In arXiv:2602.13483v2, ACC++ forms one rank-1 candidate for each pair (upstream component, singular direction of the head’s QK matrix Ω), builds the pre-softmax contribution of every candidate across competing sources, scores those candidates once with Integrated Gradients (IG) on the post-softmax weight Ads, and greedily deletes the highest fixed IG score, recomputing Ads after each deletion, until that weight falls below τ. Destination candidates use the left singular projector ukuk⊤; source candidates use vkvk⊤ and, if selected, are removed from every competing key. The paper itself calls τ a heuristic and does not uniquely fix it. It specifies a frozen IG ranking as the solver and does not claim that ranking uniquely determines the minimal removed set. The reduction to one Ω per head is derived only under an assumed full-column-rank, well-conditioned query/key factorization, checked empirically for the models studied rather than proved in general.
What counterfactual the steps are solving
Franco, Tassis, Rohr, and Crovella, in Finding Interpretable Prompt-Specific Circuits in Language Models (arXiv:2602.13483v2, 13 May 2026), define the attention-level object before any task evaluation. For a fixed head (ℓ, a) and a destination–source token pair (d, s), the pre-softmax score is the bilinear form A′ds = xd⊤Ωxs, with Ω = WQWK⊤ after the bias and RoPE reductions below. The post-softmax row is Ad = Softmax(A′d / √R), and Ads is the weight on source s. Masked attention restricts competing sources to positions j ≤ d. Upstream components are the input embedding, earlier attention heads, and feedforward modules; the residual at a token is their sum.
A signal, in this paper’s ACC++ sense, is not an arbitrary low-rank feature. It is one upstream component’s write into one singular direction of Ω, retained only if it belongs to a removed set that drives Ads below τ. There are two separate counterfactuals, and they need not return the same set. A destination intervention subtracts vectors from xd only and changes the query. A source intervention subtracts vectors from every competing key residual {xj}j≤d and leaves the query-side destination residual unchanged. Because softmax couples the row, both tests are scored on the full post-softmax row, not on the isolated pre-softmax entry A′ds. Section 2 states the four-step outline; Appendix C is the specification of the search. This account follows that v2 text. It does not treat later IOI, autointerpretation, or multilingual measurements as part of the removal rule.
Exact steps for rank-1 candidates and removal
The paper’s own order, combining §2, Figure 1, Appendix B, and the Appendix C.6 recipe, is: fold bias or RoPE into the tokens so that one Ω remains; fix the head and pair; decompose residuals; emit rank-1 (component, direction) candidates; tabulate their pre-softmax contributions; attribute Ads once with IG; greedily remove in that fixed order until the post-softmax test fires.
One Ω, then the residual decomposition
Appendix C writes the solver for a bilinear score with no bias term and no position-dependent Ω. Appendix B is the preprocessing that is supposed to make that exposition apply to GPT-2-style bias, RoPE, or both. After those token maps, the solver still sees a single head-level Ω = WQWK⊤, independent of the source position. The position dependence, when RoPE is present, sits in the transformed tokens, not in a separate Ω(d−s).
For the chosen head and pair, decompose the destination residual as xd = Σc odc, and likewise at each source position. Compute the thin SVD Ω = Σk=1R σk uk vk⊤. Here R is the query/key width, defined in Appendix A as D/H, and is also the maximum rank of Ω. The paper does not require every singular value to be used later; every direction is a candidate until removal.
Rank-1 (component, singular-direction) candidates
Destination candidates use the left singular vectors. For component c and direction k the paper sets the rank-1 projector PkU = ukuk⊤ and the candidate
sdkc = ukuk⊤ odc.
That is the object §2 calls a rank-1 candidate: one component output, one left singular direction. The cross-product is exhaustive. If C(ℓ) is the upstream set, there are Q = |C(ℓ)|·R destination candidates, not one candidate per component and not one candidate per subspace chosen independently of components. Section 2 contrasts this pairing with prior ACC, which identified subspaces and components separately.
Source candidates are the same cross-product on the right singular vectors. For each j ≤ d,
sjkc = vkvk⊤ ojc.
The candidate index remains the pair (c, k). It is not a separate candidate for each source position. Selecting (c, k) on the source side means removing what c writes along vk from every competing key.
Appendix C.2 states a limitation that the double sum does not recover the full residual. It recovers the projection of the residual into the head subspace. The authors treat that projection as the part that enters the attention score, and §2 says that summing the candidate-induced pre-softmax vectors recovers A′d. The rank-1 list is therefore a complete accounting of the score inside that subspace, not a decomposition of residual coordinates orthogonal to the head.
Contribution rows, then one frozen IG ranking
Before any deletion, each candidate becomes a row of a contribution matrix C with one column per source position j ≤ d. In the destination view the row for (c, k) is Cij = (sdkc)⊤ Ω xj. In the source view it is Cij = xd⊤ Ω sjkc. Appendix C.3 absorbs the 1/√R softmax scale into C, so the solver can write the score row as the sum of these rows and the weight as the softmax of that sum.
Attribution targets the scalar Ads, not the whole row and not the pre-softmax score. The path is the straight line from the zero score vector to the full row, z(t) = t A′d, with p(t) = Softmax(z(t)) for t in [0, 1]. The paper’s integrated-gradient assignment to candidate i is
IGi = ∫01 ps(t) (Cis − ⟨ci, p(t)⟩) dt.
The term in parentheses is the candidate’s contribution at s minus its expected contribution under the softmax distribution at t. That is how competition among j ≤ d enters the score. In practice the integral is a trapezoidal rule with T = 64 steps. Appendix C.5 then freezes those scores: they are computed once from the original full row and are not recomputed after a candidate is removed.
Greedy removal until the post-softmax test
Let S start as the full candidate index set, and let A′d(S) be the sum of the rows still in S. While the recomputed post-softmax weight on s exceeds τ, delete
i* = arg maxi∈S IGi
and recompute the weight from the remaining rows. The removed index set M is the ACC++ explanation for that view. Destination removal implements x̃d = xd − Σ(c,k)∈M sdkc, with other positions unchanged. Source removal implements x̃j = xj − Σ(c,k)∈M sjkc for every j ≤ d, with the query-side destination residual held fixed, and scores A′dj = xd⊤ Ω x̃j. The signals reported for the firing are these removed candidates, not the candidates left behind.
Two wordings of the stop rule appear in the v2 text and are not identical at equality. Section 2 and the objective statements in Appendix C say removal continues until Ads < τ. The Appendix C.5 loop is written as “while Ads(S) > τ.” The paper does not discuss the measure-zero case Ads = τ, and it does not specify a tie break if several IG scores share the maximum. The subset is the shortest prefix of this fixed order whose removal meets the implemented stop test. It is not obtained by searching all subsets.
What the paper leaves heuristic, assumed, or not uniquely determined
The three items in the question are not in the same epistemic bucket in the v2 text. The table records only the status the paper itself states or leaves open, not a ranking of which choice is better.
| Choice | What the v2 text does | Status the paper itself leaves |
|---|---|---|
| Threshold τ | §5 and Appendix D set a dynamic rule τ(d) = k/d from the empirical distribution of d·Ads, with k in [1.5, 2.5]. Experiments in §3 use k = 2.5 on IOI and k = 1.5 on multilingual IOI. | Explicitly heuristic. Not uniquely determined: the knee motivates an interval, task settings pick different k, and a dataset-agnostic rule is deferred. |
| Fixed IG ranking | Appendix C.5 computes IG once on z(t) = t A′d, freezes the scores, and deletes by arg max. The integral is replaced by a trapezoidal rule with T = 64. | Specified as the solver, not labeled a heuristic. Not shown to be the unique order or the unique minimum: exact minimal-set search under softmax is called challenging, and ties plus the T = 64 rule are solver settings. |
| One Ω per head under bias or RoPE | Appendix B folds bias into additive token terms and RoPE into position-dependent token maps Md, Ms, then uses Ω = WQWK⊤. The derivation uses W†QWQ = IR and the corresponding key identity. | Assumed, not heuristic. “Mild full-rank” and well-conditioning of WQ and WK are hypotheses; for the studied models they are supported by condition numbers below 1000, not derived for arbitrary weights. The construction is explicit under those identities; numerical equality to IR is described as very close when the matrices are well-conditioned. |
τ is a heuristic, and k is not unique
Section 5 says the method depends on τ and that the authors “use a heuristic based on the distribution of the d·Ads per model/dataset.” A principled, dataset-agnostic rule is left to future work. Appendix D motivates the scale d·Ads because a row of length d spreads a probability distribution over d attendable positions. Across the plotted model–dataset ECDFs they report a knee in 1.5 ≤ d·Ads ≤ 2.5 and therefore use τ(d) = k/d with k in that interval. They also say the impact of this global, context-size-dependent choice is unclear in other scenarios, that lower k yields larger graphs and higher k smaller graphs, and that further threshold methods are future work. The §3 choices k = 2.5 for IOI and k = 1.5 for multilingual IOI are experimental settings inside that interval, not a uniqueness proof. “Reasonable defaults” in the main text point back to this appendix; they do not replace it with a derived τ.
The frozen IG order is specified, but not a unique minimum
The paper does not call the fixed IG ranking a heuristic in the sentence it uses for τ. It does say why a ranking exists at all. Softmax makes exact identification of the minimal removed set challenging, so ACC++ uses a greedy estimate of nonlinear impact. The estimate is IG on the true weight Ads, and Appendix C.5 adds the operational restriction that those attributions stay fixed while rows are deleted and Ads is recomputed. That restriction is part of the stated algorithm. It is not derived as the only ranking that could satisfy Ads < τ.
Two further gaps are in the text rather than implied. First, the destination objective asks for a small set and the source objective asks for a minimal set, but both are solved by the same greedy loop; the algorithm is not an exact set-cover of the counterfactual. Second, T = 64 is an approximation to the integral, not a value the objective uniquely determines. An unspecified arg-max tie would also leave the next deletion non-unique. None of these points is a license to substitute another attribution method and still call the result the paper’s solver.
One Ω per head is conditional on a rank assumption
Section 2 says that, under mild full-rank assumptions, a single QK matrix per head can compute the score whether or not the model uses bias or RoPE. Appendix B states the assumption: WQ and WK, shaped D×R with D > R, are well-conditioned and therefore full column rank, so W†QWQ = IR and WK⊤(WK⊤)† = IR. When the matrices are well-conditioned, the authors say the numerical product is very close to IR. That is an assumption plus a numerical caveat, not an identity that holds for every trained head.
Under those identities the paper gives one explicit reduction, not a menu of equivalent Ω’s. For bias, cd⊤ = bQ⊤W†Q and cs = (W†K)⊤bK, and the score is (xd⊤ + cd⊤)Ω(xs + cs). For RoPE, Md = WQRdW†Q and Ms = (WK⊤)†Rs⊤WK⊤, and the score is (xd⊤Md)Ω(Msxs) with the same position-independent Ω. Bias plus RoPE composes the two maps. Appendix B.4 reports that, for the GPT-2 small, Pythia-160M, and Gemma-2 2B matrices they plot, condition numbers of WQ and WK⊤ are below 1000 and usually much lower. That check supports the assumption in those models. It does not remove the assumption, and the paper does not claim the same bound for other checkpoints.
Once the tokens have been rewritten, Appendix C says the candidate and removal procedure extends unchanged: bias adds a constant contribution and RoPE applies a per-position rotation before multiplication by Ω. The rank-1 projectors and the greedy loop do not grow a new Ω for each distance d−s.
Separated from later evaluations
The recursive circuit is a different outer loop. The paper seeds components with a direct effect on a logit direction, runs the firing-level procedure on attention events in the current trace, and walks upstream. That recursion uses the signals defined above; it does not change how a rank-1 candidate is formed or when removal stops.
IOI, Greater-Than, and Gendered Pronoun overlap tables, signal ablations, autointerpretation on cached activations, and multilingual circuit-distance comparisons are uses of traces after this solver has returned them. They are not additional constraints on the (component, singular-direction) cross-product, the frozen IG order, or the Ads < τ test. Reading a cluster, an autointerpretation score, or a cross-language correlation back into the candidate definition would add a criterion the v2 method section does not state.
What remains open inside the method, on the paper’s own account, is a non-heuristic rule for τ, a guarantee that the frozen greedy prefix is the unique minimal removed set, and any claim that the single-Ω reduction holds without the full-column-rank and well-conditioning assumption.