From Decorative to Load-Bearing: Task Difficulty Shapes the Causal Role of Chain-of-Thought

1 Introduction

• CoT load-bearingness transitions from decorative to load-bearing as model-relative task difficulty rises, with base task accuracy defining that difficulty.The paper limits its claim to the CoT → answer direction, not the inverse computation → CoT direction.Source: §1 · p. 2
• The authors introduce continuation-based causal testing: perturb one intermediate step, truncate the chain, and force continuation from the corrupted prefix.This isolates the CoT-to-answer link without allowing a separate turn to re-derive from the original question; the setup also proxies cases where a model receives a step it did not author.Source: §1 · p. 2
• 98.8% of explained deviance is attributed to task difficulty versus 0.8% to perturbation type in the matched analysis of 28,584 continuations.Across matched conditions, error propagation ranges from 6.5% for text perturbations on GSM8K to 64.5% for numerical perturbations on BBH multistep arithmetic.Source: §1 · p. 2
• Hidden-state probes predict behavioral type, but their reported accuracy is bounded by label quality and they do not provide mechanistic explanations.Switching from rule-based to LLM-judge labels shifts 3-class accuracy by about 25 percentage points at fixed features.Source: §1 · p. 2
• Additive steering offers limited control: the eight-direction extension flips roughly 25% of Type C cases at its strongest setting, leaves Type A at null, and is underpowered for Type B.Single-direction steering did not flip behavioral type at any tested layer or strength.Source: §1 · pp. 2–3

2.1 Behavioral Studies of CoT Faithfulness

• Prior behavioral studies document rationalization, unacknowledged hint reliance, and unfaithful reasoning; early answering and step removal test whether a model needs its CoT, a complementary question to whether a wrong CoT constrains continuation.The cited work also identifies limits to guaranteeing faithful CoT in general.Source: §2.1 · p. 3

2.2 Mechanistic Interpretability of Reasoning

• Mechanistic studies connect activations to computation or identify CoT-related features, while circuit tracing remains labor-intensive and limited to individual examples.This paper trades circuit-tracing granularity for scale and adapts probe-direction steering, which prior work used to elicit truthful answers.Source: §2.2 · p. 3

2.3 Self-Correction in Language Models

• Unlike prompted review or self-consistency, the paper studies mid-generation self-correction: fixing an injected error while continuing from a corrupted prefix, without a follow-up prompt or resampling.Under the production judge prompt, this occurs in 27% of perturbed continuations.Source: §2.3 · p. 3

3.1 Continuation-Based Causal Testing

• Perturbing a step, truncating the chain, and forcing continuation yields three outcomes: silent bypass and self-correction can preserve the correct answer, while error propagation produces a wrong answer.Forced continuation avoids the fresh-turn re-derivation confound; perturbation type, position, and strength are controlled axes.Source: §3.1 · pp. 3–4, Figure 2 · p. 4

3.2 Perturbation Strategies

• Seven strategies cover mathematical perturbations on GSM8K and conceptual perturbations on MMLU and BBH, with conceptual strategies also used on GSM8K in the matched-cell experiment.Strategies are applied at early, middle, and late chain positions; the reported pattern is that task difficulty dominates perturbation type.Source: §3.2 · p. 5

3.3 Three-Mode Behavioral Classification

• Type A silently bypasses the perturbation, Type B explicitly detects and corrects it, and Type C follows the perturbed step to an incorrect answer.The paper treats C-vs-non-C as the more stable label dimension because attribution of a wrong answer to uptake of the perturbed step can be ambiguous.Source: §3.3 · p. 5
• An LLM judge replaced keyword rules that misclassified correction language and missed verbal-marker-free computational shortcuts; four prompt variants shift the A/B boundary while C rates remain invariant.The n = 500 blind annotation study found perfect inter-annotator agreement on C-vs-non-C, while a BBH calibration disagreement affects within-BBH A-vs-C decomposition but not the gradient direction.Source: §3.3 · pp. 5–6

3.4 Hidden-State Probes

• Hidden-state probes use activations from three token positions across 42 Gemma layers, reduce the concatenated representation from 10,752 to 128 dimensions, and predict behavioral labels under grouped five-fold cross-validation.The probe tasks include three-class classification, bypass detection, and error-propagation detection.Source: §3.4 · p. 6

4 Experimental Setup

• Gemma-2-9B-IT is evaluated on GSM8K, MMLU, and BBH; filtering for correct answers with at least three reasoning steps yields 7,691 baselines and 21,238 labeled continuations.The datasets comprise 879 GSM8K questions, 5,000 MMLU questions, and 5,511 BBH questions; baseline accuracies are 86.5%, 74.8%, and 57.9%, respectively.Source: §4 · p. 6

5.1 Behavioral Taxonomy

• Error propagation rises with task difficulty, from 7.5% on high-school psychology to 53.7% on global facts, while GSM8K shows 94.5% silent bypass.The raw cross-dataset comparison is confounded because GSM8K alone uses numerical perturbations; matched contrasts support the difficulty effect.Source: §5.1 · p. 7, §5.2 · p. 7

5.2 Disentangling Perturbation Type from Task Difficulty

• Difficulty accounts for 98.8% of explained deviance versus 0.8% for perturbation type; error propagation reaches 64.5% on BBH multistep arithmetic, 16× the GSM8K rate.The 0.4% interaction is small but real, with numerical perturbations especially engaging arithmetic computation.Source: §5.2 · pp. 7–8, Table 2 · p. 8

5.3 Position Gradient

• Error propagation falls from 33.5% for early perturbations to 22.5% for late ones, while remaining steps and task difficulty also contribute.The middle-position dip within the 3–4-remaining-step bin departs from a simple positional gradient; difficulty is the dominant factor.Source: §5.3 · p. 8

5.4 Probe Results

• Across 42 layers, probes reach 79.4% for bypass, 75.3% for three-class behavior, and 86.2% for error propagation, but these are judge-label readouts.At fixed features, 3-class accuracy falls from 99.9% with rule labels to 75.0% with judge labels; the human-labeled pilot also reveals source-dependent performance.Source: §5.4 · pp. 8–9, Figure 4 · p. 9

5.5 Causal Intervention: Detection̸ = Control

• Additive steering only partially moves Type C: the best multi-direction setting flips 24.6%, while Type A remains largely immovable and Type B is underpowered.Single-direction steering does not flip type at tested settings; the stronger Type C result uses an eight-direction probe basis at layer 21.Source: §5.5 · pp. 9–10, Figure 5 · p. 9

5.6 Cross-Model Validation

• The difficulty gradient replicates on Llama, while DeepSeek shows lower error propagation and higher self-correction; its filtered MMLU cell complicates within-model ordering.The cross-model comparison is limited to instruction-tuned models, and DeepSeek’s ≥4-step filter makes cell-by-cell rates incomparable with Gemma’s.Source: §5.6 · pp. 10–11, Table 3 · p. 10

6.1 What the Gradient Measures

• The gradient tracks model-relative difficulty, but base-accuracy measurement conflates intrinsic task difficulty with model competence.The authors interpret load-bearingness as increasing as a model approaches its competence boundary; the within-MMLU spread still may reflect latent factors such as training-data frequency or reasoning style.Source: §6.1 · p. 11
• Bypass makes CoT monitoring uninformative on easy tasks, while error propagation on hard tasks narrows the window for intervention.Self-correction, reported at 27% overall under the production judge prompt, is the more favorable case because a monitor can verify the correction downstream.Source: §6.1 · pp. 11–12

6.2 External Validity and Methodological Lessons

• Correct-baseline filtering and asymmetric dataset conditioning constrain external validity, while the probes identify behavior rather than its cause.The authors say within-population comparisons support the gradient despite conditioning; the steering null applies only to additive interventions at tested layers.Source: §6.2 · p. 12

6.3 Limitations and Future Work

• The study is limited to small models and one-step perturbations, leaving scale and multi-step internally consistent perturbations for future work.A full pipeline and probe sweep at ≥70B exceeded the single-H100 compute budget; causal scrubbing or attention-circuit interventions could test control beyond additive steering.Source: §6.3 · p. 12
• Human consensus agrees with labels on 96.2% of C-vs-non-C rows but 76.0% of three-way rows, with both rules and the production prompt over-counting self-correction.Inter-annotator agreement is κ = 0.93 overall and 1.00 on C-vs-non-C; A/B-dependent results are reported as consistent with a prompt variant.Source: §6.3 · p. 12
• BBH rule-based Type-C labels agree with human judgments on 60% of calibration cases, versus 100% on GSM8K and MMLU.The authors identify the within-MMLU contrast as their cleanest evidence; the matched MMLU–BBH gap depends on BBH-side C labels.Source: §6.3 · p. 12

7 Conclusion

• 98.8% of explained deviance is attributed to task difficulty versus 0.8% to perturbation type across 28,584 continuations.The CoT shifts from decorative to load-bearing as model-relative task difficulty rises; reasoning-specific post-training partially offsets this axis on DeepSeek-R1-Distill.Source: §7 · p. 13
• Hidden-state probes detect behavioral position on the gradient, but tested single- and multi-direction additive steering does not reliably shift it.The authors characterize probe directions as readouts rather than controls within the additive-intervention class tested, and leave richer interventions open.Source: §7 · p. 13

Broader Impact Statement

• Perturbation strategies could inform attacks on CoT-based safety systems, while probes could also flag high-risk continuations before completion.The authors disclose the strategies, arguing defenders benefit from understanding the attack surface, and recommend independent human validation before safety-critical deployment.Source: Broader Impact Statement · p. 13

A Perturbation Examples

• In the apple example, bypass ignores the injected value, self-correction restores the original value, and propagation uses the perturbed total.The perturbed step changes the added apples from 3 to 7 and the total from 8 to 12.Source: A Perturbation Examples · p. 15
• In the mitochondria example, bypass retains the correct answer, self-correction rejects the injected Golgi claim, and propagation follows it.The perturbation asserts that the Golgi apparatus is the primary site of energy production.Source: A Perturbation Examples · p. 15

B BBH Continuation Examples

• BBH examples show error propagation when models follow corrupted ordering or object assignments, and bypass when they compute a correct Boolean result despite a wrong claim.The Boolean example is a difficult A/B boundary case: human annotators and the strict correction prompt classify it as bypass, while the production prompt classifies it as self-correction.Source: B BBH Continuation Examples · p. 16
• Self-correction reaches 66.3% on snarks and 54.4% on movie recommendation, while computational subtasks show near-zero self-correction.On computational subtasks, models either follow the perturbation or compute independently, making the Type A/B boundary harder to adjudicate.Source: B BBH Continuation Examples · p. 16

C LLM Judge Prompt

• The judge distinguishes silent bypass, self-correction, and error propagation by whether the model ignores, corrects, or follows the perturbed reasoning.For implicit correction, using correct values in subsequent computation without verbally flagging the error counts as self-correction.Source: C LLM Judge Prompt · pp. 16–17
• The four judge-prompt variants move the A/B boundary by an order of magnitude while the Type-C rate stays invariant.The passage states that the difficulty gradient depends only on the C-vs-non-C split.Source: C LLM Judge Prompt · p. 17

E Layer-Wise Probe Accuracy

• Error-propagation MLP accuracy peaks at 86.2% at layer 28, while the three-class MLP peaks at 75.3% at layer 23.The bypass linear probe reaches approximately 80% across layers 10–25, with AUROC 0.846.Source: E Layer-Wise Probe Accuracy · p. 17
• Figure 7 reports under-2pp grouped-versus-ungrouped CV gaps, a maximum 1.8pp position-ablation drop, and 15–29pp balanced-subsample margins above majority.These checks summarize probe robustness.Source: Figure 7 · p. 18

• Grouped cross-validation changes probe accuracy by less than 2pp on all tasks, indicating the signal is not a question-level artifact.Source: F Robustness and Ablation Studies · p. 18
• Removing any single feature position drops accuracy by at most 1.8pp, with the signal distributed across the perturbation neighborhood.Source: F Robustness and Ablation Studies · p. 18
• At N = 5,000, probe accuracies remain within 3pp of the N = 2,000 grouped-CV results.Reported results are 74.3% for 3-class MLP, 80.1% for binary bypass MLP, and 87.5% for binary error-propagation MLP.Source: F Robustness and Ablation Studies · p. 18
• On a proportionally stratified subsample, all probes remain 13–29pp above majority baselines despite the default sample's skewed class balance.Source: F Robustness and Ablation Studies · p. 18

• The mean answer-token logit gap is 0.017, and per-layer trajectories are visually indistinguishable, so single-token projections do not separate behavioral types.Source: G Logit Lens Analysis · p. 18

H Causal Intervention Details

• Across α, no behavioral type showed a dose-response: Type A had 0/726 errors (95% exact binomial CI [0.0%, 0.41%]) and Type C 682/726 (95% Wilson CI [92.0%, 95.5%]).The intervals rule out a small-per-α-sample explanation (n = 66); probe and random directions produced equivalent null results.Source: H Causal Intervention Details · p. 19

I Detailed Breakdowns and Additional Analyses

• The original multistep-arithmetic row is 18.6% error propagation under text perturbations, while the matched numerical cell is 64.5%.The two rates are different cells of the matched 2×2, not competing measurements of the same perturbation condition.Source: I Detailed Breakdowns and Additional… · p. 20

J Position Gradient: Remaining-Steps Confound

• Within 3–4 remaining steps, error propagation is 29.4% early, 23.1% middle, and 30.6% late; task difficulty dominates the position effect within bins.The authors note that fine-grained conditioning on remaining steps, absolute position, and difficulty is unsupported by their sample sizes.Source: J Position Gradient: Remaining-Steps… · p. 21

K Judge Validation Details

• Judge labels are temporally stable, while strategy–label association remains indirect validation and is an upper bound on genuine association.The A/B split stayed consistent from the first 500 to the final 3,948 resolved cases; the human study found many paper-B rows were human-A.Source: K Judge Validation Details · p. 22

L Experimental Setup and Reproducibility

• The core Gemma pipeline labels 21,238 of 21,242 continuations, with four unresolved; scripts, prompts, and analysis materials are released.Experiments used one NVIDIA H100 80GB GPU.Source: L Experimental Setup and Reproducibility · p. 22

M Variance Decomposition (Type × Difficulty)

• A three-level difficulty bucket yields a 97.1% / 0.9% / 2.0% partition, while reversing entry order still leaves difficulty dominant by about 33×.The four-level model breaks out BBH multistep arithmetic because it is materially harder than the BBH average.Source: M Variance Decomposition (Type ×… · p. 23, Table 5 · p. 23

N Perturbation Strength Axis

• Across tested magnitudes, within-task error-propagation spread is at most 4pp, versus a 60pp-plus gap between tasks at fixed magnitude.This argues against a simple perturbation-magnitude or competence-floor explanation in the tested range.Source: N Perturbation Strength Axis · p. 23, Table 6 · p. 24

O–R Cross-Model and Prompting Analyses

• DeepSeek probes generalize to a reasoning-specialized model, while its prompt-format diagnostic finds 38–44% greedy GSM8K accuracy across three formats.The base Gemma comparison is confounded by 0–22% structured-step compliance; Llama's text-on-GSM8K cell is not comparable because 97.7% of continuations were empty.Source: O DeepSeek-R1-Distill-Qwen-7B… · p. 24, P DeepSeek Prompt-Format Diagnostic · p. 25, Q Base Gemma-2-9B Viability and Few-Shot · p. 25

• Annotators independently labeled 500 stratified continuations with files hiding rule labels, judge variants, and sampling strata.The sample included judge-variant disagreement cells, Type-C calibration rows, and a uniform-random stratum.Source: S Human Annotation Protocol · pp. 25–26
• Inter-annotator agreement was κ = 1.000 for C versus non-C, while four-way agreement was κ = 0.927; none of 13 disagreements involved C.Twelve A-versus-B rows were adjudicated toward the annotator matching the production label, and one A-versus-U row was excluded.Source: S Human Annotation Protocol · p. 26
• Human consensus matched paper labels on 96.2% of C-versus-non-C rows but 76.0% of full labels; most three-way disagreements were paper-B rows read as human-A.On BBH, answer-scoring failures accounted for all 18 rule-C rows flipped to human-A; the gradient ordering survived under human labels, with a compressed BBH magnitude.Source: S Human Annotation Protocol · p. 26
• Among rows covered by four-variant re-judging, consensus agreement ranged from 84.2% for strict correction to 20.3% for strict bypass.The sample was stratified, so these per-variant values are a within-sample ranking rather than corpus-wide estimates.Source: S Human Annotation Protocol · pp. 26–27
• The study covers Gemma-2-9B-IT only; the cross-model B-rate comparison has not been human-validated.Source: S Human Annotation Protocol · p. 27