Successor Heads
Section titled “Successor Heads”Source. Successor Heads: Recurring, Interpretable Attention Heads In The Wild (Gould et al., 2024).
Description. One attention head maps an ordinal token to its successor, and the mechanism is a product of four weight matrices — embedding, , the head’s OV circuit, unembedding — evaluated directly, with no activation from any prompt entering it. Mod-10 features recovered from by sparse autoencoders, by a linear probe and by single-neuron ablation carry the index the head increments. Evaluated on: GPT-2, Pythia and Llama-2 from 31M to 12B for behavioral recurrence, with the feature-level case study on Pythia-1.4B L12H0 and two appendix replications. Description mode: implementational-functional.
Readings of the claim
Section titled “Readings of the claim”Successor heads. Readings of the claim, from the version the authors state to the version the field cites.
| Reading | Verdict | Missing |
|---|---|---|
| Primary. A single head’s effective OV circuit increments an ordinal token to its successor, and mod-10 features carry the index it acts on | Mechanistically Supported | Necessity measured rather than ranked (I1); a known-positive control (M5); correction for the head, feature and $$ selections (M7); and the feature account does not reproduce the greater-than bias the same weights show, which the authors state as incomplete (V2) |
| The head is specific to succession | Disconfirmed | Nothing — tested and failed: on natural text acronym and greater-than behavior take 23.8% and 18.9% of winning cases, and the paper reframes the head as interpretably polysemantic (I4) |
| The mod-10 mechanism recurs across architectures and sizes, as the abstract states | Underdetermined | Mechanistic recurrence is one case study plus two appendix replications against broad behavioral recurrence (E4); Llama-7B fails the held-out Roman-numeral task where the Pythia models do not (E4, M6), and putting the head on synthesised Roman-numeral representations drops top-1 accuracy to 0.125 (E3, E6) |
| The emergence of successor heads during training explains when the incrementation behavior appears | Underdetermined | Successor scores are tracked across checkpoints in two model families and the heads are seen to emerge, but nothing is plotted against a behavior or a loss curve, and the phase change the induction-head comparison predicts is absent (I11); no checkpoint interval shows the mechanism lapsing (I12) |
Verdict
Section titled “Verdict”Successor heads. Verdict: Mechanistically Supported. The claim stops at the rung marked Blocked; rungs below it are Reached and rungs above it are blocked by that rung. Criteria in bold are absent or adverse at that rung; only those the rung requires can block it.
| Tier | Requires | Missing |
|---|---|---|
| Proposed | construct (C1–C2) | Reached. |
| Causally Suggestive | measurement (M2); internal (I1) | Reached. |
| Mechanistically Supported | internal (I2, I4); external (E1) | Reached. |
| Triangulated | construct (C3–C4); internal (I5–I7); external (E2, E4) | Blocked. I6 double dissociation — Neither arm run, and the polysemantic head makes the converse hard |
| Validated | construct (C5–C6); measurement (M1–M6); internal (I3, I10–I12); external (E2–E6); interpretive (V1–V5) | Blocked at Triangulated. C6 complementation validity — Eight task classes scored individually and never ablated in pairs; I10 rescue reversibility — Both ablations reversible and the clean run in hand; restore never run; I12 offset coupling — The checkpoint data exists and is read in the forward direction only; M5 sensitivity — Every control a known-negative; no planted circuit recovered |
36-criterion audit
Section titled “36-criterion audit”Successor heads. Full 36-criterion audit. Status: C = Confirmed, PC = Partially confirmed, U = Untested, I = Inconclusive, D = Disconfirmed, N/A = Not applicable.
| ID | Criterion | Status | Evidence |
|---|---|---|---|
| Construct Validity | |||
| C1 | Falsifiability | C | A fixed token list and a committed threshold; some models have no head |
| C2 | Structural plausibility | C | A product of four matrices tested directly, with no prompt entering it |
| C3 | Convergent validity | C | A probe at 0.708 cosine and single neurons, sharing nothing with the SAE |
| C4 | Discriminant validity | PC | Top head eight times the runner-up; no neighboring construct attempted |
| C5 | Nomological validity | PC | Weak universality imported, tested, and partly returned (Chughtai et al., 2023) |
| C6 | Complementation validity | U | Eight task classes scored individually and never ablated in pairs |
| Measurement Validity | |||
| M1 | Reliability | PC | Modes over a hundred autoencoders; the central score has no interval |
| M2 | Baseline separation | PC | Cutting the head leaves under one percent and returns bigrams instead |
| M3 | Stability | PC | Three perturbations, one aimed at the paper’s own premise |
| M4 | Calibration | PC | A proportion against a stated threshold; importance gives rank, not magnitude |
| M5 | Sensitivity | U | Every control a known-negative; no planted circuit recovered |
| M6 | Invariance | C | Three families over three orders of magnitude, failures kept in |
| M7 | Selection correction | U | Selection at three levels, disclosed at each and corrected at none |
| Internal Validity | |||
| I1 | Necessity | PC | Winning head on all 64 prompts, which ranks the head without measuring it |
| I2 | Sufficiency | C | The OV circuit alone ranks the successor above every alternative |
| I3 | Minimality | PC | Minimal by construction; the droppable component contributes under a percent |
| I4 | Specificity | PC | Off-target function measured across five ablation regimes and found present |
| I5 | Rival mechanism exclusion | PC | Artifact and pre-head computation both excluded by built experiments |
| I6 | Double dissociation | U | Neither arm run, and the polysemantic head makes the converse hard |
| I7 | Confound control | PC | Tokenization handled by construction; letter collisions explained, not tested |
| I8 | Confounding sensitivity | U | Unattempted; nothing bounds an unmeasured confounder |
| I9 | Epistatic interaction | PC | Indirect effect bounded under both ablation values; no feature-level grid |
| I10 | Rescue reversibility | U | Both ablations reversible and the clean run in hand; restore never run |
| I11 | Onset coupling | I | Emergence observed across checkpoints; coupling to behavior is not |
| I12 | Offset coupling | U | The checkpoint data exists and is read in the forward direction only |
| External Validity | |||
| E1 | Intervention reach | C | Five intervention forms spanning removal and addition, and they agree |
| E2 | Prompt generalization | PC | A constructed token list and 128 natural contexts, the harder one genuinely hard |
| E3 | Cross-task generalization | C | Eight tasks scored individually; Roman numerals decode though never trained |
| E4 | Cross-model recurrence | PC | Behavior recurs across three families; the mechanism is single-model |
| E5 | Graded response | PC | A two-sided dose criterion, and every other ablation all-or-nothing |
| E6 | Novel prediction | C | Roman numerals fitted out of every training set and decoded anyway |
| Interpretive Validity | |||
| V1 | Level declaration | C | The level is fixed by the object: a product of four weight matrices |
| V2 | Level-evidence match | PC | Weights tested on weights; the recurrence claim rests mostly on behavior |
| V3 | Alternative level | PC | Two alternatives closed by experiment; the paper’s own third left open |
| V4 | Unlicensed labeling | C | Each label defined in the sentence that introduces its measurement |
| V5 | Scope declaration | PC | Three limits declared beside the results they bound; one gap remains |
| Total: 10 Confirmed, 18 Partially confirmed, 1 Inconclusive, 7 Untested | |||
| Verdict: Mechanistically Supported |
Exploratory Lens Analysis
Section titled “Exploratory Lens Analysis”An exploratory reading of this claim through the framework’s five lenses, written for this site and not part of the paper, is at Successor Heads — exploratory lens analysis.