Implementational Mode: Topographic

Origin	Craver (2007) — decomposition; Bechtel & Richardson (2010) — localization
Question	Which components are causally involved in this behavior?
Licensing evidence	Ablation (necessity) + locus specificity + counterfactual baseline
Interpretive-validity risk	Treating a map as an explanation — knowing where is not knowing what or how
Position in partial order	$I_{\text{top}}$ — lowest implementational commitment (a set, not a graph)

What this mode claims

A verdict tagged [implementational-topographic] identifies a set of components (heads, neurons, layers, SAE features) that participate in producing a behavior. It makes no commitment to what they do, how they are connected, or what procedure they execute. It is a map — a spatial answer to the question “where does this happen?”

This is the most commonly established claim in circuit discovery and the appropriate default for any ablation-based result. There is nothing lesser about it. Most of the field’s most important findings — the IOI heads, induction heads, Greater-Than components — were first established topographically before being characterized at higher modes.

Formal characterization

Let $C = \{c_1, \ldots, c_k\}$ be a proposed circuit (a set of attention heads and/or MLPs). A topographic claim asserts:

$\forall c_i \in C: \quad \mathbb{E}_{x \sim \mathcal{D}} \left[ m(M, x) - m(M_{\setminus \{c_i\}}, x) \right] > \delta$

where $M_{\setminus \{c_i\}}$ is the model with component $c_i$ ablated and $\delta$ is a meaningful effect threshold. Additionally, for locus specificity:

$\forall c_j \notin C \text{ (adjacent)}: \quad \mathbb{E}_{x \sim \mathcal{D}} \left[ m(M, x) - m(M_{\setminus \{c_j\}}, x) \right] < \delta$

The claim is about membership in a causal set, not about the structure of that set.

What licenses an [implementational-topographic] tag

1. Necessity — ablating each component degrades the behavior reliably, under at least two ablation methods (zero + mean, or zero + resample). Single-method ablation is insufficient because zero ablation introduces distributional shift.

2. Locus specificity — the effect is localized to the named components, not attributable to collateral disruption. Adjacent heads (same layer, not in circuit) should show negligible effect when ablated individually.

3. Counterfactual baseline — the ablation is compared against a baseline (resample, mean, or zero), and the magnitude of degradation is reported relative to the baseline, not just as a raw number.

4. Discovery procedure named — different procedures (activation patching, ACDC, EAP-IG, manual) can return different component sets for the same behavior (Conmy et al. 2023). The procedure is part of the finding.

What does NOT license a [implementational-topographic] tag

• Single-method single-example ablation. One head, one prompt, one method. Not enough for a claim about general causal involvement.
• Attribution scores without causal validation. Gradient-based attribution, DLA, or saliency maps are candidate topographies until validated by causal intervention.
• High activation ≠ causal involvement. A head can be highly active on a task without being necessary for it. Activity is a heuristic for circuit membership, not evidence.
• Complement ablation alone. Showing the circuit suffices (complement ablated, behavior preserved) is stronger than topographic — it’s a sufficiency result that supports the upgrade to connectomic.

Worked example: IOI topographic claim

Claim. Heads L5H1, L5H5, L6H9, L7H3, L7H10, L8H6, L8H10, L8H11, L9H6, and L9H9 are causally involved in IOI behavior in GPT-2 Small. [implementational-topographic]

Evidence:

• Necessity: each head shows $|\Delta \text{logit diff}| > 0.3$ under both zero and mean ablation
• Locus specificity: adjacent heads (L5H0, L5H2, L6H0, etc.) show $|\Delta| < 0.05$ under the same interventions
• Discovery procedure: activation patching (Wang et al. 2022), confirmed by ACDC (Conmy et al. 2023)

What this is not: This does not tell us that L9H9 “moves names” or that L7H3 “inhibits the subject.” Those are functional/algorithmic characterizations requiring additional evidence. The topographic claim says only: these 10 heads are where the action is.

Upgrade paths

Direction	What’s required
$I_{\text{top}} \to I_{\text{con}}$ (→ connectomic)	Path-level causal evidence of directed connections between the identified components. Path patching, not just activation patching.
$I_{\text{top}} \to I_{\text{stat}}$ (→ activation-statistical)	Characterize the distributional properties of activations at the identified components.
$I_{\text{top}} \to I_{\text{fun}}$ (→ functional)	Specify the input-output function of individual components — what each one does, not just that it matters.

Instruments that provide topographic-level evidence

• A01 (Activation patching) — single-node necessity via mean/resample ablation
• A04 (Resample ablation / complement ablation) — sufficiency of the proposed set
• A09 (Hyperparameter sensitivity) — does the discovered set change with method parameters?
• F01 (Seed variance) — does the set replicate across random seeds?
• B08 (Edge Jaccard) — agreement between discovery methods on the component set

Key references

• Wang, K., et al. (2022). “Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 Small.” ICLR 2023. — IOI circuit discovery via activation patching.
• Conmy, A., et al. (2023). “Towards Automated Circuit Discovery for Mechanistic Interpretability.” NeurIPS 2023. — ACDC; automated topographic discovery.
• Craver, C. F. (2007). Explaining the Brain. Oxford University Press. — Decomposition as the first step of mechanistic explanation.
• Bechtel, W. & Richardson, R. C. (2010). Discovering Complexity. MIT Press. — Localization strategy and its failure modes.