{"id":"f4bd7e65-9141-47cf-8906-730055499019","arxiv_id":"2505.12268","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"K-MSHC reports that Gemma-9B's grammar, arithmetic-verification, and word-problem abilities rely on largely separate, task-specific attention-head circuits with partial overlap at low-importance heads.","lead":"Researchers introduce a search algorithm that tries to find the smallest sets of attention heads in Gemma-9B that can independently sustain the model's score on grammar, arithmetic, and word-problem classification. A smart generalist might read this to see whether language models build separate circuits for language and math, or reuse the same parts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claims rest on an unvalidated linear-separability proxy; circuits may be artifacts of the EOS probe rather than of the model.","rationale":"The paper's strongest claim is that Gemma-9B organizes competence into sparse, task-specialized yet partially reusable attention-head circuits, with dedicated super-heads and characteristic layer distributions. For that claim to hold, the scoring function used to define and discover circuits must faithfully reflect whether the model can still perform the task after head ablation. The LS metric—a linear SVM on the top five PCA dimensions of the final-layer EOS token—is the sole arbiter of every ablation and threshold decision. It is never calibrated against direct behavioral accuracy. This is the load-bearing point because all downstream conclusions inherit whatever bias the probe has. If LS is insensitive to task-relevant information encoded at other positions or in nonlinear form, the reported circuits are not 'minimally sufficient head circuits' for the tasks; they are circuits that maintain linear separability of one particular representation. The theoretical analysis in Theorem 1 does not address this validity question; it assumes a stylized distribution of low-impact heads and bounds sampling error under that assumption. The limitations section acknowledges variation across trials and parameter sensitivity, but not the possibility that the metric itself is disconnected from task competence. A concrete behavioral re-run would settle this directly. Since the reader already flagged this exact weakness and issued CONDITIONAL, my analysis does not move the verdict; it reinforces the need for the conditional check before the central claims can be accepted.","tokens_in":9075,"tokens_out":3384,"duration_ms":41067,"concrete_test":"Re-run the Search-K-MSHC pipeline on Gemma-9B for at least the grammar and arithmetic-verification tasks, replacing the LS score with a direct behavioral score: for each minimal pair, compute the model's accuracy by comparing the probabilities of the correct versus incorrect continuation (e.g., P(correct sentence) > P(incorrect sentence) for grammar; P(equation true) > P(equation false) for arithmetic). Use this behavioral score in the same macro and micro ablation loops, and compare the resulting super-head sets at the 95% selection threshold with the LS-derived sets. If the Jaccard overlap between LS-based and behavior-based circuits is low, or if ablating the reported grammar circuit leaves behavioral accuracy near ceiling while LS drops from 0.86 to 0.50, then the paper's central claims are probe artifacts rather than properties of the model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—distinct task-specific head circuits with dedicated super-heads and minimal cross-task overlap—is derived entirely from the LS score of Definition 1 and Section 3.4. That score trains a linear SVM on the first D≤5 PCA components of the final-layer EOS token and is used for every ablation decision: macro layer search, stochastic head pruning, threshold checks, and all overlap statistics. No experiment compares LS against the model's actual behavioral accuracy on the same binary task (e.g., next-token log-probability or classification accuracy). The baselines are telling: arithmetic LS is 0.99, grammar 0.86, word problems 0.77. A model that solves these easy minimal pairs through non-EOS output positions or nonlinear readouts would score low on LS even when behavior is fully intact. Moreover, zeroing attention heads changes the residual stream throughout later layers; if the fixed PCA/SVM probe is even slightly misaligned with the ablated representations, LS drops without any corresponding loss of model competence. Thus the identified circuits may be exactly the heads that support linear separability of final EOS embeddings, not minimally sufficient circuits for the tasks. Theorem 1 does not rescue this: it only bounds missed prunable sets under a synthetic contamination model and assumes the LS score is the quantity that matters. The paper's own limitations section concedes imprecise triangulation and parameter sensitivity, but the more fundamental gap is that the metric itself is unvalidated against task behavior.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the (K,ε)-Minimum Sufficient Head Circuit (K-MSHC) framework and the Search-K-MSHC algorithm, which seek to identify small sets of attention heads whose K-subsets restore a task separability score. The method is applied to Gemma-9B on three binary classification tasks: grammar acceptability, arithmetic verification, and arithmetic word problems. The authors report task-specific layer distributions, threshold-dependent overlap patterns, and dedicated \"super-heads\" with minimal cross-task overlap. The paper also proves a bound on the expected number of missed prunable K-subsets under a contamination model. The main evidence throughout consists of low-dimensional linear separability (LS) scores computed on the final-layer EOS token.","tokens_in":9404,"tokens_out":4289,"duration_ms":41881,"significance":"If the reported circuits are genuinely causal, the paper offers a scalable method and a useful organizational picture of a 9B model: sparse, task-specialized but partially reusable attention heads. The formalization of K-sufficiency and the stochastic search are reasonable contributions, and the authors are transparent about several limitations. However, the significance is currently conditional: every empirical claim is mediated by the LS score, which is never validated against actual task behavior, and the theoretical bound does not close that gap. With additional validation, the framework could be a useful addition to mechanistic interpretability tooling.","major_comments":[{"comment":"The LS score is the only measure of task competence used throughout the ablation pipeline, yet it is never validated against the model's actual classification accuracy or next-token probabilities. For example, Table 1 reports baseline LS scores of 0.99, 0.86, and 0.77 for arithmetic, grammar, and word problems, but these numbers are not compared with the model's true accuracy on the same minimal pairs. If final-layer EOS linear separability diverges from behavioral competence, or if zeroing heads changes later representations in ways a fixed linear probe cannot capture, then the circuits, overlap statistics, and super-head claims in Section 4 would be properties of the probe rather than of the model. Please add a validation experiment, such as reporting model accuracy under the same head ablations or using activation patching on a held-out set, and report the agreement between LS-selected circuits and behaviorally verified circuits.","section":"Section 3.4 and Definition 1"},{"comment":"Theorem 1 bounds the expected number of missed prunable sets under a synthetic contamination model, but it does not establish that the returned set C is a (K,ε)-MSHC as defined in Definition 3. The definition of low-impact heads in Definition 4 already conditions on the threshold τ, so the \"contamination\" is defined relative to the LS threshold the algorithm uses; the bound therefore does not independently validate the algorithm's output. No theorem or experiment checks that every K-subset of the returned C achieves LS ≥ U_ε(M,B); Section 4 reports selection frequencies only. Please either prove a guarantee for the returned set under stated assumptions or empirically verify the K-sufficiency condition on the final circuits.","section":"Section 3.3 and Definition 3"},{"comment":"The overlap and super-head findings are summaries of the algorithm's own outputs, so without independent causal confirmation they are partly circular. Figure 2 and the accompanying text report Jaccard overlaps among heads selected by LS-based search, but no experiment patches or ablates the shared heads to confirm that the same heads are actually responsible for the tasks. A concrete test would be to take the union of the three circuits at the 50% threshold, ablate the shared heads, and measure the behavioral drop on each task; if the shared heads are not behaviorally important, the \"weak sharing\" conclusion would need to be revised.","section":"Section 4.3"},{"comment":"The empirical claims are based on a single model, a single hyperparameter setting (K=10, ε=0.25, W=5, p=0.75, N=10), and small evaluation sets of 50 examples per mini-batch with 20 trials. The limitations section acknowledges parameter sensitivity, but the abstract-level claims about \"distinct task-specific head circuits\" and \"dedicated super-heads\" are stated without this caveat. Please add a robustness study over K and ε, at least for the super-head and overlap conclusions, or soften the claims to match the current evidence.","section":"Section 4 and Section 5"}],"minor_comments":[{"comment":"Definitions 2 and 3 both introduce the same named object, (K,ε)-Minimum Sufficient Head Circuit; the first should be a preliminary definition or the numbering should be adjusted so each definition has a distinct name.","section":"Section 3.1"},{"comment":"Equation (2) defines U_ε(M,B), but Algorithm 1 line 15 uses U^S_DT(M,B); the superscript S and subscript DT are not defined in the text, and the notation should be made consistent.","section":"Equations (2) and Algorithm 1"},{"comment":"There is a typo in the related work section: \"V oita et al.\" should be \"Voita et al.\"; also \"numerocity\" in the grammar task description is likely \"numerosity\".","section":"Section 2 and Section 3.5"},{"comment":"The captions for Figures 1 and 2 do not specify the exact quantities plotted, such as the units for selection frequency or the precise definition of the thresholds used; please clarify these details so the heat map and overlap matrix can be interpreted independently of the prose.","section":"Figures 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The central empirical claims are conditional on the unvalidated LS proxy, so the authors should be asked to validate it against behavioral accuracy or activation patching before the paper can be considered for acceptance. The theoretical contribution is narrow but honestly scoped, and the paper fits the scope of an interpretability venue. If the validation experiments do not support the LS proxy, the reported circuits and overlap findings would need substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one thing to know: the new definition here—(K,epsilon)-MSHC, where every K-subset of a head circuit must restore the LS score to threshold—is genuinely new, and the Gemma-9B head maps are fresh measurements. But the entire empirical edifice is built on an unvalidated linear-separability proxy of final EOS embeddings, so the headline claims about task-specific circuits, super-heads, and overlap patterns are not yet supported. I'd send it to review, but it needs a serious revision.\n\nWhat's actually good. The K-sufficiency formalization is a real contribution. Most circuit-finding work asks for a sufficient set; requiring every K-subset to be sufficient builds in redundancy and gives a clean way to talk about 'dedicated' versus 'shared' heads. The Search-K-MSHC algorithm is an efficient stochastic search over heads, and the paper makes the sensible move of first finding sensitive layers, then pruning heads. The three task families are well-chosen minimal pairs: determiner-noun agreement, arithmetic verification, and word problems. The overlap analysis at three thresholds is the kind of descriptive result that can anchor later work. The limitations section is candid about parameter sensitivity and trial variance.\n\nWhere it goes soft. The LS score is used for every decision: layer ablation, head pruning, threshold checks, and all overlap statistics. The paper never shows that LS tracks the model's actual behavior on those tasks. Baseline LS scores are 0.77–0.99, but we don't know if Gemma-9B gets the examples right. Zeroing heads shifts later representations; a fixed PCA+SVM probe can drop in score even when the model still solves the task. So the 'circuits' may be circuits for linear separability of final-layer EOS tokens rather than circuits for grammar or arithmetic. Theorem 1 bounds missed prunable sets under a synthetic contamination model; it doesn't prove the returned set is a true (K,epsilon)-MSHC. And there's no comparison against existing circuit-discovery baselines like ACDC, EAP, or sparse feature circuits. Those are omissions, not crimes; the paper positions itself as a first pass. But as written, the central claim—dedicated super-heads with minimal cross-task overlap—is an artifact of the proxy until proven otherwise.\n\nWho should read it: interpretability researchers interested in circuit discovery and head redundancy. The definition is worth engaging with even if the empirical results are provisional. A referee should push for (1) validation of LS against true accuracy on the same examples, (2) a couple of behavioral checks (e.g., logit diff or next-token accuracy under head ablation), and (3) artifact release. I would not cite the empirical maps in current form, but I'd cite the K-sufficiency definition if it's sharpened.\n\nRecommendation: send to peer review. The formalization is worthwhile, and the empirical weaknesses are fixable in principle. But the current version overstates what it demonstrates.","headline":"New K-sufficiency circuit definition and a Gemma-9B head map, but the load-bearing linear-separability proxy is unvalidated, so the headline claims about task-specific circuits are shaky.","tokens_in":9894,"tokens_out":5505,"would_cite":false,"duration_ms":45880,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Gemma-9B's grammar and arithmetic skills live in mostly separate attention-head circuits.","keywords":["minimal sufficient circuits","attention heads","mechanistic interpretability","linear separability","circuit overlap","Gemma-9B","syntactic classification","arithmetic reasoning"],"falsifier":"Retrain the LS probe after each ablation and compare against direct classification accuracy on the same inputs; if ablating the reported super-heads leaves direct accuracy high while the LS score drops, the circuits disappear. Alternatively, patch in the super-heads of one task while running another task: the paper's overlap claims predict these patches should not restore performance on the second task.","tokens_in":8900,"feed_emoji":"🧠","tokens_out":5132,"duration_ms":49788,"temperature":0.7,"pith_summary":"This paper introduces a method, the (K, ε)-Minimum Sufficient Head Circuit (K-MSHC), for finding the smallest sets of attention heads that can restore a language model's performance on a classification task, and applies it to Gemma-9B on grammar acceptability, arithmetic verification, and arithmetic word problems. The authors report that each task is carried by a sparse, task-specific circuit of heads: grammar relies mostly on early layers, word problems on both shallow and deep layers, and arithmetic on a distributed set. They also report that task pairs share many weakly important heads but almost none of their strongest 'super-heads,' so even related numerical tasks do not share a single arithmetic circuit. If correct, this gives a concrete picture of how a mid-sized LLM organizes capabilities—specialized core circuits with partial resource reuse—which matters for targeted interventions and for predicting whether improving one skill transfers to another.","feed_headline":"Grammar and arithmetic run on separate head circuits in Gemma-9B","feed_subtitle":"Sparse minimal-circuit search finds task-specific super-heads, with weak heads shared across tasks.","key_machinery":"The central object is the (K, ε)-Minimum Sufficient Head Circuit: the smallest set of attention heads H such that every K-head subset of H restores the model's separability score above an ε-understanding threshold, enforcing both minimality and redundant sufficiency. To find it, the paper uses the LS (low-dimensional linear separability) score—a linear SVM trained on the top five PCA components of the final-layer EOS token—as a cheap proxy for task performance under ablation, and Search-K-MSHC, a two-phase stochastic search that first ablates layer windows to select candidate layers, then repeatedly samples K-subsets and prunes the worst-performing heads until the threshold is met. A Hoeffding-style bound (Theorem 1) states that the expected number of missed 'prunable sets' decreases exponentially with the sample count N, the subset size K, and the squared margin between the contamination rate and threshold.","core_discovery":"The reported finding is that Gemma-9B's competence on the three task families decomposes into minimal sufficient head circuits that are discoverable in practice. Grammar acceptability is concentrated in early layers (0–6); word problems show a bimodal pattern with activity in shallow (0–3) and deep (11–20) layers; arithmetic verification is broadly distributed across the network. Circuit overlap is threshold-dependent: at the 50% selection threshold, grammar and arithmetic share about 20% of heads, while at the 75% threshold arithmetic and word problems share more critical heads (about 10%) than grammar and arithmetic (about 9%); at the 95% threshold, the strongest heads barely overlap at all. The paper interprets this as each task maintaining dedicated 'super-heads' while reusing weaker heads across related tasks.","pith_inferences":["Editorial inference: if the reported super-heads are stable across random seeds and input batches, the overlap matrix at high thresholds could serve as a transfer-learning predictor: tasks with more shared strong heads should show more positive transfer, a testable claim beyond this paper.","Editorial inference: the LS probe's reliance on final-layer EOS tokens means the method finds heads whose removal changes the final-layer linear code; replacing it with direct behavioral accuracy would test whether the same circuits appear, and might reveal that some 'weak shared heads' only matter to the probe.","Editorial inference: because the circuits are found by stochastic search and the paper reports trial-to-trial variation, multiple functionally equivalent circuits likely exist; measuring invariance across seeds could turn circuit discovery from a point estimate into a distribution over circuits."],"forward_implications":["Interventions can be targeted: ablating the super-heads of one task should degrade that task much more than the others, offering a handle for controlled behavior editing.","Improvements to one capability should transfer only weakly to related tasks, because core heads are not shared even where weak heads overlap.","Arithmetic reasoning in language models is not a single circuit: verification and word problems use largely distinct strong heads, so analyses that treat 'arithmetic' as one mechanism are incomplete.","Circuit discovery with K-sufficiency is feasible on models with thousands of heads, since the stochastic search avoids enumerating all 2^{|M|} subsets.","The spatial organization of circuits (early for grammar, bimodal for word problems, distributed for verification) provides an architectural signature that can be compared across tasks and models."],"supporting_citations":[{"why":"Supplies the BLiMP minimal-pair data used to build the grammar acceptability task.","marker":"[Warstadt et al., 2020]"},{"why":"Shows that many attention heads can be pruned without major performance loss, motivating head-level ablation search.","marker":"[Michel et al., 2019]"},{"why":"Establishes linear probing of intermediate representations, the foundation of the LS score.","marker":"[Alain and Bengio, 2016]"},{"why":"Provides evidence that specialized heads do the heavy lifting, motivating task-specific circuit identification.","marker":"[Voita et al., 2019]"},{"why":"Gives computational-complexity results for circuit discovery, motivating the efficient approximate search.","marker":"[Adolfi et al., 2025]"},{"why":"Shows shared latent grammatical concepts across languages, used to interpret partial circuit overlap as a shared semantic hub.","marker":"[Brinkmann et al., 2025]"}],"fun_headline_variants":["Gemma-9B's grammar and math run on separate minimal head circuits","Minimal head circuits reveal task-specific processing in Gemma-9B","Search-K-MSHC uncovers dedicated super-heads for syntax and math","Grammar and arithmetic: separate super-heads in Gemma-9B","Task-specific head circuits: grammar early, math distributed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole pipeline assumes that the LS score—a linear SVM's accuracy on the top five PCA components of the final-layer EOS token—faithfully tracks the task competence that head ablation is meant to measure; if zeroing heads changes representations in ways the probe cannot see, the reported circuits and overlap patterns are probe artifacts rather than properties of the model.","fun_headline_variants_meta":{"raw":{"variants":["Gemma-9B's grammar and math run on separate minimal head circuits","Minimal head circuits reveal task-specific processing in Gemma-9B","Search-K-MSHC uncovers dedicated super-heads for syntax and math","Grammar and arithmetic: separate super-heads in Gemma-9B","Task-specific head circuits: grammar early, math distributed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3255,"prompt_tokens":924,"completion_tokens":2331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":2237}},"tokens_in":540,"tokens_out":2331,"duration_ms":15369,"temperature":1.0,"reasoning_tokens":2237,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:37:16.432719+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Retrain the LS probe after each ablation and compare against direct classification accuracy on the same inputs; if ablating the reported super-heads leaves direct accuracy high while the LS score drops, the circuits disappear. Alternatively, patch in the super-heads of one task while running another task: the paper's overlap claims predict these patches should not restore performance on the second task.","supporting_citations":[],"review_version":1}