Contribution
The paper measures what three ways of adapting a decoder-only model to a classification task (an instruction, demonstrations, and a trained soft prompt) do to the geometry of its internal representations. It uses manifold capacity, which links linear separability of class point clouds to their geometry. After it, there is evidence that methods of similar accuracy act at different layers and on different tokens, and that accuracy and representational separability can come apart.
Key insight
Accuracy has two parts: how separable the classes are in the residual stream, and how well the model’s own unembedding reads that separation out. Demonstrations improve the first; an instruction mostly improves the second. When few-shot prompting fails, the first is usually intact and the second has failed.
Assumptions
- Classification with a fixed label set. Manifold capacity is defined for one-vs-rest dichotomies of P classes. The authors say this “simplif[ies] the more complex tasks” and does not reach multi-token outputs.
- Capacity as defined here. For P classes, project the points randomly to D_proj dimensions and find the critical dimension D* at the inflection of the probability that a random one-vs-rest dichotomy is separable; α = P / D* (App. A.3.2). Dimension is the participation ratio (Σλ)²/Σλ² of each class’s covariance, and radius the largest pairwise distance; both are simplifications of Chou et al.’s (§3.5).
- Two embeddings. “Sentence” embeddings mean-pool the residual stream over the input sentence’s tokens only, excluding the prompt. “Last-token” embeddings take the final position (§3.4).
- Base models only. Llama-3.1-8B, with Gemma-2-2B as replication. “Instruction” names only the prompt format; no model was instruction tuned (footnote 1).
- One instruction template. “This is a text classification task. Possible categories are …” (§3.3). Wording is not varied.
- Synthetic data. The main dataset is 1000 sentences written by Claude 3.5 Sonnet, each labelled for sentiment, topic and intent (five classes each). Single-task results are replicated on AG News and TREC (coarse). Demonstrations: 1 to 40, three seeds each.
Key results
- Accuracy (§4.1, Fig. 3). The instruction beats demonstrations with ≤5 examples; more than 5 surpass it and plateau. Letter-code labels need more examples. Consistently shuffled labels (“Joy” → “Anger”) fail: the model “failed to generalize beyond pretrained associations”, low on both the target and the original labels.
- Sentence level (Fig. 4). Demonstrations, “but not abstract instruction”, raise capacity at early-to-middle layers (a peak around layer 12), by lowering manifold dimension and improving correlation structure. Gold, letter and shuffled labels give little difference: “sentence representation is primarily influenced by input distribution examples, rather than input-output mapping.”
- Last token (Fig. 5). The instruction raises capacity over the raw sentence from about layer 8 on. Demonstrations raise it further, “despite lower task performance for cases with few examples”. Letter labels give much lower last-token capacity; shuffled labels give capacity “similar to the gold label setting”.
- Sensitivity to examples (Fig. 6). At five demonstrations, accuracy across samplings runs “from below 0.1 to approximately 0.6”, while final- layer capacity barely changes, and the worst runs still exceed the instruction’s capacity with a layer profile “nearly identical” to the best. So the instability “is driven primarily by poor readout alignment rather than differences in representational geometry”.
- Multi-task (§4.2, Figs. 7–8). At the sentence level, demonstrations for a different task on the same sentences also raise capacity for the unprompted labels, less than for the prompted ones (“synergy”). At the last token, demonstrations of a different task raise capacity for the unprompted labels in early layers and lower it in late layers (“interference”, a “representational trade-off”). The coherent–incoherent difference follows center–axes correlations, not the dimension or radius of single manifolds.
- Soft prompts (§4.3, Fig. 9). A trained 5-token soft prompt beats demonstrations, with no effect of length (1 to 20). It leaves early and middle layers unchanged (no peak near layer 12), acts at late layers, raises last-token capacity above the instruction and below demonstrations, and lowers capacity for unrelated tasks more than demonstrations do.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Instruction, demonstrations and soft prompts reach similar accuracy through different representational effects | moderate | §4.1, §4.3; one main model, a replication, synthetic plus two open datasets |
| C2 | Demonstrations reshape intermediate representations of the input; an instruction does not | moderate | sentence-level capacity, Fig. 4, Figs. 13–16 |
| C3 | What reshapes them is the input sentences, not the correctness of labels | moderate | little difference across gold, letter and shuffled labels, Fig. 4 |
| C4 | Few-shot instability and failure on flipped labels are readout failures, not tangled representations | moderate | Fig. 6 and the shuffled-label capacity, §4.1 |
| C5 | Prompting for one task trades away late-layer separability for others | weak | Fig. 8, center–axes correlations, Fig. 22 |
| C6 | Soft prompts work mainly by aligning representations to the readout | weak | late-layer effects, Fig. 9; argued in §5, not tested directly |
Concepts
- Category manifold. The point cloud of embeddings of all sentences with one label.
- Manifold capacity. “The number of linearly decodable classes per dimension”: P / D*, where D* is the critical projection dimension.
- Readout alignment. How well the model’s unembedding uses the separability present in the last-token embedding.
- Coherent / incoherent. Whether the labels used to form the manifolds are the ones the prompt asks about.
Connections
It uses Chung et al.’s manifold capacity (2018) and Chou et al.’s correlated extension (2024). Its label-shuffling result echoes Min et al. (LIT-814), that correct input–label pairs matter less than the input distribution, and Pan et al.’s split of task recognition from task learning. Davidson et al. (LIT-942, read in NOTE-803) reach the same verdict on demonstrations against instructions by a causal method.
Bearing on the record
- QUESTION-024. The LIT registered this paper as “evidence that prompting methods with matched performance adapt a model through distinct representational mechanisms. That counts against a single frame induced from examples.” The first sentence is right. The second goes past the paper. It holds one instruction wording fixed and compares prompt forms; it does not vary the wording of an induced frame and ask whether the effect survives, which is what QUESTION-024 asks. What it does show that bears on the question: demonstrations change how the input itself is represented, and the change comes from the example sentences, not from their labels (C3). That is a sense in which something is induced from examples rather than told.
- The triage’s reading of the pair. The curation entry of 2026-10-14 says this paper and LIT-942 “point opposite ways on whether one induced frame is shared”. They point the same way: neither finds a mechanism common to demonstrations and instructions.
- CLAIM-029. Its “toward the defeater” item cites Lu et al.: order alone moves accuracy from near chance to near the state of the art, so the “frame” is sensitive to surface form. C4 here splits that finding. Across samplings of examples, the representation is nearly unchanged and the readout is what moves. If the frame is the representation the context sets, this paper says it is more stable than the accuracy suggests. Order was not varied here, only which examples were drawn, so it does not settle Lu et al.’s case.
- CLAIM-118. The geometry–readout split is a third axis beside context and interpreter: a context can change what is decodable without changing what the model’s own decoder reads.
- Files THEORY-251. C1–C2 are held, with LIT-942, by THEORY-264.
- The discussion proposes geometric optimisation of prompts and trained linear readouts. That is practice for the anthology and is not carried here.
Limitations
- Synthetic data, written by another language model; the open datasets replicate the single-task results only.
- Base models only, so “instruction” is a prompt format, not the instruction-following that post-training builds.
- One instruction wording, three seeds, and no statistical tests reported.
- Simplified geometry. Dimension and radius are not the anchor-point quantities in which capacity is analytically expressed.
- Readout alignment is inferred, not measured. The paper proposes, but does not run, the comparison of trained linear classifiers with the unembedding.
Open questions
- Would a trained linear readout on the “failed” few-shot runs recover the accuracy, as C4 predicts?
- Does instruction tuning change where an instruction acts?
- Does the readout account cover order effects as well as example choice?