and top-5%5\% neurons per layer. MATH-500♠ AMC♡ GPQA MMLU-Redux ISI↑ Module Method 1%1\% 5%5\% 1%1\% 5%5\% 1%1\% 5%5\% 1%1\% 5%5\% 1%1\% 5%5\% ATTN Act. Mean 88.3288.32 84.8584.85 76.1276.12 71.6471.64 38.1038.10 35.9935.99 79.9179.91 73.1873.18 0.000.00 0.000.00 Emp. Mean 85.0185.01 87.8487.84 70.9370.93 68.6668.66 45.0345.03 43.5243.52 81.5481.54 77.1277.12 0.000.00 0.000.00 Emp. SNR 86.4186.41 86.2586.25 71.6471.64 67.9167.91 44.0244.02 44.5444.54 83.4783.47 79.9579.95 0.000.00 0.000.00 Neg. CAM 85.9785.97 86.6286.62 77.6177.61 73.1373.13 47.5647.56 45.5345.53 84.0184.01 84.6784.67 0.000.00 0.010.01 RACE 85.6185.61 84.4384.43 76.8676.86 65.6765.67 47.7447.74 42.4942.49 82.6382.63 80.3580.35 0.000.00 0.000.00 MLP Act. Mean 63.8763.87 14.7114.71 21.6421.64 10.4510.45 40.4040.40 36.3636.36 81.3281.32 75.6575.65 1.471.47 1.501.50 Emp. Mean 73.9573.95 21.4321.43 25.3725.37 13.4313.43 41.4141.41 38.8938.89 80.8180.81 73.4673.46 1.351.35 1.501.50 Emp. SNR 76.8976.89 44.5444.54 23.8823.88 18.6618.66 23.7423.74 21.7221.72 52.8152.81 46.2146.21 0.000.00 0.200.20 Neg. CAM 92.4092.40 97.4897.48 79.8579.85 79.8579.85 52.0252.02 50.5150.51 84.3284.32 85.1685.16 0.000.00 0.000.00 RACE 56.8056.80 6.726.72 14.1814.18 7.467.46 39.8339.83 37.7937.79 82.7282.72 77.7977.79 1.651.65 1.731.73 OLMo-3.1-32B-it 86.40 79.85 48.6 84.70 — Observation: RACE consistently outperforms gradient-based methods, confirming RDA as a reliable gradient-free evidence source. The pronounced OOD degradation further validates RACE scores as an effective target-domain proxy. RSF substantially improves neuron suppression tolerance, with target performance dropping more steeply under higher budgets—indicating successful isolation of domain-specific neurons. Figure 2 confirms this specificity: for both code and math, RACE-selected suppression under RSF hurts target-domain more than non-target performance. However, effects vary by module: MLP interventions cause significant same-domain OOD drops, while attention interventions show weaker domain-specific degradation. This asymmetry is clearer when comparing vanilla selection to RSF: Under vanilla selection, ATTN interventions produce marked target drops even at low budgets, but RSF’s ISI for ATTN remains substantially lower than for MLP, especially on the 32B model. We attribute this to ATTN’s high-scoring neurons being sparse yet broadly reusable: RSF removes these general-purpose neurons via reference filtering, leaving a weaker domain-specific signal. This aligns with known sparsity in ATTN’s OW_O (Michel et al., 2019), suggesting ATTN’s target-domain high-activation neurons also serve broader linguistic functions. 3.3.2 Distributional Verification Table 4: Distributional disruption on Qwen3-4B-it under Dtar∖WikiText-2R_D_tar -2, after suppressing the top-1% neurons within the selected module at each layer. (a) MBPP+∖WikiText-2R_MBPP+ -2 (b) MATH-500∖WikiText-2R_MATH-500 -2 Metric Module MBPP+♠ MATH-500 WikiText-2 MBPP+ MATH-500♠ WikiText-2 ΔPPL _PPL ATTN +6.17+6.17% +6.18+6.18% −2.49-2.49% +4.52+4.52% +7.29+7.29% −2.74-2.74% MLP +77.31+77.31% +23.98+23.98% +2.76+2.76% +25.41+25.41% +129.04+129.04% +2.60+2.60% D¯KL D_KL ATTN 0.100.10 0.080.08 0.030.03 0.090.09 0.090.09 0.030.03 MLP 0.580.58 0.220.22 0.020.02 0.220.22 0.860.86 0.030.03 Beyond coarse accuracy drops, we examine the mechanism of disruption at the token-distribution level using relative perplexity degradation ΔPPL _PPL (Eq. 27) and mean forward Kullback–Leibler divergence D¯KL D_KL (Eq. 28). The distributional metrics (Table 4) reveal a drastic asymmetric disruption, particularly for MLP interventions. Suppressing RACE-selected MLP neurons triggers severe target-domain distribution collapse (ΔPPL _PPL reaching 77.31% on MBPP+ and 129.04% on MATH-500, alongside massive D¯KL D_KL shifts), while leaving the reference corpus (WikiText-2) almost entirely unperturbed. This targeted disruption validates the effectiveness of RACE in decoupling and localizing function-specific neurons without collapsing the model’s fundamental linguistic competence. As a byproduct, ATTN interventions exhibit significantly weaker distributional contrasts, suggesting lower functional sparsity and weaker sensitivity to perturbations compared to the MLP module. Appendix M reports w/o-RSF distributional disruption results. 3.3.3 Fine-Grained Behavioral Steering Table 5: Fine-grained suppression of Python comprehension generation on Qwen3-4B-it. Records: outputs containing ≥1≥ 1 comprehension. Total: aggregate comprehension instances. Pass%: pass@1 functional correctness on the subset of generated solutions that use comprehensions. MBPP+ HumanEval+ Strategy Records Total Pass% Records Total Pass% Qwen3-4B-it 5555 5959 92.7392.73 3636 4545 86.1186.11 MBPP+∖WikiText-2R_MBPP+ -2 4444 5656 97.7397.73 3333 4646 81.8281.82 PyComp-1K∖WikiText-2R_PyComp-1K -2 16−70.9%16_ [rgb]0.5,0,0 -70.9\% 18−69.5%18_ [rgb]0.5,0,0 -69.5\% 87.5087.50 14−61.1%14_ [rgb]0.5,0,0 -61.1\% 19−57.8%19_ [rgb]0.5,0,0 -57.8\% 85.7185.71 To test RACE’s resolution on narrow stylistic behaviors, we target Qwen3-4B-it’s generation of Python comprehensions. These are semantically optional but syntactically idiomatic constructs. Using PyComp-1K as the narrow scoring set, we apply PyComp-1K∖WikiText-2R_PyComp-1K -2 and MBPP+∖WikiText-2R_MBPP+ -2 with a small intervention budget of k=10k=10 neurons per layer. Suppressing PyComp-1K neurons drastically reduces comprehension usage (-70.9% on MBPP+, -61.1% on HumanEval+) while largely preserving functional correctness (Table 5). This specific decline without catastrophic forgetting confirms that RACE-selected neurons steer behavior rather than cause mere disruption. The generation-level effect of this targeted perturbation suggests that RACE can use a deliberately designed dataset to localize a specific model behavior. Appendix J provides a comparison of model outputs before and after perturbation. 3.4 Computational Efficiency Table 6 reports the additional FLOPs required by each scoring method, measured with a profiler on Qwen3-4B-it using 6464 input tokens and 1616 generated tokens. Appendix H gives the profiling protocol. Table 6: Profiler-measured scoring overhead relative to forward inference on Qwen3-4B-it. Method Extra FLOPs Forward-equivalent overhead GxAct 18.7718.77 TFLOPs 144.29×144.29× AttnLRP 18.8218.82 TFLOPs 144.62×144.62× RDA 81.581.5 GFLOPs 0.63×0.63× 3.5 Discussion We analyze how scoring-set size and variance regularization affect RACE, using Qwen3-4B-it with MBPP+ unless stated otherwise. Scoring-set sample size N. The scoring-set sample size N controls how reliably RACE estimates intervention-worthy neurons from the target domain. Figure 3 evaluates this effect through downstream ISI rather than ranking correlation: for each N, we construct MBPP+∖WikiText-2R_MBPP+ -2, suppress the top-1%1\% MLP neurons per layer, and compare RACE with Emp. Mean. The ISI curves are non-monotonic under small and mid-sized scoring sets, but both methods improve as more MBPP+ examples are used and reach their strongest selectivity on the full scoring set (N=378N=378). Notably, under the default prior, the posterior mean is a monotonic rescaling of the empirical mean (μn,j∝e¯j _n,j e_j). The finite-sample difference between RACE and empirical averaging therefore arises from CAM’s uncertainty penalty. Figure 3: Sample efficiency on Qwen3-4B-it under MBPP+∖WikiText-2R_MBPP+ -2: ISI after suppressing top-1%1\% MLP neurons per layer as the scoring-set size N varies, comparing RACE with Emp. Mean. Bayesian variance regularization. The sample-size behavior clarifies that RACE’s low-data advantage comes from calibrated uncertainty rather than a different mean estimator. Sparse module evidence traces can make empirical variance-based ratios over-rank weak nonzero fluctuations, consistent with the poor MLP intervention specificity of Emp. SNR in Table 2. CAM avoids this failure mode by retaining a finite uncertainty margin under the default NIG prior. It becomes prior-insensitive once the induced evidence count n is large. Appendix P gives the short derivation. 4 Related Work Current research in mechanistic interpretability primarily focuses on analyzing individual instances through gradient-based methods, residual stream analysis (Belrose et al., 2023; Ghandeharioun et al., 2024), and circuit discovery (Conmy et al., 2023; Gu et al., 2025). While sparse autoencoders (Bricken et al., 2023; Templeton et al., 2024) offer detailed feature-level analysis, scaling these methods for dataset-level evaluation remains computationally challenging. Recent work has identified specific neural patterns linked to particular skills (Wang et al., 2022; Song et al., 2024; Leng and Xiong, 2025), languages (Tang et al., 2024; Zhang et al., 2024), and factual knowledge (Dai et al., 2021; Meng et al., 2022; Chen et al., 2024). However, existing approaches often struggle to reliably separate meaningful patterns from random noise (Niu et al., 2024), limiting their effectiveness for applications like model pruning (Ma et al., 2023; Sun et al., 2024) or behavior control (Turner et al., 2024; Zou et al., 2023). RACE addresses these limitations with a more robust statistical framework for model auditing. 5 Conclusion To assess functional consistency, RACE recasts neuron analysis as inference over latent alignment distributions. The resulting posterior isolates neurons whose domain contributions are directionally aligned and statistically stable. Targeted perturbations confirm this population is causally load-bearing. Designed with linear computational complexity, the approach remains practical for real-world applications. Limitations We discuss the currently identified limitations in the formulation of RACE. The extraction of alignment evidence relies on linear residual-stream projections, which means the framework may miss complex non-linear synergies among multiple neurons or distributed polysemantic features that require non-linear decoding. Additionally, our experimental results indicate that RACE, alongside other evaluated baselines, demonstrates limited effectiveness in identifying functionally consistent neurons within attention modules. This suggests that attention mechanisms may encode information in a fundamentally different manner than MLPs, making the development of statistical audit strategies specifically tailored to attention an important direction for future work. Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grants 52202496, 52442218, and U2433216; The Key Research and Development Project of Nantong City, China (Special Project for Prospective Technology Innovation, No. GZ2024001); and the Key Laboratory of Target Cognition and Application Technology (2023-CXPT-LC-005). References Achtibat et al. (2024) R. Achtibat, S. M. V. Hatefi, M. Dreyer, A. Jain, T. Wiegand, S. Lapuschkin, and W. Samek Attnlrp: attention-aware layer-wise relevance propagation for transformers. arXiv preprint arXiv:2402.05602. Cited by: §1, §3.1. AI at Meta (2024) AI at Meta The llama 3 herd of models. arXiv preprint arXiv:2407.21783. Cited by: §3.1. Austin et al. (2021) J. Austin, A. Odena, M. Nye, M. Bosma, H. Michalewski, D. Dohan, E. Jiang, C. Cai, M. Terry, Q. Le, and C. Sutton Program synthesis with large language models. arXiv preprint arXiv:2108.07732. Cited by: 1st item. Belrose et al. (2023) N. 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Fredrikson, J. Z. Kolter, and D. Hendrycks Representation engineering: a top-down approach to AI transparency. arXiv preprint arXiv:2310.01405. Cited by: §4. Appendix A Why Use a Module-Local Axis Instead of a Global Logit Axis A natural alternative to RDA is to score every neuron against a single global direction derived from the final next-token prediction. For example, one could replace the module-local axis in Eq. (7) with a normalized unembedding or contrastive logit direction ^t g_t, and score a neuron by ejglobal(t)=aj(t)j⊤^t,e^global_j(t)=a_j^(t)w_j g_t, (22) where jw_j is the neuron’s output vector and aj(t)a_j^(t) is its activation at observation t. This resembles vocabulary-projection analyses such as the logit lens and direct logit attribution. RACE instead uses the module-local residual update Δl,m(t) _l,m^(t) as the evaluation direction because the global-logit alternative answers a different question. It asks whether a neuron already points toward the final prediction, whereas RACE asks whether the neuron consistently contributes to the computation being performed by its own module at its own depth. Depth-local computations are not exchangeable. Prior work repeatedly shows that Transformer layers are functionally stratified rather than interchangeable. In BERT, lower layers capture phrase-level or surface information, middle layers capture syntactic structure, and upper layers capture more semantic information (Jawahar et al., 2019). Tenney et al. similarly find a localized progression resembling the classical NLP pipeline, from POS tagging and parsing through semantic roles and coreference (Tenney et al., 2019). For decoder language models, Geva et al. show that FFN memories differ across depth: lower layers tend to match shallower patterns, while upper layers encode more semantic patterns and more directly induce output-vocabulary distributions (Geva et al., 2021). Follow-up work further views FFN outputs as additive updates to a changing vocabulary distribution (Geva et al., 2022), and factual-recall analyses identify distinct early-MLP enrichment and later information-routing phases (Geva et al., 2023). These results imply that an intermediate neuron can be important because it constructs, routes, erases, or transforms information that is not yet aligned with the final answer token. A final-logit axis therefore imposes a late-stage semantic criterion on layers whose local role may be lexical, syntactic, relational, or preparatory. Raw logit projections are diagnostic tools, not layer-invariant objectives. Lens-style methods are useful precisely because they expose how predictions evolve across depth, but their behavior also reveals the limits of using the final unembedding as a universal coordinate system. The tuned lens was introduced as a refinement of the logit lens because the raw logit lens is often brittle; the tuned lens learns a separate affine translator for each layer and is reported to produce more predictive, reliable, and less biased intermediate predictions (Belrose et al., 2023). Patchscopes reaches a similar conclusion from another direction: many vocabulary-projection methods can be viewed as special cases of a broader representation-inspection framework, and their shortcomings include failures in early layers and limited expressivity (Ghandeharioun et al., 2024). If intermediate states required no layer-specific decoding correction, these layer-wise translators or richer patching contexts would be unnecessary. Thus, raw alignment with the final unembedding is best interpreted as a probe of whether a layer has become linearly readable in vocabulary space, not as a universal measure of neuron importance. Direct logit attribution can mis-rank intermediate components. Direct logit attribution projects intermediate residual components onto final logit directions, but this projection ignores the fact that later layers can overwrite, rotate, or erase earlier residual directions. Janiak et al. provide a concrete adversarial example in GELU-4L: the model uses a memory-management mechanism in which later heads and MLPs remove directions written by earlier heads, and DLA becomes misleading because it does not account for this erasure (Janiak et al., 2024). For RACE, this failure mode is especially relevant. A globally positive logit projection at layer l may be a transient direction that later modules remove, while a globally weak projection may be a necessary intermediate feature that later modules transform into the final prediction. Ranking neurons by final-logit projection would therefore mix three factors: local functional contribution, survival through subsequent computation, and proximity to the output head. Global axes conflate alignment magnitude with depth. Because later representations are closer to the output distribution, a global logit axis tends to favor late-layer neurons whose effects have already been rotated into vocabulary-readable directions. This creates a numerical depth bias: the sorted list of “important” features can be dominated by late-layer units even when earlier and middle layers contain indispensable local computations. Such a ranking confounds two quantities that RACE keeps separate: the strength of a neuron’s contribution to its module’s current update, and the downstream fate of that update after many additional nonlinear transformations. Implication for RACE. The module-local axis does not claim that local alignment alone proves final-output causality. It deliberately produces a stable, layer-appropriate observation variable for population-level statistical inference. Final behavioral relevance is then tested separately through targeted suppression and reference-set filtering. This separation matches the evidence from prior work: intermediate layers perform different computations, raw logit projections require careful layer-specific interpretation, and direct logit attribution can be misleading when later layers erase or transform earlier residual directions. Appendix B Why Residual Alignment Measures Functional Role We first state two formal properties of the RDA evidence that anchor the discussion, then supply the conceptual premises that make them carry functional meaning and close the gap between per-observation geometry and the population-level definition of functional consistency. Two formal properties of the alignment evidence. Writing each neuron’s contribution as j=ajjv_j=a_jw_j gives the exact linear sum Δ=∑j = _jv_j; with the realized output direction ^=Δ/∥Δ∥2 d= / _2 (Eq. 7), the RDA score is ej=⟨j,^⟩e_j= _j, d . Because the decomposition is exact, the scores satisfy a completeness identity—each write splits into an aligned part and an orthogonal part that cancels in aggregate: j _j =ej^+j, =e_j d+q_j, j _j ⟂^, d, (23) ∑j _jq_j =, =0, ∑jej _je_j =∥Δ∥2, = _2, so the scores do not approximate importance but partition it: ∑jej _je_j exhausts the module’s output norm. The score is also the local-sensitivity of that norm to rescaling the neuron: with Δj(α)=Δ+(α−1)j _j(α)= +(α-1)v_j, ∂α∥Δj(α)∥2|α=1=ej, . ∂α _j(α) _2 |_α=1=e_j, (24) i.e., eje_j is the first-order effect of the neuron on the strength of its module’s realized update. The residual stream is the sole functional medium. Modules communicate only through a shared residual stream, so a neuron’s entire downstream effect is mediated by its write jv_j, the natural unit of functional contribution—not the scalar pre-activation aja_j (Elhage et al., 2021; Geva et al., 2021; Geva et al., 2022). This licenses the projection: a neuron’s contribution to its module’s computation is exactly how much of its write survives into the net update, and d is the axis along which that survival is measured. Raw activation |aj||a_j| answers a weaker question—how strongly the channel fired—silent on whether the fired vector supports, opposes, or is orthogonal to the computation the module performs. The cancelling orthogonal component is superposition interference. The completeness decomposition (Eq. 23) has a direct reading under feature superposition (Elhage et al., 2022). When many features are packed into fewer dimensions, individual neuron writes are rarely aligned with the net update; most of their energy lives in the orthogonal component jq_j and is cancelled by opposing writes from other neurons before it reaches the residual stream. Scoring by eje_j deliberately discards this cancelled energy and retains only the component that the module’s combinatorics let through. This is the geometric reason RDA is robust to the polysemantic firing that inflates activation-based scores: a neuron may fire vigorously on a domain input yet contribute nothing to the module’s domain update because its write is spent in directions the rest of the module cancels. The sign of eje_j further separates a neuron that reinforces the realized update from one that counteracts it and must be overpowered for the net Δ to emerge. From per-observation alignment to population functional role. Functional role is not a property of a single observation; it is a population property—a neuron serves a domain function to the extent that it contributes consistently across that domain. The generative model of Eq. 2 is built so that this population property is exactly what the posterior recovers. A neuron whose write reliably reinforces the module’s domain-relevant update produces a stream of positive ej(t)e_j^(t) with small spread, yielding a high posterior mean μn,j _n,j together with low dispersion SSjS_j; both enter CAM, the former through the location and the latter through the credible-bound penalty in Eq. 19. The two functional-consistency desiderata of §2.1—magnitude and stability—are therefore not imposed after the fact but are inherited from the alignment geometry: magnitude is the signed share of executed module work (Eq. 23), and stability is the cross-observation reproducibility of that share. Neurons that participate only on a handful of domain inputs, or whose write reinforces the update on some observations and opposes it on others, see positive and negative shares cancel, inflating SSjS_j and shrinking CAM regardless of their peak activation. Alignment supplies the per-observation evidence; Bayesian aggregation promotes a reproducible positive share into the statistical notion of a role. The match is by construction, since ej(t)e_j^(t) is precisely the quantity whose stable positive population mean defines “this neuron reliably drives the module’s domain computation.” Why the module increment, not the full residual. The completeness and sensitivity identities (Eqs. 23–24) hold because the evaluation axis is the module’s own increment Δ rather than the full residual in+Δr_in+ . The full residual is dominated by the accumulated output of all prior layers, so projecting onto it would credit a neuron for agreement with computation performed upstream rather than for the work this module performs at its own depth. Using the increment keeps ∑jej=∥Δ∥2 _je_j= _2 an identity of the module’s local job, so that alignment measures contribution to what the module itself does—leaving final-output relevance to be tested separately by suppression and reference-set filtering (Appendix A). Local alignment alone does not establish causality for the final model output; it deliberately produces a stable, layer-appropriate observation variable for population-level statistical inference. Appendix C RACE Evidence Computation Strategies RACE supports two primary evidence tracking strategies based on the nature of the evaluation corpus: • Generative Evidence (Autoregressive): For task-solving benchmarks like MBPP+ and MATH-500, we perform full autoregressive generation. Only the output tokens generated by the model are logged as evidence. The prompt/prefill tokens provide necessary conditioning but are not counted in the target-domain evidence, focusing the scoring on the model’s active generation behavior. • Non-Generative Evidence (Teacher Forcing): For continuous text corpora (WikiText-2) or fixed structural behaviors (PyComp-1K), we execute a single forward pass over the provided text sequences using teacher forcing. In this setting, all input tokens within the sequence are tracked as valid evidence. This strategy relies on the intrinsic sequence structure without requiring the model to sequentially produce new tokens. Under the generative evidence strategy, the Qwen3-4B-it auditing runs induce token-position evidence counts of n=705,426n=705,426 for MATH-500 and n=98,560n=98,560 for MBPP+. Appendix D Cross-Domain Consistency of RACE-Selected Neurons This appendix analyzes whether the top-k neurons identified by RACE are shared across distinct task domains. The goal is to distinguish broadly shared selected neurons from neurons selected primarily for a single target distribution. We evaluate three domains: mathematical reasoning, code generation, and general language modeling, as summarized in Table 7. Table 7: Domains used for cross-domain consistency analysis. Domain Type Description Math-500 Reasoning Mathematical problem solving MBPP+ Code Python programming tasks WikiText-2 Language General text modeling Analyzed Modules. For each configuration, we analyze both ATTN and MLP. For each layer and module, RACE ranks neurons by CAM. We then select the top-k neurons under thresholds k∈1%,5%,10%k∈\1\%,5\%,10\%\. Metrics. Let Sd(l,m,k)S_d^(l,m,k) denote the top-k neuron set selected for domain d, layer l, and module m. For each pair of domains d1,d2d_1,d_2, we compute the layer-wise Jaccard similarity: J⁡(d1,d2)=|Sd1(l,m,k)∩Sd2(l,m,k)||Sd1(l,m,k)∪Sd2(l,m,k)|.J(d_1,d_2)= |S_d_1^(l,m,k)∩ S_d_2^(l,m,k)||S_d_1^(l,m,k)∪ S_d_2^(l,m,k)|. (25) We additionally compute an all-domain overlap score that requires a neuron to be shared by all three domains: Jall=|Smath500∩Smbpp_plus∩Swikitext2||Smath500∪Smbpp_plus∪Swikitext2|.J_all= |S_math500∩ S_mbpp\_plus∩ S_wikitext2||S_math500∪ S_mbpp\_plus∪ S_wikitext2|. (26) All reported values are averaged over layers, with standard deviations computed across layers. Main Finding. Attention neurons are substantially more domain-general than MLP neurons. Under CAM with a Top-1%1\% threshold, attention neurons obtain an all-domain Jaccard of 0.2640.264 (std=0.099std=0.099), meaning that roughly 26%26\% of the selected attention neurons are shared across all three domains. In contrast, MLP neurons obtain an all-domain Jaccard of only 0.0850.085 (std=0.052std=0.052), meaning that only about 8.5%8.5\% of the selected MLP neurons are shared. This gives a 3.1×3.1× gap between attention and MLP modules. Table 8: All-domain Jaccard similarity of top-k neurons under MATH-500R_MATH-500, MBPP+R_MBPP+, and WikiText-2R_WikiText-2. Attention neurons are consistently more shared across domains than MLP neurons. Configuration Attention MLP Ratio CAM, Top-1%1\% 0.2640.264 0.0850.085 3.1×3.1× CAM, Top-5%5\% 0.2490.249 0.1100.110 2.3×2.3× CAM, Top-10%10\% 0.2580.258 0.1370.137 1.9×1.9× Table 8 shows that the gap is robust across both scoring metrics and all top-k thresholds. As k increases, the MLP all-domain overlap rises moderately, but attention remains consistently higher. This suggests that attention output channels contain a larger population of reusable cross-domain routing or integration neurons, whereas MLP down-projection neurons are less shared across tasks and show narrower response patterns under RACE scoring. Figure 4: Layer-wise cross-domain overlap of top-k neurons (CAM score, top-1%1\%). Heatmaps show Jaccard similarity between domain pairs across transformer layers for the selected module, attribution metric, and top-k threshold. Each cell reports the overlap between the two domains’ top-ranked neuron sets in that layer, with higher values indicating greater cross-domain consistency in the identified important neurons. Top: attention neurons exhibit substantially higher cross-domain overlap, consistent with the all-domain Jaccard of 0.2640.264. Bottom: MLP neurons show markedly lower inter-domain agreement, reflecting more domain-specific response patterns (all-domain Jaccard 0.0850.085). Appendix E Math Domain Intervention with Reference-Set Filtering on Llama-3.1-8B-it This appendix reports additional k=50k=50 ATTN and MLP suppression results for MATH-500∖WikiText-2R_MATH-500 -2 on Llama-3.1-8B-it in Table 9. Neurons are scored on MATH-500, and the intervention set removes neurons that overlap with a reference RACE result computed on WikiText-2 before suppression. Table 9: Math domain with MATH-500∖WikiText-2R_MATH-500 -2: accuracy on Llama-3.1-8B-it after suppressing top-k=50k=50 neurons per layer. Module Method MATH-500♠ ↓ AMC♡ ↓ GPQA MMLU-Redux ARC ISI↑ ATTN Neg. CAM 47.0047.00 15.6715.67 28.7928.79 71.3571.35 86.1086.10 2.022.02 RACE 27.0027.00 6.726.72 22.7322.73 67.3567.35 86.1686.16 1.651.65 MLP Neg. CAM 51.0051.00 23.1423.14 31.3131.31 72.6172.61 86.2286.22 0.980.98 RACE 8.808.80 2.242.24 24.7524.75 69.8969.89 85.7185.71 2.362.36 Llama-3.1-8B-it 50.4050.40 24.6324.63 29.8029.80 72.9172.91 86.1686.16 — Appendix F Model, Generation, and Evaluation Details This appendix summarizes the model configurations, decoding parameters, and benchmark evaluation protocol used in the experiments. Benchmark evaluation is conducted with EvalScope 1.5.0 using vLLM 0.15.1 as the inference backend. Unless otherwise specified, benchmark scores follow the official metric implementation exposed by EvalScope. We report the resulting benchmark score as Domain Accuracy (DA) in the main tables. Model generation settings, architectural dimensions, benchmark roles, EvalScope benchmark metadata, and corpus statistics are reported in Tables 10, 11, 12, 13, 14, and 15. Table 10: Model and decoding settings used for benchmark evaluation. Parameters not listed are left at the evaluator or backend default; “—” indicates that the parameter is unset. Model Scale Temperature Top-p Top-k Qwen3-4B-it-2507 4B 0.7 0.8 20 Llama-3.1-8B-it 8B 0.6 0.9 — OLMo-3.1-32B-it 32B 0.6 0.95 — Table 11: Architectural dimensions of all evaluated models. Model Layers ATTN neurons / layer MLP neurons / layer Qwen3-4B-it-2507 36 2560 9728 Llama-3.1-8B-it 32 4096 14336 OLMo-3.1-32B-it 64 5120 27648 Table 12: Benchmark evaluation setup. Benchmark Role Reported score Evaluator MBPP+ Code scoring / in-distribution validation EvalScope benchmark score EvalScope + vLLM HumanEval+ Code out-of-distribution validation EvalScope benchmark score EvalScope + vLLM MATH-500 Math scoring / in-distribution validation EvalScope benchmark score EvalScope + vLLM AMC Math out-of-distribution validation EvalScope benchmark score EvalScope + vLLM ARC Non-target retention / reasoning control EvalScope benchmark score EvalScope + vLLM MMLU-Redux General-knowledge retention EvalScope benchmark score EvalScope + vLLM GPQA General-knowledge retention EvalScope benchmark score EvalScope + vLLM Table 13: EvalScope benchmark metadata for all benchmark datasets used in the paper, extracted from the corresponding EvalScope benchmark cards (EvalScope, 2026e; EvalScope, 2026c; EvalScope, 2026d; EvalScope, 2024; EvalScope, 2026a; EvalScope, 2026f; EvalScope, 2026b). Benchmark EvalScope key Task type Split Metric / aggregation MBPP+ mbpp_plus Python code generation test acc; mean_and_pass_at_k HumanEval+ humaneval_plus Python code generation test acc; mean_and_pass_at_k MATH-500 math_500 Mathematical problem solving test acc AMC amc Competition math multiple choice N/A acc ARC arc Multiple-choice science QA test acc MMLU-Redux mmlu_redux Multiple-choice knowledge QA test acc GPQA gpqa_diamond Expert-level science MCQ train acc Table 14: Dataset statistics for all EvalScope benchmarks used in the paper. Prompt lengths are measured in characters as reported by the EvalScope benchmark cards (EvalScope, 2026e; EvalScope, 2026c; EvalScope, 2026d; EvalScope, 2024; EvalScope, 2026a; EvalScope, 2026f; EvalScope, 2026b). Benchmark Samples Mean prompt length Min / max prompt length Subset / coverage notes MBPP+ 378 375.53 222 / 2801 Python programming tasks with expanded tests HumanEval+ 164 609.57 274 / 1519 Original HumanEval problems with enhanced tests MATH-500 500 266.89 91 / 1804 Levels 1–5: 43 / 90 / 105 / 128 / 134 examples AMC 134 324.58 98 / 1218 AMC22 / AMC23 / AMC24: 43 / 46 / 45 examples ARC 3548 424.43 253 / 1157 ARC-Easy / ARC-Challenge: 2376 / 1172 examples MMLU-Redux 5700 600.81 255 / 5082 57 subjects; 100 examples per subject GPQA 198 841.15 340 / 5845 GPQA-Diamond expert science subset Table 15: Non-EvalScope corpora and locally constructed datasets appearing in the paper. WikiText-2 provides DrefD_ref for Dtar∖WikiText-2R_D_tar -2, The Stack (Kocetkov et al., 2022; BigCode, 2026) is the source corpus for constructing PyComp-1K, and PyComp-1K provides DtarD_tar in the fine-grained PyComp-1K∖WikiText-2R_PyComp-1K -2 intervention. Dataset / corpus Role in paper Size / statistics Construction or source notes WikiText-2 (Merity et al., 2016) General-language RSF reference 2,088,628 train tokens; 217,646 validation tokens; 245,569 test tokens Wikipedia Good/Featured articles; vocabulary of 33,278 words The Stack (Kocetkov et al., 2022; BigCode, 2026) Source corpus for PyComp-1K construction 6.4 TB permissively licensed source code across 358 programming languages Only Python files are streamed for PyComp-1K extraction before AST filtering PyComp-1K Narrow code-behavior scoring corpus 1000 AST-verified Python statements Extracted from bigcode/the-stack (Kocetkov et al., 2022; BigCode, 2026); targets ListComp, SetComp, DictComp, and GeneratorExp nodes Appendix G Computational Architecture This appendix reports the compute environments used for the main experiments. Experiments on Qwen3-4B-it were executed on the 8-GPU NVIDIA RTX 4090 machine. Experiments on Llama-3.1-8B-it and OLMo-3.1-32B-it were executed on the 8-GPU NVIDIA A100 80GB machine. The full hardware and system configuration is given in Table 16. Table 16: Computational architecture used in the experiments. CPU, memory, operating system, and GPU information are reported from the execution environments. Environment System CPU Memory GPU / experiments RTX 4090 server Ubuntu, Linux 6.8.0-107-generic Intel Xeon Gold 6138 125GiB RAM 8× NVIDIA GeForce RTX 4090; Qwen3-4B-it A100 server Linux 4.18.0-147 AMD EPYC 7713 368GiB RAM 8× NVIDIA A100 80GB; Llama-3.1-8B-it and OLMo-3.1-32B-it Appendix H Profiler-Based Scoring Cost Measurement We measure the computational-efficiency numbers in Table 6 with torch.profiler(with_flops=True) on Qwen3-4B-it. The forward reference is the profiled cost of the same 6464-input/1616-output window, including the prefill-to-generation increment and the selected lm_head projections. This reference costs 130.100130.100 GFLOPs. Protocol. All methods use the same WikiText-2 token window, target positions, target token ids, and target modules. We use one lookahead token to define the next-token target for the 16th generated position, but do not increase the model input length. For GxAct and AttnLRP, we batch all target layers together by default, avoiding artificial repetition of the same forward/backward computation once per layer. The profiler records only the extra scoring computation; CAM, NIG updates, HDF5 writes, and CPU aggregation are excluded. We use eager attention so that PyTorch’s FLOP profiler can observe the attention matrix multiplications instead of hiding them inside fused kernels. Table 17: Profiler-measured FLOPs for Qwen3-4B-it under the 6464-input/1616-output protocol. Measurement Profiler FLOPs Relative to forward Forward reference 130.100130.100 GFLOPs 100.000%100.000\% RDA extra 81.54581.545 GFLOPs 62.679%62.679\% GxAct extra 18.77218.772 TFLOPs 14428.812%14428.812\% AttnLRP extra 18.81518.815 TFLOPs 14461.987%14461.987\% Appendix I PyComp-1K Dataset To evaluate model behaviors on specific Python structural patterns, we utilize the PyComp-1K dataset. This dataset contains 1,000 Python statements extracted from the Python subset of bigcode/the-stack (Kocetkov et al., 2022; BigCode, 2026). Each example consists of the nearest enclosing Python statement around one or more comprehension expressions. I.1 Construction Process The dataset construction process streamed Python files from The Stack (Kocetkov et al., 2022; BigCode, 2026) and parsed each source file using Python’s ast module. The extraction logic specifically targeted four types of AST comprehension nodes: ListComp, SetComp, DictComp, and GeneratorExp. For each unique comprehension span identified, the nearest enclosing statement was extracted and saved. Rows that could not be parsed as valid Python source were skipped. I.2 Dataset Statistics The 1,000 extracted statements exhibit the following distribution of comprehension types (note that a single statement may contain multiple comprehensions): Table 18: Distribution of comprehension types in the Python Comprehension Statements 1K dataset. Comprehension Type Count List Comprehension (ListComp) 754 Generator Expression (GeneratorExp) 181 Dictionary Comprehension (DictComp) 76 Set Comprehension (SetComp) 9 Each row in the dataset provides the extracted statement, the corresponding line numbers in the original source, the set of AST comprehension types present, and the source metadata from The Stack (Kocetkov et al., 2022; BigCode, 2026) (e.g., commit hash, repository name, and file path). Appendix J Qualitative Outputs under the PyComp-1K Perturbation This section reports five representative Python-generation samples under the PyComp-1K perturbation setting. For each pair, the first block shows a correct unmodified model output, while the second block shows an incorrect output after perturbing RACE-selected PyComp-1K neurons. Example 1: square individual list elements. Original model (correct). ⬇ ‘python def square_nums(nums): """ Takes a list of numbers and returns a new list containing the square of each element. Args: nums (list): A list of integers or floats Returns: list: A list where each element is the square of the corresponding element in the input list Examples: >>> square_nums([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] >>> square_nums([10, 20, 30]) [100, 400, 900] >>> square_nums([12, 15]) [144, 225] """ return [num ** 2 for num in nums] ‘ Perturbed model (incorrect). ⬇ ‘python def square_nums(nums): """ Takes a list of numbers and returns a new list with the square of each element. Args: nums (list): A list of integers or floats Returns: list: A list containing the square of each element from the input list """ return [num**2] ‘ Example 2: extract strings of a specified length. Original model (correct). ⬇ ‘python def extract_string(string_list, size): """ Extract strings from a list that have a specified length. Args: string_list: List of strings to filter size: The length to check for (strings of this exact length will be returned) Returns: List of strings that have the specified length """ return [s for s in string_list if len(s) == size] ‘ Perturbed model (incorrect). ⬇ ‘python def extract_string(string_list, size): """ Extract strings from a list that have the specified length. Args: string_list: List of strings to filter size: The length to filter by (only strings of this exact length are returned) Returns: List of strings that have the exact specified length """ return [string for len(string) == size] ‘ Example 3: sort a dictionary by value. Original model (correct). ⬇ ‘python def sort_counter(d): """ Sort a dictionary by value in descending order and return a list of tuples. Args: d (dict): A dictionary with keys and values to be sorted Returns: list: A list of tuples (key, value) sorted by value in descending order """ return sorted(d.items(), key=lambda item: item[1], reverse=True) ‘ Perturbed model (incorrect). ⬇ ‘python def sort_counter(dict_input): """ Sort a dictionary of key-value pairs by value in descending order. Args: dict_input (dict): A dictionary where each key-value pair is a tuple or key-value pair Returns: list: A list of tuples sorted by value in descending order """ # Convert the dictionary to a list of tuples and sort by value (second element) in descending order return sorted(dict_input, key=lambda x: x[1], reverse=True) ‘ Example 4: sorted union of two tuples. Original model (correct). ⬇ ‘python def union_elements(tuple1, tuple2): """ Find the union of elements from two tuples and return them in sorted order. Args: tuple1: First tuple of elements tuple2: Second tuple of elements Returns: A tuple containing the union of elements from both tuples, sorted in ascending order """ # Convert tuples to sets to find union (automatically removes duplicates) union_set = set(tuple1) | set(tuple2) # Convert back to sorted tuple sorted_union = tuple(sorted(union_set)) return sorted_union ‘ Perturbed model (incorrect). ⬇ ‘python def union_elements(tuple1, tuple2): """ Find the union of elements from two tuples and return them in sorted order. Args: tuple1: First tuple of elements tuple2: Second tuple of elements Returns: A tuple containing the union of elements from both input tuples, sorted in ascending order """ # Convert tuples to sets to find union (remove duplicates) set1 = set(tuple1) set2 = set(tuple2 # Find union of both sets union_set = set1.union(set2) # Convert back to sorted tuple sorted_union = tuple(sorted(union_set)) return sorted_union ‘ Example 5: retrieve the k-th array element. Original model (correct). ⬇ ‘python def kth_element(arr, k): """ Find the kth element in the given array using 1-based indexing. Args: arr: List of integers k: 1-based index of the element to find Returns: The kth element in the array Raises: IndexError: If k is out of bounds (k > length of array) """ if k < 1: raise IndexError("k must be a positive integer") if k > len(arr): raise IndexError(f"k (k) is greater than array length (len(arr))") return arr[k - 1] # Convert 1-based index to 0-based index ‘ Perturbed model (incorrect). ⬇ ‘python def kth_element(arr, k): """ Find the kth element in the given array using 1-based indexing. Args: arr: List of numbers k: 1-based index of the element to find Returns: The kth element in the array Raises: IndexError: If k is out of bounds (k < 1 or k > len(arr)) """ # Validate input if k < 1: raise IndexError("k must be a positive integer (1-based indexing)") if k > len(arr): raise IndexError(f"k (k) is greater than array length (len(arr)") # Return the kth element (k-1 index in 0-based indexing) return arr[k - 1] ‘ Appendix K Robustness to Prior Settings and Confidence Levels When the sample size is sufficiently large (n≫λ0n _0), the posterior distribution is dominated by empirical statistics, rendering the choice of prior parameters largely inconsequential. Empirically, across a wide range of prior configurations (λ0,β0∈[10−3,103] _0, _0∈[10^-3,10^3], α0∈[0.5,50] _0∈[0.5,50]), the top-1%1\% selected neurons remain highly stable (Jaccard similarity ≥0.98≥ 0.98). Regarding the CAM confidence level γ, it primarily acts as an absolute verification threshold rather than altering the relative ranking of the top candidates. Consequently, the sets of top-k neurons selected under fixed budgets remain largely identical across different γ values. In practice, adopting a more stringent confidence level (e.g., tightening γ from 0.050.05 to 0.0010.001) effectively filters out marginal candidates, shrinking the pool of valid positive-CAM neurons from 81.75%81.75\% to 62.24%62.24\% (at N=10N=10), without displacing the most prominent neurons. Appendix L Distributional Disruption Metrics The main paper evaluates token-distribution-level disruption using relative perplexity degradation ΔPPL _PPL and mean forward Kullback–Leibler divergence D¯KL D_KL. ΔPPL _PPL is reported in percentage form: ΔPPL=PPLsup−PPLbasePPLbase. _PPL= PPL_sup-PPL_basePPL_base. (27) This aggregates the increase in per-token surprisal and quantifies the overall deterioration in predictive quality. D¯KL D_KL is computed as D¯KL D_KL =1T∑t=1TDKL(Pbase(⋅∣<t)∥ = 1T _t=1^TD_KL\! (P_base(· _<t)\,\| (28) Psup(⋅∣<t)). P_sup(· _<t) ). This measures the mean distributional shift per token position and can detect behavioral changes even when aggregate likelihood is largely preserved. Appendix M Without RSF (w/o-RSF) Distributional Disruption on Qwen3-4B-it This appendix reports distributional disruption results w/o-RSF for Qwen3-4B-it. All checkpoints use top-k=5k=5 RACE-selected neurons per layer with the suppression operation. They are evaluated by direct model comparison against the unmodified model on the first 100 samples per dataset in order. The resulting ΔPPL _PPL and D¯KL D_KL measurements are summarized in Table 19. Table 19: w/o-RSF domain perplexity degradation (ΔPPL _PPL, Eq. 27) and mean per-token forward KL divergence (D¯KL D_KL, Eq. 28) on Qwen3-4B-it after suppressing top-k=5k=5 RACE-selected neurons per layer. Columns report interventions under MBPP+R_MBPP+ and MATH-500R_MATH-500; all evaluations use direct model comparison against the unmodified model. (a) MBPP+R_MBPP+ (b) MATH-500R_MATH-500 Metric Module MBPP+♠ MATH-500 WikiText-2 MBPP+ MATH-500♠ WikiText-2 ΔPPL _PPL ATTN +30.16+30.16% +25.97+25.97% +5.82+5.82% +28.43+28.43% +27.50+27.50% +4.84+4.84% MLP +263.82+263.82% +170.24+170.24% +740.63+740.63% +16.86+16.86% +34.57+34.57% +58.56+58.56% D¯KL D_KL ATTN 0.29130.2913 0.27930.2793 0.20360.2036 0.27930.2793 0.28230.2823 0.20270.2027 MLP 1.40311.4031 1.12461.1246 2.45492.4549 0.17210.1721 0.32800.3280 0.34760.3476 Appendix N Code Domain Intervention with Reference-Set Filtering on Qwen3-4B-it This appendix reports the code-domain Qwen3-4B-it results summarized by the radar plot in Figure 2. Neurons are selected with MBPP+∖WikiText-2R_MBPP+ -2, and Table 20 provides the full benchmark and ISI values. Table 20: Code domain with MBPP+∖WikiText-2R_MBPP+ -2: accuracy (%) and ISI on Qwen3-4B-it after suppressing top-1%1\% ATTN or MLP target-selected neurons per layer. Module Method MBPP+♠ ↓ HumanEval+♡ ↓ MATH-500 AMC MMLU-Redux GPQA ISI↑ ATTN GxAct 8.738.73 7.937.93 17.2017.20 2.242.24 47.6747.67 21.7221.72 0.260.26 AttnLRP 0.000.00 0.000.00 2.602.60 0.750.75 6.076.07 5.055.05 0.050.05 Act. Mean 81.7581.75 86.5986.59 90.0390.03 75.3775.37 80.6780.67 46.9746.97 0.000.00 Emp. Mean 66.4066.40 78.6678.66 88.3488.34 63.4363.43 78.5478.54 40.4040.40 0.000.00 Emp. SNR 68.5268.52 84.1584.15 93.4493.44 76.8676.86 79.4979.49 40.4040.40 0.090.09 Neg. CAM 81.7581.75 82.9382.93 92.6192.61 83.5883.58 81.5381.53 45.4545.45 0.000.00 RACE 65.8765.87 80.4980.49 90.3090.30 68.6668.66 79.0779.07 44.4144.41 0.290.29 MLP GxAct 23.8123.81 19.5119.51 79.2379.23 46.2746.27 78.4978.49 40.4040.40 1.291.29 AttnLRP 0.000.00 3.053.05 61.3761.37 35.0735.07 76.3976.39 31.8231.82 1.071.07 Act. Mean 11.3811.38 12.2112.21 88.7388.73 72.0972.09 82.6182.61 40.9840.98 2.222.22 Emp. Mean 5.825.82 2.442.44 86.4486.44 63.4363.43 82.3082.30 55.0555.05 2.262.26 Emp. SNR 83.3383.33 84.7684.76 93.8393.83 69.4069.40 78.9878.98 38.8938.89 0.000.00 Neg. CAM 68.2568.25 84.7684.76 93.2393.23 81.3481.34 81.7281.72 48.9948.99 1.041.04 RACE 3.173.17 1.221.22 85.4185.41 75.1175.11 82.3582.35 52.5352.53 2.652.65 Qwen3-4B-it 82.2882.28 83.5483.54 94.4094.40 87.3187.31 81.3781.37 45.4545.45 — Appendix O Math Domain Intervention on Qwen3-4B-it This appendix reports additional math-domain suppression results on Qwen3-4B-it in Tables 21 and 22. Table 21: Math domain with MATH-500R_MATH-500: accuracy (%) on Qwen3-4B-it after suppressing top-k=5k=5 neurons per layer. Module Method MATH-500♠ AMC♡ MBPP+ HumanEval+ MMLU-Redux GPQA ISI↑ ATTN GxAct 80.0080.00 47.0147.01 59.7959.79 49.3949.39 78.3978.39 37.3737.37 0.270.27 AttnLRP 89.6089.60 73.8873.88 48.1548.15 47.5647.56 76.9876.98 43.9443.94 0.000.00 Act. Mean 92.8092.80 86.5786.57 70.1170.11 63.4163.41 79.5479.54 45.9645.96 0.000.00 Emp. Mean 93.6093.60 86.5686.56 72.2272.22 56.7156.71 79.1479.14 44.9544.95 0.000.00 Emp. SNR 94.2094.20 84.3384.33 76.7276.72 49.3949.39 80.0780.07 41.4141.41 0.000.00 Neg. CAM 94.2094.20 85.8285.82 73.2873.28 83.5483.54 79.5679.56 43.9443.94 0.000.00 RACE 94.2094.20 84.3384.33 73.2873.28 51.8351.83 80.1880.18 42.4242.42 0.000.00 MLP GxAct 89.6089.60 70.1570.15 80.4280.42 71.3471.34 80.0780.07 41.4141.41 0.460.46 AttnLRP 90.4090.40 80.6080.60 83.6083.60 83.5483.54 81.4281.42 43.4343.43 1.041.04 Act. Mean 92.0092.00 80.6080.60 78.5778.57 71.9571.95 81.1981.19 53.0353.03 0.000.00 Emp. Mean 57.4057.40 35.0835.08 78.3178.31 85.3785.37 80.3080.30 42.4242.42 2.472.47 Emp. SNR 93.8093.80 80.6080.60 71.9671.96 83.5483.54 80.3080.30 42.9342.93 0.000.00 Neg. CAM 56.0056.00 35.0735.07 79.6379.63 85.3785.37 79.9879.98 40.4040.40 2.312.31 RACE 63.6063.60 29.1129.11 79.8979.89 85.9885.98 80.1480.14 44.4444.44 2.932.93 Qwen3-4B-it 94.4094.40 87.3187.31 82.2882.28 83.5483.54 81.3781.37 45.4545.45 — Table 22: Math domain with MATH-500∖WikiText-2R_MATH-500 -2: accuracy (%) on Qwen3-4B-it after suppressing top-1%1\% ATTN or MLP target-selected neurons per layer. Module Method MATH-500♠ AMC♡ MBPP+ HumanEval+ MMLU-Redux GPQA ISI↑ ATTN GxAct 4.004.00 2.242.24 1.061.06 0.000.00 16.9816.98 7.077.07 0.050.05 AttnLRP 1.001.00 1.491.49 0.000.00 0.000.00 4.334.33 2.532.53 0.000.00 Act. Mean 73.2073.20 41.6041.60 78.0478.04 78.6678.66 75.2375.23 36.8736.87 1.281.28 Emp. Mean 92.2092.20 73.8873.88 69.3169.31 83.5483.54 78.3378.33 40.4040.40 0.020.02 Emp. SNR 91.8091.80 78.3678.36 82.5482.54 84.7684.76 79.6179.61 43.9443.94 1.011.01 Neg. CAM 91.2091.20 82.8482.84 79.8979.89 79.8879.88 81.2681.26 42.9342.93 0.000.00 RACE 90.6090.60 72.3972.39 71.4371.43 84.7684.76 78.5478.54 40.9140.91 0.320.32 MLP GxAct 30.8030.80 9.709.70 69.0269.02 51.3951.39 77.1977.19 41.4141.41 1.461.46 AttnLRP 46.0046.00 31.3431.34 53.7053.70 33.5433.54 79.1679.16 43.4343.43 0.780.78 Act. Mean 58.4058.40 47.7647.76 65.6165.61 48.1748.17 79.7979.79 40.9140.91 0.750.75 Emp. Mean 27.2027.20 12.6912.69 61.6461.64 55.4955.49 77.7477.74 39.3939.39 1.361.36 Emp. SNR 95.0095.00 86.5786.57 74.0774.07 82.9382.93 80.7080.70 39.9039.90 0.000.00 Neg. CAM 95.6095.60 85.0885.08 80.6980.69 85.9885.98 81.4481.44 44.4444.44 0.000.00 RACE 29.0029.00 11.9411.94 70.2070.20 67.4967.49 77.7977.79 40.4040.40 1.761.76 Qwen3-4B-it 94.4094.40 87.3187.31 82.2882.28 83.5483.54 81.3781.37 45.4545.45 — Appendix P Bayesian Variance Regularization in Low-Data Regimes This appendix expands the variance-regularization argument summarized in §3.5. Let N=|Dc|N=|D_c| denote the number of input samples. Let n=|Tc|n=|T_c| denote the induced token-position evidence count for a fixed scoring set. Under the default RACE prior μ0=0 _0=0, λ0=1 _0=1, α0=1 _0=1, and β0=1 _0=1, the NIG update in Eq. (16) becomes βn,j=1+SSj2+ne¯j22(1+n)≥1. _n,j=1+ S_j2+ n e_j^22(1+n)≥ 1. (29) This lower bound holds even when the empirical dispersion collapses to SSj=0S_j=0. Since αn=1+n/2 _n=1+n/2 and λn=1+n _n=1+n, the posterior scale for mean evidence satisfies σμ,j=βn,jαnλn≥1(1+n/2)(1+n)>0. _μ,j= _n,j _n _n≥ 1(1+n/2)(1+n)>0. (30) Thus CAM retains a finite conservative margin against weak or sparsity-induced evidence at finite n. This prevents near-zero empirical variance from eliminating the uncertainty penalty. As n grows, this lower bound decays, matching the main-text observation that the prior becomes negligible in large-evidence regimes.

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