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REVIEW 3 major objections 6 minor 63 references

Probing for Arithmetic Errors in Language Models

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A language model's arithmetic errors show up in its internal activations before the answer is produced: simple probes decode the intended digit and the correct digit, and their disagreement flags the mistake.

desk verdict Useful, mostly careful probing study whose central claim that the correct answer is internally encoded is undercut by an unaddressed confound, but the error-detection and self-correction results survive. read the letter →

arxiv 2507.12379 v1 pith:AUD7NZHT submitted 2025-07-16 cs.CL cs.AI

Yucheng Sun , Alessandro Stolfo , Mrinmaya Sachan This is my paper
classification cs.CLcs.AI
keywords probingarithmeticerrordetectionlanguagemodelinterpretabilityinternalactivationschain-of-thoughtreasoningcircularrepresentationsself-correctionGSM8K
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether a language model's internal activations expose its arithmetic errors before it commits to an answer. In a controlled setting of 3-digit addition, the authors show that simple probes, most notably a circular probe that reads a digit from the angle a hidden state makes on a plane, can decode both the digit the model intends to output and the digit of the correct sum, each with accuracy above 90% in the deepest layers. Because the two readings disagree precisely when the model is wrong, an error detector built on them predicts model correctness with over 90% accuracy on a balanced dataset. The same probes, trained only on isolated sums, transfer without any retraining to structured chain-of-thought steps in GSM8K addition problems, keeping roughly 85% accuracy, which the authors take as evidence that digit representations are consistent across contexts. The paper closes by using the probes as weak oracles that flag erroneous reasoning steps for selective re-prompting, correcting up to 11.8% of flagged errors while preserving essentially all correct steps, and concludes that arithmetic errors can be anticipated from internal activations alone.

What carries the argument

The carrying object is the circular probe: a pair of weight vectors $(w_1, w_2)$ that project the residual-stream hidden state $x_l$ at the equals-sign token onto a plane, with the probe's digit output read off the angle $\theta = \operatorname{atan2}(w_1^\top x_l, w_2^\top x_l)$ scaled as $\hat{y} = \theta \cdot 10 / (2\pi)$, trained with a smooth $\ell^1$ loss. The angle acts as a clock face for the digit value, which is why a probe matched to circular geometry succeeds where a linear probe fails; logistic and MLP probes serve as nonlinear baselines that confirm the same information is accessible. Error detection is carried by running two such readings side by side: one probe trained on the model's predicted digit, one on the ground-truth digit, with the angular difference between them, passed through a sigmoid or compared directly, serving as the correctness signal. All probes read activations at the equals sign, the token position immediately before the model produces its output, which prior work identified as carrying the computed result.

What would settle it

Re-run the cross-setting transfer with the label distributions matched: train the pure-arithmetic probes with supervision covering all ten digits 0 through 9, or restrict the GSM8K evaluation to steps whose predicted and true hundreds digits both lie in the original 2 through 9 range. If the reported ~85% transfer accuracy collapses toward the 50% majority baseline once the ranges match, the transfer result is an artifact of the label-range mismatch rather than evidence of consistent internal representations.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that at the equals sign of an addition computation, a language model's residual stream encodes two distinct quantities even when the model's output is wrong: the digit it is about to produce and the digit of the correct sum, and that simple probes can extract both. A probe matched to the observed geometry, a circular probe that projects the hidden state onto two directions and reads the resulting angle as a digit, recovers each with more than 90% accuracy at deeper layers, while a purely linear probe fails, and the probes' outputs become geometrically structured, clustered and then circularly arranged, as depth increases. A control experiment decoding the input operands at an early layer shows the probes are not merely solving the arithmetic themselves; instead, the authors argue, the model internally refines partial information about the correct result that its output head sometimes fails to decode, in line with prior evidence that models know more than they show. The central practical claim is that error detectors built from the disagreement of the two readings generalize from isolated sums to structured chain-of-thought steps on GSM8K without retraining, reaching about 85% accuracy, and can act as weak oracles that flag individual steps for selective re-prompting, correcting up to 11.8% of flagged errors with essentially no disruption to correct steps. The paper's overarching claim is that arithmetic errors can be anticipated from internal activations alone, making simple probes a viable lightweight path toward model self-correction.

Load-bearing premise

The roughly 85% cross-setting transfer, probes trained on isolated 3-digit sums applied without retraining to chain-of-thought steps, is only evidence of consistent internal representations if the mismatch between the probes' training labels (hundreds digits 2 through 9) and the chain-of-thought sums' leading digits (which can be 1) is not what the probes are exploiting.

Editorial extensions

If this is right

  • Arithmetic errors become predictable before the answer token is generated, because the disagreement between the internally decoded prediction and the internally decoded ground truth is already present in the residual stream at the equals sign.
  • Error detection at over 90% accuracy costs only a few weight vectors per layer, so monitoring arithmetic correctness is feasible as a lightweight, side-by-side tool rather than a full classifier.
  • Probes trained on isolated sums transfer to chain-of-thought steps without retraining, implying a single reusable probe set could monitor multi-step arithmetic reasoning across tasks.
  • Selective re-prompting guided by the probes corrects up to 11.8% of flagged wrong steps while preserving all correct ones in most prompt configurations, offering a low-risk intervention for improving task accuracy.
  • The appendix results extend the pattern to subtraction, operands up to five digits, free-form chain-of-thought traces, and a second model (Phi-3), suggesting the mechanism is not specific to the main test bed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The angular margin of the probe outputs, how far the predicted digit's angle sits from the ground-truth angle, is a natural graded confidence score for each reasoning step; the paper uses only a binary disagreement, but thresholding on that margin could rank steps for revision rather than simply flag them.
  • Because the two probes read the hidden state with a handful of vectors, the same reading could be applied during decoding as a steering signal, nudging the residual stream so the model-prediction angle rotates toward the ground-truth angle, an intervention the paper does not test.
  • The paired-probe logic, decode predicted value, decode ground-truth value, compare, generalizes in principle to any task with a checkable ground truth and a structured internal encoding, such as factual recall or date arithmetic, though the paper only demonstrates it for arithmetic.
  • The free-form chain-of-thought result reported in Appendix H.3, with the best detector near 72% accuracy, suggests the structured <a+b=c> format helps but is not required; an ablation measuring how much accuracy is lost per unit of format freedom would tell practitioners whether enforcing structure is worth the cost.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper investigates whether residual-stream activations of an instruction-tuned language model encode enough information to detect arithmetic errors. In a controlled 3-digit addition setting (Gemma 2 2B IT, with a Phi-3 replication), the authors report that circular, logistic, and MLP probes reach roughly 92% accuracy in decoding both the model's predicted digit and the ground-truth digit, and that error detectors combining these signals exceed 90% accuracy on a balanced dataset. They then apply probes trained in the pure arithmetic setting to structured chain-of-thought steps from an addition-only GSM8K subset, reporting about 85% transfer accuracy. Finally, they use an MLP error detector as a weak oracle to select steps for re-prompting, reporting correction of up to 11.8% of true-positive errors with near-perfect preservation of correct steps.

Significance. If the results hold, the paper makes a useful practical and scientific contribution: it demonstrates that lightweight probes can serve as cheap correctness monitors for arithmetic reasoning, and it provides evidence for cross-context consistency in how numerical information is encoded in LLM activations. The controlled setting, balanced datasets, operand-decoding control, multiple probe architectures, and replication on a second model are strengths, and the detailed appendices make the experimental protocol unusually transparent. The main scientific interpretation, however, depends on discriminating between an internally represented correct answer and a probe that learns to correct the model's predicted digit from the correlation structure of the training set; that discrimination is not yet established. Similarly, the cross-setting transfer claim needs a precise statement of label ranges and per-digit behavior. The paper's practical error-detection results are more robust than its representational interpretation; with additional controls and a re-analysis of Section 4.4, the contribution would be solid.

major comments (3)
  1. [§3.2, Appendix B] The ground-truth probing result in §3.2 does not yet rule out a learned correction of the predicted digit. Appendix B balances the dataset by the model's predicted hundreds digit and by correctness, but it does not balance by ground-truth digit conditional on predicted digit. Since every correct example has ground-truth digit equal to predicted digit, a probe can reach high accuracy by decoding the predicted digit (which §3.1 shows is highly decodable) and then applying a correction for the systematic errors in the incorrect examples. The operand-decoding control in Table 1 only tests whether operand information is available at layer 5; it does not test whether the ground-truth probe is exploiting the predicted-output/ground-truth correlation that is present by construction. I request an additional control: train the ground-truth probe on data balanced by ground-truth digit within each predicted-digit class, or report ground-truth probe accuracy separately for correct and incorrect examples, and compare against a baseline that post-processes the predicted-digit probe with a learned correction. Without this, the abstract's wording that the correct answer is decoded from hidden states goes beyond what is established, and the §4.4 transfer claim inherits the same ambiguity.
  2. [§4.4, Appendices B and D] The cross-setting transfer experiment is under-specified with respect to label ranges. The pure arithmetic probes are trained on sums below 1000 and on hundreds digits in {2,...,9} (Appendix B), whereas the CoT steps in Appendix D sample operands in [100,999] and can produce 4-digit results (the example in Appendix D includes 771+611=1382). The paper never states whether the CoT target is the leading digit or the hundreds digit of each step result, nor how a probe with label set {2,...,9} is applied when the relevant digit is 1 or 0. If such values occur in the 685-step evaluation set, the ~85% transfer accuracy in Figure 5 could reflect nearest-label behavior rather than consistent internal representations. Please specify the exact label mapping, report the label distribution of the CoT evaluation set, and give per-digit transfer accuracies for the affected digits.
  3. [§5, Appendix E] The self-correction section reports TP Correction and FP Preservation but does not report end-to-end task accuracy before and after re-prompting, nor the detector's precision/recall at the operating point used to obtain 178 true positives and 22 false positives. Since the abstract promises 'improving task accuracy with minimal disruption,' an accuracy comparison on the same 685-step set (or per-problem accuracy) is needed to support that claim; as written, correcting 11.8% of flagged errors could have a negligible or negative effect on overall task accuracy once the 22 false positives and any new errors from re-prompting are accounted for.
minor comments (6)
  1. [§3.3] In §3.3, the text refers to 'Section 4.1' and 'Section 4.2' when describing probes for the model output and ground truth; these should be §3.1 and §3.2.
  2. [§3.1, Appendix B] The training description says '10,000 epochs' for a dataset of 800 examples; please clarify whether this means optimization steps and describe early stopping or validation, since overfitting could inflate the reported test accuracies.
  3. [§2.2, Eqs. (1)-(2)] The circular probe's loss is not circular: predictions near 9 and near 0 for the same angle are treated as far apart; a circular loss or explicit angle wrapping would better match the stated geometric motivation.
  4. [Appendix H.3, Tables 3 and 4] Tables 3 and 4 report 'maximum accuracy across all layers' without the layer index or the layer-selection rule, which makes the free-form CoT results difficult to reproduce.
  5. [Figure 7 caption] The Figure 7 caption contains a typo: 'on an wider operand range' should read 'on a wider operand range.'
  6. [Table 2] Table 2 reports FP Preservation of 100.00% for most prompts, but the raw counts are small (22 false positives); reporting counts per prompt or confidence intervals would help interpret these numbers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all headline results are supervised empirical findings, not derivations from fitted constants or self-citation chains.

full rationale

The paper's claimed derivation chain is empirically self-contained rather than circular. The digit probes are supervised classifiers trained on external labels—the model's emitted hundreds digit or the arithmetic ground truth—and are evaluated on held-out splits, so no fitted parameter is renamed as a prediction. The error detectors in §3.3 and §4.3 are either supervised binary classifiers or a post hoc disagreement rule between two independently trained probes; equation (8) is a comparison, not a construction that forces the reported accuracy. The cross-setting generalization result (§4.4) is an actual transfer evaluation: probes trained on simple arithmetic are applied without further training to CoT activations and measured against ground-truth labels, so the ~85% figure is not implied by training. The circular-probe architecture is motivated by an unsupervised PCA visualization, but the paper also tests linear, logistic, and MLP probes, so the main conclusion does not rest on a self-fulfilling ansatz. Self-citations (e.g., Stolfo et al. 2023a for the equals-sign position, Stolfo et al. 2023b and Opedal et al. 2024 for dataset abstraction) are contextual rather than load-bearing, and the cited works are themselves empirical. The skeptical concern that the ground-truth probe may exploit the predicted-output/ground-truth correlation is a validity or confounding risk, not a circular reduction: the probe is trained on true digits, and the authors themselves concede in §3.2 that the evidence 'does not fully justify the stronger claim that probes recover a clean representation of the correct answer.' Similarly, the potential CoT label-range mismatch (hundreds digit 1 falling outside the 2–9 training classes) would be a data-distribution issue, not a circularity. No circular step can be exhibited from the paper's equations or citations. Hence score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a few domain assumptions about where arithmetic information lives in the residual stream and what the balanced accuracy numbers mean, plus three hand-selected experimental choices (layer, prompt, epoch count). No new entities are posited.

free parameters (3)
  • Self-correction detector layer = 25
    The best-performing error detector was selected by evaluating all 26 layers on the same evaluation data; this post hoc choice is reported without variance or error bars.
  • Correction prompt wording = "That step looks suspicious. Let's re-do just this step:"
    The 'suspicious' prompt achieved the highest true-positive correction (11.8%) among five manually designed messages; the other prompts performed worse, so the headline result depends on this selection.
  • Probe training epochs = 10000
    All probes are trained for 10,000 epochs on datasets as small as 560 training samples; the effect of early stopping or fewer epochs is not reported.
assumptions (4)
  • domain assumption The residual stream activation at the equals-sign token encodes the computed arithmetic result
    This is the basis for all probing experiments, inherited from prior work (Stolfo et al., 2023; Nikankin et al., 2025) rather than established within this paper.
  • domain assumption The hundreds digit is a sufficient target for detecting arithmetic errors
    Appendix F verifies this for pure arithmetic (~100% of errors involve the hundreds digit), but the same assumption is applied to CoT steps where sums can have a leading digit of 1, without re-verification.
  • domain assumption Balanced sampling of correct and incorrect examples gives a meaningful measure of detector performance
    Error detectors are reported on a 50/50 balanced dataset, while the natural error rate is about 1%; precision and recall in deployment are not directly characterized.
  • domain assumption Decoding the ground-truth digit implies the model internally represents the correct answer rather than the probe computing from operands
    The paper tests this with an early-layer operand-decoding control, but the interpretation that the residual stream contains a latent correct answer is not conclusively established.

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Cite this review

Pith. "Pith review of Probing for Arithmetic Errors in Language Models." pith.science (2026). https://pith.science/paper/AUD7NZHT

@misc{pith2026250712379,
  author       = {Pith},
  title        = {Pith review of: Probing for Arithmetic Errors in Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUD7NZHT}},
  note         = {Machine review of arXiv:2507.12379}
}
read the original abstract

We investigate whether internal activations in language models can be used to detect arithmetic errors. Starting with a controlled setting of 3-digit addition, we show that simple probes can accurately decode both the model's predicted output and the correct answer from hidden states, regardless of whether the model's output is correct. Building on this, we train lightweight error detectors that predict model correctness with over 90% accuracy. We then extend our analysis to structured chain-of-thought traces on addition-only GSM8K problems and find that probes trained on simple arithmetic generalize well to this more complex setting, revealing consistent internal representations. Finally, we demonstrate that these probes can guide selective re-prompting of erroneous reasoning steps, improving task accuracy with minimal disruption to correct outputs. Our findings suggest that arithmetic errors can be anticipated from internal activations alone, and that simple probes offer a viable path toward lightweight model self-correction.

Figures

Figures reproduced from arXiv: 2507.12379 by the authors.

Figure 1
Figure 1. Detecting Arithmetic Errors from Hidden States. We investigate whether internal activations in a language model reveal when its arithmetic predictions are incorrect. We train simple probes to decode both the model’s output and the correct answer. The probes’ output can serve as a reliable signal of model error. and Tegmark, 2025; Zhu et al., 2025). However, despite increasing insight into the structure of these repr… view at source ↗
Figure 2
Figure 2. PCA of Residual Stream Activations. PC projections colored by the hundreds digit of the ground￾truth result. Representations become more structured with depth, showing clear digit clusters and a circular layout in deeper layers. digit encodings at deeper layers, we treat circular probing as a natural baseline. However, we also explore alternative probing approaches to evaluate different ways of extracting numerical … view at source ↗
Figure 3
Figure 3. Probing 3-Digit Arithmetic Queries. (a) Probes recover the model’s output with high accuracy in deeper layers; linear probes perform poorly. (b) Ground-truth digits are similarly decodable, suggesting correct answers are often internally represented. (c) Error detectors show that model correctness can be inferred from hidden states. The dashed lines indicate the accuracy of the majority class baseline. 3 Probing 3-D… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Probing Structured Chain-of-Thought Reasoning. (a) Probes recover the model’s predicted digit with increasing accuracy across layers; non-linear probes reach over 90% in the final layers. (b) Ground-truth digits remain decodable, though slightly less accurately than in…
Figure 5
Figure 5. Figure 5: Cross-Setting Error Detection. Accuracy of probes trained on simple arithmetic queries evaluated on GSM8K problems. Probes generalize well to the structured CoT setting, reaching up to 85% accuracy and indicating consistent internal representations. 4.4 Do Probes Gener…
Figure 6
Figure 6. Figure 6: Probing 3-Digit Arithmetic Queries (Subtraction). Probing and error detection results on subtraction tasks. As with addition, probes reliably decode both model predictions (a) and ground-truth answers (b), and error detectors achieve high accuracy in predicting correct…
Figure 7
Figure 7. Figure 7: Probing 3-Digit Arithmetic Queries (Wider Operand Range). Probing and error detection results on an wider operand range. As with addition, probes reliably decode both model predictions (a) and ground-truth answers (b), and error detectors achieve high accuracy in predi…
Figure 8
Figure 8. Figure 8: Probing 3-Digit Arithmetic Queries (Phi-3). Probing and error detection results on Phi-3. As with Gemma 2 2B IT, probes reliably decode both model predictions (a) and ground-truth answers (b), and error detectors achieve high accuracy in predicting correctness (c), con…
Figure 9
Figure 9. Figure 9: Probing Structured Chain-of-Thought Reasoning (Phi-3). (a) Probes accurately recover the model’s prediction in deeper layers. (b) Ground-truth digits are similarly decodable. (c) Error detectors achieve strong performance, confirming that findings generalize to structu…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.