{"id":"cf541a9a-1c27-4771-9d4a-60b5b0d55897","arxiv_id":"2506.07824","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In LLaMA-3-8B-Instruct, addition information becomes linearly decodable in a fixed layer-wise order: formula structure, carries and sums, result digits, then output generation.","lead":"This study dissects how LLaMA-3-8B-Instruct solves multi-digit addition by decoding its hidden states at every layer. It identifies a consistent four-stage sequence: recognizing the expression structure, computing carries and sums, forming abstract digit representations, and finally generating the output token.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Probe-onset order is read as a processing timeline without controls for depth-dependent separability, leaving the four-stage trajectory as a descriptive finding rather than evidence of computation.","rationale":"Good faith reading: the paper is a careful descriptive probing study; the release of code and data is real support, and the stage-wise accuracy curves are reproducible. The descriptive finding—that different arithmetic attributes reach high linear decodability from the last-token state at different depths—is credible. The concern is strictly about the interpretive step from those curves to a 'computation-like process' over memorization. The controls suggested above would distinguish a general depth trend from an addition-specific trajectory. The paper's Limitations section already concedes the causal gap, so the conditional verdict is appropriate; the central claim should be reframed as a descriptive ordering unless control and causal evidence are added. I agree with the reader that the load-bearing assumption is the equation of probe-onset depth with processing order; the threshold and readout-position issues are facets of the same gap. One further internal flag: the Appendix D.2 captions note a data anomaly for the 'a+b=500' slice ('correct samples are only ~10% of the total'), which is not reconciled with the main-text sum-range results and should be clarified before the core-computation stage is treated as established. I recommend UNCHANGED, not because the concern is minor, but because the reader's verdict already conditions acceptance on addressing it.","tokens_in":16268,"tokens_out":11146,"duration_ms":149040,"concrete_test":"Run the identical layer-wise probe pipeline (§3.3) on the same LLaMA-3-8B-Instruct hidden states with two controls: (i) random labels generated by a fixed permutation of each probe's class labels (null expectation: chance at every layer), and (ii) a matched 10-way control label that is prompt-determined but arithmetic-irrelevant, e.g., the hundreds digit of the first addend, trained and evaluated with the same splits. If control (ii) becomes linearly decodable at the same or later layers than the result-digit probes, the late digit onset is not addition-specific.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central inference—that the layer at which each linear probe first reaches high accuracy (Section 4) reflects the order in which the model computes addition-related information (Section 7)—is not secured. The compared probe tasks differ in class count, baseline accuracy, and intrinsic difficulty, and the paper does not define an onset criterion or include any control probe. It is therefore possible that deeper last-token states simply become more linearly separable for any sufficiently complex prompt feature, or that the observed sequence tracks how long information takes to aggregate at the final token position, not when the model computes it. The paper's own Limitations section states that decodability does not imply necessity and that causal interventions are required, yet the conclusion still calls the observations 'strong evidence' for a computation-like process. This gap is load-bearing: if the ordering is an artifact of task difficulty or readout position, the four-stage trajectory does not distinguish computation from memorization, and the central claim reduces to a restatement of probe accuracy curves.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multi-digit addition in LLaMA-3-8B-Instruct by training linear probes on the last-token hidden state at every layer and by applying the logit lens. It reports a consistent layer-wise ordering of linearly decodable information: formula structure first, then sum-range and carry information, then individual result digits, and finally output-aligned top-1 commitment. The authors interpret this ordering as a four-stage information-processing trajectory and conclude that it supports a computation-like process over rote memorization. They also report similar curves for several 7B-scale models and release code and data.","tokens_in":16435,"tokens_out":5157,"duration_ms":64890,"significance":"The descriptive part of the paper is solid: probes are trained on a frozen model, splits are disjoint, results are averaged over five seeds, and the formula-structure probe is evaluated out-of-distribution across digit lengths. The cross-operation transfer of the hundreds-digit probe is a useful sanity check, and the release of code and data supports reproducibility. However, the paper's central significance depends on reading the layer-wise ordering of probe accuracy as a processing trajectory. That reading is not yet secured: the compared probe families differ in class count, baseline accuracy, and intrinsic difficulty; no control probes calibrate how linear separability of last-token states grows with depth per se; and the primary model was selected for showing the clearest four-stage ordering. The work therefore provides a useful descriptive map of where addition-related information is linearly accessible, but the claimed support for a computation-like process over memorization remains an interpretation rather than a demonstrated result.","major_comments":[{"comment":"The central inference that the layer at which a probe saturates reveals the order in which the model computes addition-related information is not operationalized or controlled. The four probe families are not commensurable: formula-structure probes are 3-way with chance 0.33, carry probes are binary with chance 0.50, sum-range and digit probes are 10-way with chance 0.10, and the logit-lens 'first top-1' metric has no chance baseline at all. No onset threshold is defined for 'near-perfect and stable' in §3.4, and no control probe (e.g., an unrelated prompt attribute or random-label probes) measures how linear separability at the last-token position grows with depth independently of task content. The observed ordering could therefore reflect task difficulty or the gradual aggregation of input information at the final token position rather than a computation trajectory. This is load-bearing because §7 uses the ordering as 'strong evidence' for a computation-like process; the paper should either provide matched control tasks and a quantitative onset criterion or substantially weaken that conclusion.","section":"§3.4, §4, §5"},{"comment":"LLaMA-3-8B-Instruct was selected because it exhibited the clearest four-stage ordering (Section 3.1), yet Section 7 claims the trajectory is robust across architectures. Appendix D does not fully support this generalization: for Qwen2.5-Math-7B and AceMath, the first-correct-token layer is reported at layer 19 for essentially all samples (Figs. 25–26), and the sum-range probes reach only low generalization accuracy for Qwen (GenAcc 0.20–0.37 in Fig. 13). The apparent cross-model consistency should be established with the same onset criterion applied to all models, or the generalization claim should be restricted to the primary model.","section":"§3.1, §7, Appendix D"},{"comment":"The Conclusion's characterization of the observations as 'strong evidence' for a computation-like process over rote memorization exceeds what the correlational design can establish, and the Limitations section explicitly concedes that decodability does not imply necessity and that causal interventions are required. Since the ordered probe decodability is the only evidence offered for the computation-first interpretation, the conclusion should be reframed as a descriptive layer-wise map of linearly decodable information, with the computation-first reading presented as a hypothesis requiring causal verification.","section":"§7, Limitations"}],"minor_comments":[{"comment":"The caption says accuracy plateaus in 'early-to-mid layers' while the main text says 'mid-to-late layers'; these should be reconciled.","section":"Figure 3 caption vs. §4.1"},{"comment":"Table 1 reports 98.18% overall accuracy for LLaMA-3-8B-Instruct while Table 2 reports 97.29% for the same model; the discrepancy is not explained.","section":"Tables 1 and 2"},{"comment":"Appendix C.2 says the task is to 'predict the exact sum; each class is one sum value', while §4.2 describes a 10-way bin task such as '500-509'; the task definition and the relationship between classes and sum values should be clarified.","section":"§4.2, Appendix C.2"},{"comment":"The captions for these figures refer to layers 23–32 while the plots show layers 19–28; the captions and axes should be made consistent.","section":"Figures 25–26"},{"comment":"The model name appears as LLAMA, LLaMA, and Llama in different places; the spelling should be standardized.","section":"Throughout"},{"comment":"The line 'Status. This paper has been accepted to EMNLP 2025' is not scientific content and should be removed or moved to a footnote.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The descriptive probing results are reproducible and worth publishing, but the interpretive step from probe-accuracy curves to a computation trajectory needs substantive additional support. A revision that adds control probes (or an equivalent calibration) and tempers the conclusion would be suitable; without that, the central claim is not yet supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful thing here is the descriptive result: in LLaMA-3-8B-Instruct, addition-related signals become linearly decodable in a consistent layer order—formula structure first, then sum range and carry indicators, then individual result digits, then output-aligned top-1 ranking. That integrated ordering is not explicitly in prior work, and the paper documents it cleanly with 5-seed probes, disjoint splits, OOD tests, and a code/data release. The logit-lens observation that the gold token reaches top-1 only in layers 23–32 is a nice confirmation of the late-informativeness point, and the cross-operation generalization (addition-trained probe transfers to subtraction and, less strongly, to multiplication) is a genuinely interesting empirical addition.\n\nThe paper's own Limitations section is admirably candid: probes are correlational, decodability does not imply use, and causal interventions are left to future work. But the conclusion still calls the observations \"strong evidence\" for a computation-like process, and that is an overclaim. The stress-test concern lands: the four probe families differ in class count, baseline accuracy, and intrinsic difficulty, and there is no control probe (e.g., an unrelated feature or random labels) to rule out that deeper last-token states simply become more linearly separable for any sufficiently complex attribute. The model selection also cuts against the cross-model claim—LLaMA-3-8B-Instruct was chosen because it showed the clearest ordering, and the appendix shows other models with varied onset patterns.\n\nNone of this makes the four-stage trajectory false. As a description of where certain attributes become linearly readable, the data support it. But the computation-versus-memorization conclusion needs causal evidence—patching, ablation, or circuit analysis—before it can carry that weight. The paper would be stronger if the \"trajectory\" were framed as a footprint consistent with computation rather than evidence for it.\n\nThe related work is well covered, and the citation pattern looks fair. This paper deserves a serious referee: the probing methodology is careful, the findings are reproducible, and the gap between description and causal interpretation is exactly what peer review should push on. I'd engage with it as a descriptive study and ask the authors to either soften the conclusion or add the missing controls. I would not cite it as evidence about internal algorithms, but I would use it as a careful example of layer-wise probe curves and their limits.","headline":"A careful, reproducible probe study with a plausible four-stage layer-wise ordering, but the conclusion overreaches: probe accuracy curves do not by themselves show the model computes rather than memorizes.","tokens_in":16939,"tokens_out":2865,"would_cite":false,"duration_ms":35081,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Multi-digit addition in LLaMA-3-8B-Instruct follows a fixed four-stage layer-wise order, from equation structure to output token.","keywords":["linear probing","logit lens","multi-digit addition","LLM arithmetic","mechanistic interpretability","carry propagation","layer-wise representation","computation vs memorization"],"falsifier":"Train layer-wise linear probes on the same hidden states but with labels that have nothing to do with the arithmetic—for example, random shuffles of the digit classes or unrelated binary labels—and compare the accuracy curves; if any control probe matches the high-accuracy plateaus and the four-stage ordering, the trajectory is an artifact of depth-dependent linear separability rather than a signature of arithmetic computation.","tokens_in":16045,"feed_emoji":"🧮","tokens_out":7911,"duration_ms":79994,"temperature":0.7,"pith_summary":"The paper tries to establish that when LLaMA-3-8B-Instruct solves multi-digit addition, its internal representations pass through four recognizable stages in a fixed order: first the structure of the equation, then the column sums and carries, then the individual digits of the result, and finally the organization of the output token. It matters because arithmetic is a compact battleground for the question of whether LLMs genuinely compute or mostly retrieve memorized patterns; a fixed internal order of operations would support the computation side. The evidence comes from training linear probes on the hidden state of the last prompt token at every layer and from logit-lens inspection of the same states. The paper argues this staging is strong evidence for a computation-like process rather than rote memorization, while carefully limiting the claim to the models, prompt format, and in-distribution base-10 addition tested.","feed_headline":"Probes show LLM addition follows a fixed four-stage order","feed_subtitle":"In LLaMA-3-8B, four signal types surface in a fixed depth order, pointing to internal computation.","key_machinery":"The load-bearing device is the layer-wise linear probe: for each layer state $l \\in \\{0,\\ldots,32\\}$ of LLaMA-3-8B-Instruct, a separate linear classifier is trained on the last-input-token hidden state $h_S^{(l)}$ to predict a task-specific arithmetic attribute (equation type, carry bit, sum bin, or result digit), and its held-out accuracy is plotted against depth. Complementing this, the logit lens projects the same hidden state through the unembedding matrix, $\\ell^{(l)} = h_S^{(l)} W_U^\\top$, to track when the correct next token first becomes top-1. The ordering of the resulting accuracy curves is what carries the argument: each signal family has a characteristic depth at which it becomes linearly accessible, and those depths fall in a consistent sequence across tasks and, qualitatively, across other 7B-scale models.","core_discovery":"The central discovery is a layer-wise ordering of linearly decodable arithmetic information in LLaMA-3-8B-Instruct. Formula-structure signals (whether a prompt is a+b, b+a, or a+a) become decodable first, peaking in early-to-mid layers. Core computational features—carry indicators at each decimal position and the sum-range bin of the answer—emerge next, beginning around layer 14 and saturating around layer 19. Result-level numerical abstractions, namely the ones, tens, and hundreds digits of the sum, become reliably decodable later, saturating after roughly layer 28. Finally, logit-lens inspection shows the correct first answer token reaches rank 1 only in layers 23–32, peaking at layer 30, with no test case failing to reach top-1. The paper interprets this sequence as evidence that the model performs a structured, computation-like process rather than surface pattern matching, and further shows that a hundreds-digit probe trained on addition transfers to subtraction (~0.9 peak accuracy) and multiplication (~0.8).","pith_inferences":["The paper's ordering is correlational; a natural extension would be causal interventions—patching or ablating the carry or sum-range representations at mid layers to see whether late-layer digit decoding and final answers depend on them.","If control probes with non-arithmetic labels match the same layer-wise accuracy curves, the four stages would be an artifact of linear separability, not a processing timeline.","The cross-operation transfer result hints at a shared numerical subspace; one could test whether the degree of subspace alignment predicts transfer to division, mixed-base arithmetic, or longer operands.","The fixed ordering could be exploited practically: layer-specific probes could locate where an arithmetic error is introduced, or identify which layers to edit to change a model's arithmetic behavior."],"forward_implications":["In LLaMA-3-8B-Instruct, addition is not a single late readout: structure, carries, sum ranges, digit identities, and output commitment become decodable at different, ordered depths.","The logit lens is most informative only in the final third of the network because the correct next token first reaches top-1 in layers 23–32, peaking at layer 30.","Carry and sum-range information consolidates in mid layers (around L14–L19), before individual result digits become abstracted (after ~L28), implying that the model forms an intermediate result representation before the final digits.","A digit probe trained on addition generalizes to subtraction (~0.9) and multiplication (~0.8), suggesting the model's numerical representations are partly operation-independent.","The qualitative four-stage ordering recurs in other 7B-scale models, so the order of processing may be a general feature of these LLMs even if exact layer indices vary."],"supporting_citations":[{"why":"It supplies the linear-probing method used to build every layer-wise classifier.","marker":"Belinkov, 2022"},{"why":"It introduces the logit lens used to track when the gold next token becomes top-1.","marker":"nostalgebraist, 2020"},{"why":"It defines LLaMA-3-8B-Instruct, the model whose hidden states are probed throughout.","marker":"Grattafiori et al., 2024"},{"why":"It provides the prior finding of component specialization in arithmetic that motivates the layer-wise analysis.","marker":"Yu and Ananiadou, 2024"},{"why":"It states the opposing 'bag of heuristics' account that the four-stage ordering is meant to answer.","marker":"Nikankin et al., 2024"},{"why":"It shows that Transformer-scale models can learn carry propagation, making the staged computation plausible.","marker":"Kruthoff, 2024"},{"why":"It documents position-specific digit encodings, which aligns with the late digit-abstraction stage.","marker":"Levy and Geva, 2025"},{"why":"It reports that digit values are linearly recoverable, justifying the use of linear probes for numeric attributes.","marker":"Zhu et al., 2025"}],"fun_headline_variants":["LLM addition: formula, carry, digits, token in fixed order","Four stages: How LLMs compute addition internally","LLaMA-3-8B addition: a fixed four-stage depth trajectory","Probes map LLM addition: from structure to answer","Addition in LLMs: hierarchical computation, not memorization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the assumption that the layer at which a probe first decodes a piece of information tells us when the model is actually processing that information; if hidden states simply become more linearly separable with depth for reasons unrelated to arithmetic, the observed ordering would not describe a processing trajectory.","fun_headline_variants_meta":{"raw":{"variants":["LLM addition: formula, carry, digits, token in fixed order","Four stages: How LLMs compute addition internally","LLaMA-3-8B addition: a fixed four-stage depth trajectory","Probes map LLM addition: from structure to answer","Addition in LLMs: hierarchical computation, not memorization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000886,"raw_usage":{"total_tokens":3815,"prompt_tokens":922,"completion_tokens":2893,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":2806}},"tokens_in":538,"tokens_out":2893,"duration_ms":25264,"temperature":1.0,"reasoning_tokens":2806,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:24:17.081815+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train layer-wise linear probes on the same hidden states but with labels that have nothing to do with the arithmetic—for example, random shuffles of the digit classes or unrelated binary labels—and compare the accuracy curves; if any control probe matches the high-accuracy plateaus and the four-stage ordering, the trajectory is an artifact of depth-dependent linear separability rather than a signature of arithmetic computation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the prior finding of component specialization in arithmetic that motivates the layer-wise analysis."},{"cited_title":"Carrying over algorithm in transformers","cited_arxiv_id":"2401.07993","evidence_quote":"It shows that Transformer-scale models can learn carry propagation, making the staged computation plausible."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It reports that digit values are linearly recoverable, justifying the use of linear probes for numeric attributes."}],"review_version":1}