Soft positional priors can help small Transformers learn retrieval circuits, but it is unclear whether the resulting circuits remain functional once the prior is removed. We test this with an annealable soft-prior Transformer whose attention biases can be learned, faded, or zeroed during training and evaluation. On associative recall, unforced models perform well with the prior active (0.772±0.020) but collapse at zero gate (0.095±0.009). Smooth fade-to-zero training preserves high zero-gate accuracy (0.734±0.028), whereas forced-zero training, hard switching, and post hoc continuation fail to recover the same effect. The pattern also appears on Markov induction. Linear regression ICL provides a boundary case because zero-gate training can learn that task directly. Mechanistic traces show that circuit consolidation occurs after the gate reaches zero, even though the responsible heads vary across seeds. These results suggest that circuit removability in small discrete retrieval tasks depends on the training trajectory, not just the final architecture.
We present a three-step recipe for identifying attention-head circuits in pretrained transformers. A per-head spectral signal -- the time-integrated participation ratio of each head's attention output -- ranks heads doing sustained content-dependent computation without labels or attribution gradients. A task-pattern screen filters this general indicator into a task-specific candidate circuit, and group ablation against a matched-random control completes the causal claim. We validate across an 8x parameter range (51M to 1B-active / 7B-total), two architecture families (dense, mixture-of-experts), and four pretraining pipelines. The recipe ports: a 2-6 head induction circuit is causally necessary in every model tested, with a 94-100% drop in synthetic-induction top-1 after ablation. The spectral signal is predictive without supervision: on six independent seeds of a 51M-parameter probe model, the same computation identifies the seed-specific circuit on each seed. The fraction of heads doing identifiable specialized computation is conserved at 17-19% across the Pythia family (124M to 410M), while specific induction circuits stay 3-11 heads -- sublinear in total head count. This paper is the methodology anchor of a three-paper program; companion papers extend the recipe to developmental trajectories during pretraining and to composed-task circuits where pattern selectivity decouples from task-causal structure.
We provide a systematic recipe for translating ReLU approximation results to softmax attention mechanism. This recipe covers many common approximation targets. Importantly, it yields target-specific, economic resource bounds beyond universal approximation statements. We showcase the recipe on multiplication, reciprocal computation, and min/max primitives. These results provide new analytical tools for analyzing softmax transformer models.
Mechanistic interpretability seeks to explain transformer behavior through circuits: sets of internal components that causally support a behavior. However, self-repair creates a blind spot: ablating a primary component can activate a dormant backup, so a circuit that explains behavior in the intact model can become incomplete under the intervention used to test it. We formulate this gap as conditional circuit completion: given a primary set, identify components that become causally important after its removal. We introduce conditional co-ablation (CoAx), which ranks candidates by growth in ablation effect after primary-set removal. We show that a perfectly dormant backup can be indistinguishable from an irrelevant component to per-unit intact-state scores, whereas its conditional effect change exactly aggregates all interaction orders linking it to the removed set. On GPT-2-small's Indirect Object Identification (IOI) circuit, CoAx recovers the documented backup heads at 0.941 ROC-AUC, versus 0.815 for the strongest intact-state attribution baseline and 0.758 for the matched conditional-energy control. Recovery drops to 0.40 +/- 0.13 AUC for alternative component sets matched in behavioral effect, output displacement, and depth, showing that recovery is specific to the removed circuit. Beyond recovery, the CoAx-selected heads are causally load-bearing: freezing them after primary removal sharply reduces the IOI margin, while adding them to the incomplete circuit reduces incompleteness from 0.75 to 0.21. More broadly, conditional growth aligns with intervention-derived repair in 11/12 held-out instances across 4 mechanism clusters, and CoAx completions outperform matched random completions on all 8 non-GPT-2 models spanning 6 architecture families. Together, causal explanations of self-repairing transformers must account for backup circuitry when primary components fail.