Correctness Conditioning as Selection in Chain-of-Thought Faithfulness Evaluation

Yueer Pan; Yuexi Pan

cs.AIOct 27, 2026

Correctness Conditioning as Selection in Chain-of-Thought Faithfulness Evaluation

Authors: Yueer Pan, Yuexi Pan

Organizations: Zhejiang University · University of Science and Technology of China

Abstract

Restricting chain-of-thought (CoT) faithfulness evaluation to correctly answered instances changes the population being measured. We study this selection step in a seven-variable structural causal model and derive the exact observed gap $Δ= \operatorname{Cov}(A, C)/P(C=1)$ . The model distinguishes difficulty confounding from shortcut-induced collider bias and states the conditional-independence assumption needed to express the gap through latent response propensities. A protocol review of 18 papers distinguishes answer-correctness restrictions from agreement-based selection, benchmark construction, and training-data filtering. We evaluate four sub-ceiling deployments on arithmetic tasks and observe positive selection gaps in all four. The primary GLM-5.3 run yields a +7.4 percentage-point gap ( $n = 1000$ , $p < 0.0001$ ), and pooled Cochran-Mantel-Haenszel analysis gives OR = 1.70 ( $p < 10^{-6}$ ). An assumption-free covariance decomposition assigns 42.8% of the GLM-5.3 gap to between-stratum variation in interventionable-operation count, leaving the majority unattributed. When counterfactual compliance is observable for every instance, evaluating all traces directly estimates population faithfulness without correctness filtering.

Figures & tables

Panel of Figure 1 — Figure 1: CoTsel overview. Conceptual task intervention and correctness selection (top), with measured faithfulness and selection gaps on arithmetic DAGs (bottom).

Figure 2: Correct factual answers can conceal different intervention responses. The two rules agree before the edit and diverge afterward.

Figure 3: Structural causal model before and after correctness conditioning. Both panels show all dependencies in Definition 1 ; the dashed line marks a selection-induced association.

Figure 4: Selection rules in evaluation protocols. Answer correctness, behavioral agreement, and data construction define different populations.

Figure 5: Procedural task families. Each example pairs the original task with an intervention and a deterministic counterfactual answer.

Figure 6: All-trace intervention protocol. On this illustrative arithmetic task, a value intervention changes the oracle answer and a semantics-preserving edit leaves it unchanged.

Figure 7: Faithfulness and selection gaps across deployments. Open markers denote all traces; filled markers denote selected traces. Intervals show uncertainty in the selection gap.

Boundary controls ( $P(C{=}1)=1$ , filter vacuous):
Model	$n$	Acc.	$F_{\mathrm{pop}}$	$F_{\mathrm{sel}}$	$\Delta$ (pp)	95% CI (pp)	$p$
glm-5.2	100	79.0%	0.440	0.456	+1.6	[ $-$ 3.4, +6.6]	0.540
deepseek-v4-pro	200	66.5%	0.555	0.579	+2.4	[ $-$ 2.6, +7.2]	0.337
glm-5.3	1000	51.5%	0.507	0.581	+7.4	[+4.3, +10.4]	$\bm{<}$ 0.0001 ∗
deepseek-v4-flash	300	48.3%	0.527	0.586	+6.0	[+0.2, +11.8]	0.046 ∗
qwen3.8-max	20	100%	0.600	0.600	0.0	N/A	N/A

Table 1: Faithfulness across serving deployments. All-trace and correctness-conditioned estimates, with ceiling controls shown separately.

Figure 8: Operation-count decomposition. GLM-5.3 faithfulness across strata and the corresponding covariance split. The split is descriptive; the within-stratum component remains unattributed.

Figure 9: Sensitivity to adjustment and response handling. Weighted selected estimates are compared with all traces; deployment gaps are compared under two null-response conventions. Markers show point estimates.

Model	$F_{\mathrm{pop}}$	Acc.	Implied $p_{s}$	Predicted	Observed	Ratio
	( $=p_{r}$ )	( $=P(C{=}1)$ )		$\Delta$ (pp)	$\Delta$ (pp)
glm-5.2	0.440	79.0%	$+0.625$	11.7	1.6	$7.3\times$
deepseek-v4-pro	0.555	66.5%	$+0.247$	28.0	2.4	$11.7\times$
GLM-5.3	0.507	51.5%	$+0.016$	47.7	7.4	$6.4\times$
deepseek-v4-flash	0.527	48.3%	$-0.092$	56.3	6.0	$9.4\times$

Table 2: Observed and predicted selection gaps. Predictions use the shortcut-mixture special case; the flash prediction lies outside the valid probability regime.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Family	$n$	$p_{r}$	$F_{\mathrm{pop}}$	$F_{\mathrm{sel}}$	Gap (pp)
Arithmetic DAG	30	0.33	0.333	0.417	+8.3
Boolean circuit	50	0.58	0.580	0.659	+7.9
Graph reachability	50	0.52	0.520	0.605	+8.5
Mean					+8.2

Appendix

Table 3: Implementation consistency. Three procedural task families evaluated under a common shortcut prevalence.

Figure 10: Selection bias surface. Closed-form predictions with simulated points overlaid on the marginal slices.

Paper	Year	Procedure	Reporting population	Source
Lanham et al.	2023	Trace truncation, mistakes, paraphrase	Collected traces; length-weighted aggregation	v1 ; §2.1–2.6
Turpin et al.	2023	Biased input cues	Main input-bias tests; correctness-selected qualitative examples	v2 ; §3.1; Tab. 7
Lyu et al.	2023	Deterministic solver	Test-set accuracy; correct outputs for human plausibility	v3 ; §4.2, 6.3
Radhakrishnan et al.	2023	Truncation and corruption	All reasoning samples; gold-label subsets for bias tests	v2 ; §3.2–3.3
Wang et al.	2023	Retrieval and trace construction	Test-set task success; correct-answer training pool	v2 ; Alg. 1; Tab. 1
Bentham et al.	2024	With/without-CoT agreement	Dataset average; no answer-correctness screen in metric	v3 ; §3, Eq. 1

Appendix

Table 4: Reporting populations in prior work. Protocol-specific evidence from 18 papers.

Achieved: glm-5.3 at $n=1000$ : $\Delta=+7.4$ pp, $p<0.0001$ .
Model	$n$ (pilot)	Pilot $\Delta$ (pp)	Power at $n$	$n$ for 80% power
glm-5.2	100	+1.6	0.06	$>$ 5,000
glm-5.3	100	+8.8	0.25	$\approx$ 400
deepseek-v4-flash	100	+3.8	0.09	$\approx$ 1,800
qwen3.8-max	20	0.0	—	—
Achieved: deepseek-v4-flash at $n=300$ : $\Delta=+6.0$ pp, $p=0.046$ .

Appendix

Table 5: Power projections. Pilot-based estimates for the scaled deployment runs.

Steps	$n$	$a$	$b$	$c$	$d$	Acc.	$F_{\mathrm{pop}}$	$F_{\mathrm{sel}}$	$\Delta$ (pp)
1 (simple)	115	81	5	25	4	75%	0.922	0.942	$+2.0$
2 (easy)	104	51	18	27	8	66%	0.750	0.739	$-1.1$
3 (moderate)	232	53	77	35	67	56%	0.379	0.408	$+2.8$
4	0	0	0	0	0	—	—	—	—
5 (medium)	223	56	57	39	71	51%	0.426	0.496	$+7.0$
6	0	0	0	0	0	—	—	—	—

Appendix

Table 6: GLM-5.3 results by operation count. Joint outcome counts and selection gaps, with empty strata retained for completeness.

Model	$n$	$a$	$b$	$c$	$d$	$\Delta$ (pp)	OR
GLM-5.3	1000	299	216	208	277	$+7.4$	1.84
deepseek-v4-flash	300	85	60	73	82	$+6.0$	1.59
deepseek-v4-pro	200	77	56	34	33	$+2.4$	1.33
glm-5.2	100	36	43	8	13	$+1.6$	1.36
Boundary controls ( $P(C{=}1)=1$ ; $c=d=0$ by construction):
qwen3.8-max	20	12	8	0	0	$0.0$	—

Appendix

Table 7: Joint compliance and correctness counts. Ceiling controls are excluded from the pooled analysis.

Figure 11: Worked selection calculation. An illustrative intervention and the joint counts connecting oracle compliance to all-trace and correctness-selected estimates.

Model	$n$	$\Delta$ (pp)	Between-stratum	Within-stratum
GLM-5.3	1000	7.36	42.8% [26.8, 69.1]	57.2%
deepseek-v4-flash	300	5.95	21.9% [ $-39.0$ , 112.2]	78.1%

Appendix

Table 8: Covariance decomposition by deployment. Brackets report bootstrap intervals; the within-stratum component remains unattributed.

Steps	$n$	$a$	$b$	$c$	$d$
1 (simple)	36	21	1	12	2
2 (easy)	42	9	9	9	15
3 (moderate)	63	12	21	9	21
5 (medium)	59	15	12	17	15
7 (hard)	33	5	8	8	12
8 (very hard)	37	13	5	11	8

Appendix

Table 9: DeepSeek-v4-flash strata. Joint counts by interventionable-operation count for covariance decomposition.

Figure 12: Ranking reversals at

\pi_{10}=0.001

. The orange region contains candidates with lower population faithfulness but higher selected-trace scores than reference

\mathcal{B}

$\pi_{10}$	Reversal fraction	Reversal count
0.001	40.561%	16,224
0.05	3.568%	1,427
0.1	0.0075%	3
0.2	0.0150%	6

Appendix

Table 10: Reversal sensitivity. Counts and percentages on the same 39,999-candidate grid, with the reference and other correctness parameters held fixed.

Removed	Remaining $n$	$z$	$p$ (two-sided)
GLM-5.3	600	2.24	0.025
deepseek-v4-flash	1300	4.80	$<$ 0.0001
deepseek-v4-pro	1400	5.17	$<$ 0.0001
GLM-5.2	1500	5.17	$<$ 0.0001

Appendix

Table 11: Leave-one-out analysis. Pooled association after omitting each deployment.

Figure 13: Deployment-level and pooled odds ratios. Square areas encode inverse-variance weights; horizontal lines show individual confidence intervals and the diamond the pooled interval.

Model	$n$	Acc.	$F_{\mathrm{pop}}$	$F_{\mathrm{sel}}$	$\Delta$ (pp)	$\chi^{2}$ ( $p$ )
GLM-5.3	50	78%	0.560	0.590	$+3.0$	0.206 (0.650)
deepseek-v4-flash	50	44%	0.220	0.182	$-3.8$	0.055 (0.815)
Breakdown by cue correctness: flash correct-cue $+1.31$ pp ( $n{=}36$ ), incorrect-cue $-21.43$ pp ( $n{=}14$ );
GLM-5.3 correct-cue $+4.50$ pp ( $n{=}36$ ), incorrect-cue $+1.43$ pp ( $n{=}14$ ).

Appendix

Table 12: Selection gaps under adversarial framing. Estimates from the earlier run, separate from the subsequent exploratory probes.

Figure 14: Selection gap versus accuracy. All confidence intervals are shown; asterisks mark descriptive

p<0.05

CoT Faithfulness

Chain-of-thought faithfulness measures whether a model’s stated reasoning causally reflects and supports the answer it gives.

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Measuring Faithfulness Depends on How You Measure: Classifier Sensitivity in LLM Chain-of-Thought Evaluation

Richard J. Young

University of Nevada, Las Vegas, Department of Management, Entrepreneurship and Technology, Lee Business School, Las Vegas, NV, USA · DeepNeuro AI, Las Vegas, NV, USA

CoT Faithfulness LLM Evaluation Faithfulness of Language Model Explanations

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Recent work on chain-of-thought (CoT) faithfulness reports single aggregate numbers (e.g., DeepSeek-R1 acknowledges hints 39% of the time), implying that faithfulness is an objective, measurable property of a model. This paper provides evidence that it is not. Three classifiers (a regex-only detector, a regex-plus-LLM pipeline, and a Claude Sonnet 4 judge) are applied to 10,276 influenced reasoning traces from 12 open-weight models spanning 9 families and 7B to 1T parameters. On identical data, these classifiers produce faithfulness rates of 74.4%, 82.6%, and 69.7%. Per-model gaps range from 2.6 to 30.6 percentage points; all pairwise McNemar tests are significant (p < 0.001). The disagreements are systematic: Cohen's kappa ranges from 0.06 ("slight") for sycophancy hints to 0.42 ("moderate") for grader hints, and the asymmetry is pronounced: for sycophancy, 883 cases are classified as faithful by the pipeline but unfaithful by the Sonnet judge, while only 2 go the other direction. Classifier choice can also reverse model rankings: Qwen3.5-27B ranks 1st under the pipeline but 7th under Sonnet; OLMo-3.1-32B moves from 9th to 3rd. Different classifiers operationalize faithfulness at different levels of stringency (lexical mention versus epistemic dependence), yielding divergent measurements on the same behavior. These results indicate that published faithfulness numbers cannot be meaningfully compared across studies using different classifiers, and that future evaluations should report sensitivity ranges across multiple classification methodologies.

Abstract

CoT Faithfulness

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