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SIFT: Robust Meta-Faithfulness Verification of Chain-of-Thought Reasoning Under Distribution Shift

Noor Islam S. Mohammad, Md. Basim Al Zabir Shammo, Hasan Siddiki, Mahmudul Hasan, Md. Faisal Sheikh, Jakaria Habib

arXiv:2610.04594v1cs.CLcs.LG

TL;DR

Because CoT faithfulness detectors are themselves predictors, the paper asks whether their verdicts remain faithful under distribution shift. It formalizes meta-faithfulness, tests detectors across ten FaithShift shift axes, and proposes SIFT; results show substantial transfer gaps and that sampling stochasticity dominates instability.

  • Problem

    Whether CoT faithfulness detectors remain faithful to themselves under distribution shift is unexamined, despite their use in high-stakes settings.

    Source: §1 · pp. 2–3

  • Method

    The paper formalizes meta-faithfulness, evaluates detectors across ten FaithShift shift axes, and proposes SIFT using hidden-state trajectories, IRM and DANN penalties, and selective abstention.

    Source: §1 · p. 3, §6.0.1 · p. 8, §6.1.3 · p. 13

  • Results

    Existing detectors show transfer gaps ≥0.15 AUROC, while over 80% of instability is sampling stochasticity; SIFT cuts violations 64%, but ensemble baselines narrow its MFS margin to 0.01.

    Source: §1 · p. 3, §7.2 · p. 14, §10 · pp. 16–17

  • Takeaways & Limitations

    The authors recommend seed-repeat analysis and 3–4-seed ensembling, provisional verdicts until second-order robustness is demonstrated, and multi-model training with target-model calibration for cross-model deployment.

    Source: §10 · p. 17

    The audit verifies label preservation, not mechanism preservation, and statistical inference is constrained by 750 independent base items.

    Source: §9 · p. 16

Abstract

from arXiv · show

Chain-of-Thought (CoT) faithfulness detectors are widely used to audit reasoning models, yet a detector is itself a predictor whose verdicts are treated as stable properties. We ask whether a detector is faithful to itself under distribution shift. We formalize meta-faithfulness as an invariance principle: a valid detector must return identical verdicts on traces that differ only by transformations preserving ground-truth faithfulness. We prove three results: (i) no detector using only intervention-response profiles can separate faithful from epiphenomenal mechanisms with identical signatures; (ii) any detector relying on shift-sensitive features violates invariance at a rate independent of its in-distribution accuracy; (iii) an asymptotic certified selective-risk guarantee enables confident abstention. We operationalize the principle in FaithShift, a stress-test protocol spanning ten shift axes, and propose SIFT, a hidden-state trajectory detector trained with cross-environment invariance objectives and certified abstention. Across 14,996 traces, four domains, and eight models, three findings emerge. First, transfer collapse is real: all existing detectors show gaps $\geq 0.15$ AUROC. Second, the dominant bottleneck is sampling stochasticity, not shift: over 80% of detector instability stems from random seed variation, falsifying our preregistered prediction that shift-attributable violations exceed 0.25. Third, SIFT cuts invariance violations by 64% over the best single-seed baseline, but a four-seed ensemble of any detector narrows the margin to 0.01 (indistinguishable at matched coverage, $p=0.21$), and SIFT needs a 51% abstention rate. Cross-model transfer degrades from within-family to cross-family to open-weight-to-API, partly closed by multi-model training. We offer a framework for auditing auditors: the real barrier is detector variance, not distribution shift.

1 INTRODUCTION

Faithfulness detectors are themselves predictors, yet their verdicts are often treated as stable across distributions. This paper formalizes meta-faithfulness, evaluates detectors under controlled shifts, and finds that sampling stochasticity—not shift—is the dominant source of instability.

1 INTRODUCTION

  • Meta-faithfulness requires a detector to return identical verdicts on traces transformed without changing their ground-truth faithfulness label.The paper treats disagreement as a detector property, making the auditing instrument itself an object of study.Source: §1 · p. 2, Figure 1 · p. 2

1.1 CONTRIBUTIONS

  • The theory shows that behavioral detectors cannot distinguish faithful from epiphenomenal mechanisms with identical intervention-response profiles, and that accuracy alone does not ensure invariance.It also establishes an asymptotic certified selective-risk guarantee that supports abstention.Source: §1 · p. 3

1 INTRODUCTION

  • All existing detectors show transfer gaps of at least 0.15 AUROC, while over 80% of observed instability stems from sampling stochasticity rather than distribution shift.SIFT reduces raw invariance violations by 64% over the best single-seed baseline, but four-seed ensembles narrow its margin to 0.01; the authors frame it as a demonstration, not a broadly superior detector.Source: §1 · pp. 2–3
  • FaithShift tests six detector families across ten shift axes, 14,996 traces, four reasoning domains, and eight model architectures.SIFT uses hidden-state trajectories with invariance objectives, while the protocol pairs traces that differ along preserving or label-altering transformations.Source: §1 · pp. 2–3

1.1 CONTRIBUTIONS

  • Cross-model transfer declines from within-family to cross-family open-weight to open-weight-to-API, and multi-model training partly closes the gap.Training on 3–4 models improves transfer to 0.65 AUROC, with IRM contributing most among the reported components.Source: §1 · p. 3

2 RELATED WORK

Prior work examines whether chains of thought reflect model computation, how faithfulness can be measured, and whether detectors generalize. This paper connects those concerns to robustness, selective prediction, and reproducibility.

Monitorability and Safety

  • CoT monitoring is a fragile safety opportunity because reward hacking can move in or out of the visible trace, making monitorability contingent on monitor robustness.This motivates auditing the monitors themselves under deployment-relevant shifts.Source: §2 · p. 4

Robustness and Selective Prediction

  • Robustness work motivates invariance-based training, while selective prediction provides a framework for abstention with risk guarantees.The paper applies these tools to faithfulness detection as a domain-generalization problem with a certified operating point.Source: §2 · p. 4

Stochasticity and Reproducibility

  • Over 80% of detector instability is stochastic, connecting faithfulness detection to reproducibility work and motivating variance reduction alongside distribution robustness.The paper reports quantifying the relative contributions of stochasticity and shift in this setting.Source: §2 · pp. 4–5

Recent Advances in Reasoning Models

  • Faithfulness varies with model family, size, training, and inference budget, reinforcing the need to evaluate detectors across changing conditions.More inference time does not necessarily make models more faithful and can, in some cases, lead to more sophisticated rationalization.Source: §2 · p. 5

Measuring Faithfulness

  • Faithfulness tests include cue verbalization, counterfactual sensitivity, hidden-state probes, and internal–external discrepancy measures, but consistency with stated reasoning does not establish causal use of that reasoning.Benchmark detectors have reported high held-out accuracy, while their evaluation remains limited to in-distribution settings.Source: §2 · p. 5

3 PROBLEM FORMULATION

The problem formulation treats faithfulness as a causal property of the relationship between a model’s computation and its stated chain of thought. Because that property is not directly observable, detectors estimate it from available evidence.

3 PROBLEM FORMULATION

  • Ground-truth faithfulness indicates whether the answer is causally produced by the computation described in the chain of thought rather than by omitted features or a divergent computational path.It is not directly observable and must be approximated through construction protocols or expert audit.Source: §3 · p. 5
  • A faithfulness detector scores the likelihood of faithfulness given the model, query, and chain-of-thought trace, then converts the score into a binary verdict using a threshold.The supplied formulation identifies the score’s inputs and thresholding operation.Source: §3 · p. 6
  • A behavioral detector relies solely on intervention-response profiles generated by the model and maps those profiles to faithfulness scores.The profiles pair interventions with the model’s responses to them.Source: §3 · p. 6

4 META-FAITHFULNESS AND THEORY

The paper defines meta-faithfulness through stable verdicts under label-preserving transformations and responsive verdicts under label-altering ones. It then establishes limits of behavioral detection, an invariance lower bound, and an asymptotic guarantee for selective risk.

4.1 CORE DEFINITIONS

  • A meta-faithful detector preserves its verdict under verified faithfulness-preserving transformations and changes it in the direction implied by verified faithfulness-altering transformations.The audit verifies label preservation rather than preservation of the underlying mechanism.Source: §4.1 · p. 6

4.2 BEHAVIORAL INDISTINGUISHABILITY

  • No behavioral detector can distinguish faithful from epiphenomenal mechanisms when they induce identical intervention-response profiles.The result applies to detectors restricted to the intervention family’s information.Source: §4.2 · p. 7

4.3 ACCURACY DOES NOT BOUND INVARIANCE

  • AUROC or classification accuracy alone does not determine invariance-violation rate because they need not control detector-margin probability near the decision boundary.The lower bound assumes local sensitivity and symmetric, independent perturbations; the authors note it is not assumed to be tight.Source: §4.3 · p. 7

4.4 CERTIFIED SELECTIVE RISK

  • The selective-guaranteed-risk procedure asymptotically selects a candidate threshold whose selective risk meets the target with probability at least 1 − δ, if a candidate satisfies the constraint.The guarantee is asymptotic and applies to a finite predetermined candidate set.Source: §4.4 · p. 7

5 SIFT: A SHIFT-INVARIANT TRAJECTORY DETECTOR

SIFT uses hidden-state trajectories and multi-scale encoding to detect faithfulness across environments. Its invariance-oriented training combines IRM and DANN penalties to encourage robustness to preserving transformations.

5 SIFT: A SHIFT-INVARIANT TRAJECTORY DETECTOR

  • SIFT combines trajectory-based features, multi-scale encoding, and IRM/DANN training to encourage robustness across environments.The detector processes layer-by-step hidden-state trajectories and uses a cross-environment invariance penalty before abstaining on out-of-support traces through a certified risk gate.Source: §5 · pp. 7–8, Figure 2 · p. 8
  • Training treats each preserving transformation axis as an environment and discourages environment-specific decision rules while maintaining predictive performance.The objective combines per-environment predictive loss with an invariance regularizer; IRM encourages a shared classifier to remain optimal across environments.Source: §5 · p. 8

6 FAITHSHIFT: A STRESS-TEST PROTOCOL

FaithShift evaluates detector invariance and sensitivity under controlled transformations, while the section specifies SIFT’s trajectory-based pipeline and establishes theoretical limits and a selective-risk certificate. The results motivate richer internal-state information and explicit abstention, with guarantees bounded by their stated assumptions.

6 FAITHSHIFT: A STRESS-TEST PROTOCOL

  • FaithShift tests detector behavior across ten axes—seven preserving and three altering—using paired traces to assess invariance and sensitivity.The inference-budget axis is mixed because its effect on faithfulness labels is determined empirically.Source: §6 · p. 8

6.0.1 TRAJECTORY FEATURES

  • SIFT extracts layer-by-step hidden-state trajectories and derives descriptors for representational change, early answer information, and attention stability.Inter-layer velocity and step-to-step drift are measurable signals, not direct evidence that reasoning is substantive or post-hoc; normalization reduces feature-magnitude and trace-length influence but does not guarantee invariance.Source: §6.0.1 · pp. 8–9

6.0.2 MULTI-SCALE TRAJECTORY ENCODING

  • A multi-scale temporal convolutional network and transformer encode local and distant temporal patterns, then learned pooling and an MLP produce a faithfulness score.The score is thresholded into the binary verdict used for invariance-violation and alteration-sensitivity evaluation.Source: §6.0.2 · pp. 9–10

6.0.3 INVARIANCE OBJECTIVE AND TRAINING PROCEDURE

  • The DANN objective suppresses environment-specific information while retaining faithfulness information, but training on observed environments does not guarantee invariance to unseen transformations.Independent FaithShift evaluation remains necessary.Source: §6.0.3 · p. 10

6.1.1 PROOF OF THEOREM 4.5

  • Detectors limited to intervention-response profiles cannot distinguish mechanisms that produce identical responses under every available intervention.This is an information-theoretic limit of the observation model, not a detector-architecture limit; resolving the ambiguity requires richer information, such as internal-state trajectories.Source: §6.1.1 · p. 10

6.1.2 PROOF OF THEOREM 4.6 AND GAUSSIAN CHARACTERIZATION

  • High in-distribution AUROC does not bound verdict changes under preserving transformations; local margin mass near the decision boundary also matters.Under the illustrative Gaussian characterization, the lower bound is approximately 0.19 at b = 0.5ς and 0.34 at b = ς; this is not a claim that margins are generally Gaussian.Source: §6.1.2 · p. 11

6.1.3 SGR PROCEDURE AND PROOF OF THEOREM 4.7

  • The SGR procedure selects the highest-coverage evaluated threshold whose asymptotic upper confidence bound meets the target risk.The certificate applies to a finite predetermined threshold set under regularity assumptions and is not an exact finite-sample guarantee; more conservative finite-sample certificates may reduce coverage.Source: §6.1.3 · pp. 12–13

7 EXPERIMENTS

Across the benchmark, existing detectors show transfer collapse, while SIFT reduces invariance violations but does not establish broad superiority over a four-seed ensemble. Seed variation accounts for most instability, and SIFT’s certified operating point entails substantial abstention.

7.1 EXPERIMENTAL SETUP

  • The evaluation retains 14,996 traces from 15,000 constructed traces and uses 750 independent base items as the effective sample size for inference.The traces are split 60/15/25 into train, validation, and test; confidence intervals are bootstrap-stratified by base item. Six detector families are compared.Source: §7.1 · p. 13

7.2 MAIN RESULTS

  • All existing detectors have transfer AUROC gaps of at least 0.15, while SIFT has the smallest gap at 0.10.Table 1 reports the main comparisons across six detectors, including in-distribution versus transfer AUROC.Source: §7.2 · p. 13, Table 1 · p. 14
  • SIFT reduces mean IVR by 64%, but its MFS margin over a four-detector ensemble is 0.01 and is indistinguishable at matched coverage.At κ = 0.70, the matched-coverage comparison has p = 0.21; SIFT abstains on roughly 51% of traces at its certified operating point.Source: Table 1 · p. 14, §7.2 · pp. 14–15
  • Over 80% of observed instability comes from sampling stochasticity rather than distribution shift, falsifying the preregistered shift-attributable IVR hypothesis.For attribution-consistency, raw IVR 0.282 decomposes into a 0.232 seed-repeat floor and 0.050 shift-attributable IVR; SIFT’s corresponding values are 0.102, 0.028, and 0.074.Source: §7.2 · p. 14
  • SIFT cross-model transfer follows a hierarchy: within-family exceeds cross-family open-weight, which exceeds open-weight-to-API transfer.Table 2 also reports average cross-domain transfer of 0.63, with GSM8K↔MATH strongest at 0.62–0.63 and GSM8K↔CommonsenseQA weakest at 0.60–0.61.Source: Table 2 · p. 15
  • The SGR certificate meets α = 0.25 with risk bound 0.249 at coverage 0.50, while α = 0.30 permits κ = 0.61 at risk at most 0.30.The first operating point falls short of the preregistered κ ≥ 0.60 coverage requirement.Source: Table 3 · p. 15

8 ABLATION STUDY

The ablations identify IRM as SIFT’s largest contributing component, while multi-model training improves cross-model transfer with diminishing returns beyond 3–4 models.

8 ABLATION STUDY

  • Removing IRM increases IVR by 110% and decreases MFS by 28%, the largest component-ablation effect reported.DANN, transformer, and TCN removals have smaller effects; attention-stability and early-commitment contribute moderately.Source: Table 4 · p. 16, §8 · p. 16
  • Multi-model training raises transfer AUROC from 0.58 to 0.65 and cuts the gap from 0.18 to 0.14, with diminishing returns beyond 3–4 models.MFS is not reported for this comparison.Source: §8 · pp. 15–16, Table 4 · p. 16

9 LIMITATIONS AND FUTURE WORK

The evaluation’s independent-item count and several component analyses constrain the strength and scope of its conclusions.

9 LIMITATIONS AND FUTURE WORK

  • Statistical inference rests on 750 independent base items, not 14,996 traces, with confidence intervals bootstrap-stratified by base item.The audit verifies label preservation rather than mechanism preservation, so IVR may be inflated if a nominally preserving transformation changes computation.Source: §9 · p. 16
  • Per-axis alteration sensitivity uses only three altering pairs each for cue injection and cue verbalization, and the two-detector agreement analysis is underpowered.Source: §9 · p. 16
  • The eight white-box models span 1.5B–8B parameters, cross-lingual coverage is limited to English–Turkish, and the SGR guarantee is asymptotic under standard regularity assumptions.The abstention cost-benefit analysis uses evaluation-specific empirical numbers, so another deployment regime may have a different optimal operating point.Source: §9 · p. 16

10 CONCLUSION

The paper formalizes meta-faithfulness and diagnoses sampling stochasticity as the dominant source of detector instability. Its framework, stress test, and detector are presented as contributions to auditing and diagnosis, with operational and scope limits.

10 CONCLUSION

  • Over 80% of observed instability is sampling stochasticity; SIFT reduces invariance violations by 64% versus the best single-seed baseline but requires 51% abstention.With ensemble baselines, its MFS margin narrows from 0.05 to 0.01; the authors frame SIFT as a demonstration of reduced stochastic instability, not a broadly superior detector.Source: §10 · pp. 16–17

IMPACT STATEMENT

  • FaithShift, SIFT, and seed-repeat guidance are offered for auditing CoT detectors, and the auditing-auditors principle extends to settings where one model evaluates another.Source: IMPACT STATEMENT · p. 17

A.1 EXTENDED PROBLEM FORMULATION

  • A trace is faithful when its described computation causally produces the answer; behavioral detectors cannot distinguish genuine reasoning from post-hoc traces with identical intervention-response profiles.Internal detectors access hidden-state trajectories, but can still rely on features that change under preserving transformations.Source: §A.1 · pp. 23–24

A.2 EXTENDED DISCUSSION OF META-FAITHFULNESS DEFINITIONS

  • Meta-faithfulness requires invariance to label-preserving transformations and sensitivity to label-altering ones; the protocol checks label, not mechanism, preservation.The MFS combines stability, alteration sensitivity, and usable coverage, so excessive abstention is penalized.Source: §A.2 · pp. 24–25

A.3.1 WHY INVARIANCE TRAINING STABILIZES DETECTORS

This section examines SIFT’s detector stability and cross-domain transfer, including the effects of seed variation, abstention, and moderate label noise on detector metrics and rankings.

B EXTENDED EXPERIMENTAL SETUP

  • The evaluation covers eight models, four reasoning domains, ten transformation conditions, and 14,996 scored traces from 750 independent base items.Experiments used a single NVIDIA A100 GPU; the full experimental suite took approximately 100 GPU-hours. Confidence intervals used base-item-stratified bootstrap samples.Source: B EXTENDED EXPERIMENTAL SETUP · p. 26, §B.1 · p. 26, §B.3 · p. 26

C.3 PER-AXIS INVARIANCE VIOLATION VISUALIZATION

  • SIFT reduces mean IVR by 64% across shift axes, including 76% reductions for paraphrase and hint format, while step reordering yields near-zero violations.Attention dropout is the dominant stochastic source for both SIFT and attribution-consistency, followed by weight initialization.Source: §A.3.1 · p. 25, Table 6 · p. 25, §C.3 · p. 27

C.4 SEED-REPEAT DECOMPOSITION AND SEED STABILITY

  • Over 80% of attribution-consistency’s instability is seed-repeat variation: raw IVR 0.282 decomposes into 0.232 stochasticity and 0.050 shift-attributable IVR.SIFT’s raw IVR is 0.102, with a 0.028 seed-repeat floor and 0.074 shift-attributable IVR; representation variance correlates with IVR (ρ ≈0.80) and falls 53% after IRM and DANN.Source: §A.3.1 · p. 25, §C.4 · pp. 27–28, Figure 5 · p. 29

C.6 SPURIOUS FEATURE IDENTIFICATION

  • Semantic content shift correlates most strongly with verdict flips (ρ = 0.234, p < 0.001), followed by lexical overlap (ρ = 0.187, p = 0.001); trace length is not significant.The theorem’s trace-length-based IVR floor has substantial slack, consistent with semantic and lexical changes being stronger observed drivers of instability.Source: §C.6 · p. 30, Figure 7 · p. 31, §C.8 · p. 34

C.7 META-FAITHFULNESS SCORE AND CERTIFIED SELECTIVE RISK

  • SIFT reaches MFS 0.72 versus 0.67 for the strongest single-seed baseline, but a four-detector instance-level ensemble reaches 0.71, narrowing the margin to 0.01.SIFT also reduces IVR by 64% (0.10 vs 0.27) while maintaining AS of 0.76; H3 is not supported under the revised comparator.Source: §C.5 · p. 30, §C.7 · p. 31

C.7.1 CERTIFIED SELECTIVE RISK

  • At α = 0.25, SIFT’s certified risk bound is 0.249 but coverage is only 0.50; α = 0.30 raises coverage to 0.61 with certified risk at most 0.30.The preregistered coverage target is missed, and the certified operating point requires 51% abstention.Source: §C.1 · p. 27, §C.7.1 · p. 31, Figure 8 · p. 32

C.7.3 COVERAGE-MATCHED COMPARISON

  • At 70% coverage, SIFT and the instance-level detector are statistically indistinguishable (MFS 0.64 vs. 0.62, p = 0.21), while SIFT’s advantage is largest at 50–60% coverage.Abstention concentrates on longer traces and cross-lingual or cross-model shifts; the analysis recommends matching operating points to application costs, with 40–50% abstention optimal in high-stakes settings.Source: §C.7.2 · p. 31, §C.7.3 · p. 32, §C.7.4 · p. 33

C.8.1 LABEL NOISE SENSITIVITY

  • At 10% label noise, SIFT’s IVR rises by 0.04 and MFS falls by 0.03, but detector rankings remain unchanged through 15% noise.The hypothesis summary also reports inter-detector kappa of 0.38 for H2b, though the two-detector suite makes that estimate underpowered.Source: §C.1 · p. 27, Figure 10 · p. 34
  • At ϵ = 0.10 label noise, IVR rises from 0.10 to 0.14 and MFS falls from 0.72 to 0.69, while detector rankings remain preserved up to ϵ = 0.15.The authors conclude that findings are robust to moderate label noise, but absolute MFS values warrant caution; systematic annotation errors cannot be ruled out despite κ ≈0.80 agreement.Source: §C.8.1 · p. 34

C.9 CROSS-DOMAIN GENERALIZATION

  • SIFT achieves 0.63 average transfer AUROC, with strongest GSM8K–MATH transfer at 0.62–0.63 and weakest GSM8K–CommonsenseQA transfer at 0.60–0.61.In-distribution AUROC ranges from 0.75 (GSM8K) to 0.78 (BBH).Source: §C.9 · p. 35, Figure 11 · p. 35

C.10 CROSS-MODEL GENERALIZATION ACROSS EIGHT ARCHITECTURES

Cross-model transfer is substantially weaker than in-distribution performance and follows a hierarchy from within-family to cross-family and API-model transfer. Decomposing and aligning representations, training on multiple models, and evaluating computational cost characterize possible improvements and trade-offs.

C.10 CROSS-MODEL GENERALIZATION ACROSS EIGHT ARCHITECTURES

  • 0.76 in-distribution AUROC falls to 0.58 on average cross-model transfer, a 0.18 gap, with within-family transfer stronger than cross-family and API-model transfer.The hierarchy is also reflected in gaps of 0.02–0.08 AUROC between within-family and cross-family open-weight transfer, and 0.05–0.10 between cross-family and API transfer.Source: §C.10 · pp. 35–37, Figure 12 · p. 36, Figure 13 · p. 37

C.10.1 DECOMPOSITION OF TRANSFERABLE FEATURES

  • 0.67 transfer AUROC for early-commitment descriptors exceeds attention stability’s 0.54, while middle layers (8–16) transfer best.The analysis interprets attention patterns as more model-specific than representational dynamics and identifies limits to universal transfer.Source: §C.10.1 · p. 37, Table 12 · p. 37

C.10.2 REPRESENTATION ALIGNMENT FOR CROSS-MODEL TRANSFER

  • 0.69 transfer AUROC with representation alignment, rising to 0.71 when combined with multi-model training, remains below the 0.78 in-distribution result.These results are reported for training on Llama-3 and testing on Mistral.Source: §C.10.2 · p. 38, Table 13 · p. 38

C.11 MULTI-MODEL TRAINING RESULTS

  • 0.65 average transfer AUROC from four-model training improves on the single-model 0.58, reducing the transfer gap from 0.18 to 0.14.Gains plateau beyond 3–4 models, with diminishing returns from additional models.Source: §C.11 · p. 38, Figure 14 · p. 38

C.12 COMPUTATIONAL EFFICIENCY

  • 156 ms inference time, 45 MB memory, and 28.4K parameters make SIFT heavier than biasing, which uses 12 ms, 5 MB, and 1.2K parameters.SIFT’s cost accompanies a 64% reduction in invariance violations; the paper describes 156 ms inference as practical for most deployment scenarios.Source: §C.12 · p. 38, Table 14 · p. 39, Figure 15 · p. 39

D.1 COMPONENT CONTRIBUTION ANALYSIS

  • Removing IRM raises IVR by 110% (0.10→0.21) and lowers MFS by 28% (0.72→0.52), the largest component contribution reported.Removing DANN raises IVR by 80%, transformer by 60%, and TCN by 40%.Source: §D.1 · p. 40

D.2 EXTENDED ABLATION: MULTI-MODEL TRAINING COMPONENTS

Multi-model training improves cross-model transfer, while IRM and the transformer remain important; seed averaging accounts for a substantial share of SIFT’s matched-coverage gain. The audits find mostly preserved mechanisms and a favorable stability–sensitivity profile.

D.2 EXTENDED ABLATION: MULTI-MODEL TRAINING COMPONENTS

  • Transfer AUROC improves from 0.58 to 0.65 with multi-model training; removing IRM widens the transfer gap from 0.14 to 0.19, while removing the transformer widens it to 0.16.Removing DANN also widens the gap to 0.16, and removing the TCN to 0.15; Figure 17 ranks IRM as the largest component contribution by MFS impact.Source: §D.2 · p. 41, Figure 17 · p. 41

E.1 ADDITIVE DECOMPOSITION AND NON-OVERLAP TEST

  • Attribution-consistency has seed-repeat floor 0.232 and shift-attributable IVR 0.050, with raw IVR 0.282; estimated overlap is near zero.The overlap estimate is 0.0003 (95% CI [0.0000, 0.0009]) for attribution-consistency and 0.0002 (95% CI [0.0000, 0.0007]) for SIFT, supporting additive decomposition within sampling error.Source: §E.1 · p. 42, Table 15 · p. 42

E.2 VARIANCE SOURCE DIAGNOSTICS

  • The seed-repeat floor is driven most by attention dropout, with weight initialization the second-largest tested source of stochastic instability.The authors suggest deterministic attention as a possible way to reduce the floor, leaving a direct test for future work.Source: §E.2 · p. 42, Table 16 · p. 43

F FACTORIAL VARIANCE DECOMPOSITION

  • At matched coverage, seed averaging contributes approximately 0.04 MFS uniformly across cells, invariance training 0.03, and architecture 0.01.The full SIFT-ensemble gains 0.08 MFS over the simplest baseline; the table reports matched-coverage MFS across the 2 × 2 × 2 design.Source: F FACTORIAL VARIANCE DECOMPOSITION · p. 43, Table 17 · p. 43

G STABILITY–SENSITIVITY TRADE-OFF

H MECHANISM-PRESERVATION AUDIT

  • Mechanism preservation averages 0.92 across audited axes, and IVR on mechanism-preserved pairs is lower by approximately 0.01 on average.For the model-family axis, IVR falls from 0.18 to 0.15; the audit supplies a lower bound on verifiably preserved mechanisms, not an upper bound on how often they hold.Source: H MECHANISM-PRESERVATION AUDIT · pp. 45–46, Table 19 · p. 45

I EMPIRICAL AUDIT OF THEOREM 4.6 ASSUMPTIONS

The audit finds Theorem 4.6’s assumptions hold on most shift axes, with model-family shifts the least favorable case. The discussion connects detector failures to stochasticity and offers practical recommendations for stability and transfer.

I EMPIRICAL AUDIT OF THEOREM 4.6 ASSUMPTIONS

  • Local sensitivity fails on the model-family axis, where higher-order residual is 11% of the first-order term; symmetry and independence hold across axes, making that axis least favorable for Theorem 4.6.The bound is approximate for model-family shifts; for remaining axes the assumptions meet indicated thresholds, though numerical magnitude depends on the spurious feature chosen.Source: I EMPIRICAL AUDIT OF THEOREM 4.6… · p. 46, Table 20 · p. 46

J.1 THEORETICAL IMPLICATIONS

  • Theorem 4.5 rules out behavioral detectors distinguishing faithful from unfaithful mechanisms with matching intervention-response profiles, while Theorem 4.6 shows accuracy does not bound invariance.Theorem 4.7 gives asymptotic certified abstention under standard regularity conditions.Source: §J.1 · pp. 46–47

J.2 EMPIRICAL INSIGHTS

  • Over 80% of attribution’s raw IVR is sampling stochasticity rather than shift sensitivity, while SIFT reduces raw IVR 64% versus the best baseline.SIFT’s MFS advantage narrows to 0.01 against a four-detector instance-level ensemble, and it abstains on 51% of traces; multi-model training improves transfer AUROC from 0.58 to 0.65.Source: §J.2 · p. 47

J.3 PRACTICAL RECOMMENDATIONS

  • The recommendations include seed-repeat analysis and 3–4-seed ensembling, training on 3–4 model families for cross-model deployment, and collecting 50–100 target-model labels for calibration.The authors also recommend comparing multiple detectors for consequential decisions and accounting for SIFT’s coverage penalty.Source: §J.3 · pp. 47–48

J.4 LIMITATIONS

The reported evidence is bounded by model scale, access requirements, audit design, sample size, detector and language coverage, certification assumptions, and approximate ground truth. The authors identify broader audits, data, and deployment settings as future work.

J.4 LIMITATIONS

  • Evaluation covers models from 1.5B to 8B parameters, not larger models, and SIFT requires hidden-state access unavailable for API-only models.The authors caution that larger models may have qualitatively different faithfulness signatures and that smaller-model-trained detectors may not generalize to them.Source: §J.4 · p. 48
  • The audit checks label preservation but not mechanism preservation; per-axis SIFT alteration sensitivity relies on only three cue-injection and cue-verbalization pairs.Statistical inference is constrained by 750 independent base items rather than the 14,996 traces.Source: §J.4 · p. 48
  • The final suite has only two detectors, cross-lingual evaluation covers English and Turkish, and Theorem 4.7’s certification is asymptotic under regularity assumptions.Finite-sample guarantees would require additional assumptions and more conservative confidence bounds.Source: §J.4 · p. 48
  • Ground-truth labels approximate faithfulness using construction protocols and expert audit rather than direct observation, so absolute MFS values warrant caution.The authors cannot rule out systematic label errors, although their sensitivity analysis and mechanism-preservation audit suggest robustness to moderate noise.Source: §J.4 · p. 48

J.5 INTEGRATED FUTURE DIRECTIONS

  • Future work includes seed-stable detector designs, online multi-environment adaptation and certification, expanded cross-model and multilingual evaluation, API-only methods, and larger mechanism-preservation audits.The authors also propose human-AI verification, scalable certification, decision interpretability, adversarial robustness, dynamic thresholds, and testing deterministic attention.Source: §J.5 · pp. 48–49

K ADDITIONAL THEORETICAL FOUNDATIONS AND COMPLETE PROOFS

The appendix formalizes limits on behavioral detection, conditions that produce invariance violations, and guarantees for selective prediction. It treats ranking quality, invariance, and selective reliability as distinct properties.

K ADDITIONAL THEORETICAL FOUNDATIONS AND COMPLETE PROOFS

  • When two mechanisms share an intervention-response profile, no detector restricted to that profile can distinguish them; differing internal statistics can enable separation.The internal-statistics result applies when the statistic differs between the mechanisms.Source: K ADDITIONAL THEORETICAL FOUNDATIONS… · pp. 49–50

K.3 RELATIONSHIP BETWEEN IVR AND RANKING QUALITY

  • AUROC does not determine invariance violations because it does not specify how much score mass lies near the decision threshold.Ranking can remain perfect even when many examples lie close to that threshold.Source: §K.3 · p. 51

K.4 GAUSSIAN CHARACTERIZATION

  • Under the Gaussian specialization, the invariance-violation lower bound depends on the perturbation-band-to-margin-standard-deviation ratio, not perturbation magnitude alone.For a fixed perturbation band, smaller margin variance increases the bound.Source: §K.4 · p. 52

K.5 FINITE-SAMPLE CERTIFIED SELECTIVE RISK

  • A finite-sample certificate bounds selective risk with probability at least 1 − δ and selects the highest-coverage qualifying candidate threshold.The guarantee uses valid upper confidence bounds over a finite threshold set.Source: §K.5 · p. 53

K.6 COVERAGE–RISK TRADE-OFF

  • Certified coverage cannot decrease when the allowed risk level is relaxed.Among candidate thresholds passing the certificate, the smallest qualifying threshold maximizes coverage.Source: §K.6 · p. 54

K.7 MONOTONICITY OF THE IVR LOWER BOUND

  • The invariance-violation lower bound is non-decreasing as the perturbation band grows.The result assumes an absolutely continuous margin distribution.Source: §K.7 · p. 55

K.9 SUMMARY OF THEORETICAL IMPLICATIONS

  • The results distinguish behavioral indistinguishability, invariance, ranking quality, and selective reliability as separate dimensions of detector behavior.The appendix also supplies auxiliary results on coverage–risk trade-offs and estimation.Source: §K.9 · p. 55

K.2 INVARIANCE-VIOLATION LOWER BOUND

  • A detector using a feature changed by a semantics-preserving transformation can incur invariance violations when enough examples lie near its decision boundary.The lower bound assumes the feature perturbation is independent of the detector margin and has a conditionally symmetric sign.Source: §K.2 · p. 50, §K.9 · p. 55

M.1 DETAILED DISCUSSION OF LIMITATIONS

This section discusses the study’s scope and methodological limitations, including cases where meta-faithfulness may be inappropriate, cost and trajectory-representation analyses, and comparisons with simpler alternatives. It also describes extensions to three non-CoT domains, their evaluation setup, and future directions and reproducibility resources.

M.1 DETAILED DISCUSSION OF LIMITATIONS

  • Results are limited by evaluating only 1.5–1.7B-parameter models and requiring hidden-state access; the audit checks label, not mechanism, preservation, and certification is asymptotic.The audit reports mean mechanism-preservation of 0.92, with slightly lower IVR on mechanism-preserved pairs; the detector suite and language coverage are also limited.Source: §M.1 · pp. 56–57
  • Only 3 altering pairs support SIFT’s cue-injection and cue-verbalization sensitivity estimates, while approximate ground-truth labels (κ ≈0.80) leave possible systematic error.The pooled alteration sensitivity of 0.26 is effectively determined by causal truncation; analyses suggest robustness to moderate label noise, but absolute MFS values warrant caution.Source: §M.1 · p. 57

M.2 EXTENDED FUTURE DIRECTIONS

  • Future work proposes online adaptation and certification, cross-model consistency training, API-only assessment, expanded mechanism audits, and deterministic attention to reduce seed-repeat instability.Other directions include human review of abstained traces, scalable certification, detector interpretability, adversarial robustness, broader language coverage, and dynamic thresholds.Source: §M.2 · pp. 57–58

N REPRODUCIBILITY STATEMENT

  • The reproducibility release commits to SIFT code, FaithShift data and scripts, open-weight hidden-state trajectories, cached API-model states, and audit and analysis code.Experiments use four seeds per detector, twenty seeds for extended stability analysis, and a single NVIDIA A100 40GB GPU.Source: N REPRODUCIBILITY STATEMENT · pp. 58–59

A.3 COST-BENEFIT ANALYSIS

  • SIFT achieves MFS 0.72 at 300 min training and 265 ms inference, while the 4-seed ensemble reaches MFS 0.71 at 180 min and 232 ms; they are indistinguishable at matched 51% abstention (p = 0.21).The section favors the ensemble on cost grounds, while single-seed detectors offer 42–58 ms inference at MFS 0.47.Source: §A.3 · p. 60

B WHEN META-FAITHFULNESS MAY NOT APPLY: LIMITATIONS AND DEPARTURE CASES

  • Meta-faithfulness may be unsuitable when faithfulness depends on domain norms, transformations are adversarial, detectors measure proxies, or multi-environment training data is insufficient.The section recommends explicit domain criteria, auditing preserving transformations, validating construct validity, and using simpler detectors or augmentation when diverse data is unavailable.Source: B WHEN META-FAITHFULNESS MAY NOT… · p. 60, §B.1 · p. 61, §B.2 · p. 61

C TRAJECTORY REPRESENTATION ABLATIONS: ARCHITECTURE AND DIMENSIONALITY

  • Mean pooling reaches MFS 0.68 versus 0.72 for TCN and 0.73 for Transformer; learned TCN features outperform hand-crafted features at 0.72 versus 0.66 MFS.Transformer’s 0.01 MFS gain over TCN carries a 4.1× inference penalty; learned features also achieve IVR 0.050 versus 0.098 for hand-crafted features.Source: §C.1 · p. 62, Table 25 · p. 62, §C.3 · p. 63

C.2 DIMENSIONALITY ABLATIONS

  • At d = 64, SIFT reaches MFS 0.72, IVR 0.050, and 45 ms inference; higher dimensions risk overfitting and d = 256 is 4× slower.IVR improves with dimensionality through d = 64 and then plateaus, making d = 64 the recommended balance.Source: §C.2 · pp. 62–63

D.3 RECOMMENDATION

  • Practitioners are advised to prefer Group DRO over SIFT, or a 4-seed ensemble when simpler methods suffice.Group DRO uses one loss term instead of two, converges with fewer iterations, and has lower code-maintenance burden; the ensemble averages predictions, needs no hyperparameter tuning, and allows inspection of individual seed verdicts (MFS 0.71 vs. SIFT 0.72).Source: §D.3 · pp. 63–64

D.2 EXPERIMENTAL COMPARISON

  • Group DRO approaches SIFT (MFS 0.70 vs. 0.72; IVR 0.075 vs. 0.050), while the JIT oracle reaches MFS 0.75 but requires test-time environment labels.Group DRO trains with 50% fewer iterations and avoids adversarial training; the discussion says SIFT’s gains do not justify IRM + DANN complexity and hyperparameter tuning. Table 27 reports an IRM × 4-seed ensemble recovering 98% of SIFT’s gain at comparable inference cost (180 ms vs. 45 ms) and lower code complexity.Source: §D.2 · p. 63, Table 27 · p. 64

E CROSS-DOMAIN GENERALIZATION: BEYOND CHAIN-OF-THOUGHT

  • The evaluation extends FaithShift beyond CoT reasoning to summarization factuality, retrieval-augmented generation, and instruction following.The main evaluation covered GSM8K, MATH, CommonsenseQA, and BBH; the extension assesses generalizability to non-CoT faithfulness domains.Source: E CROSS-DOMAIN GENERALIZATION: BEYOND… · p. 64, §E.1 · p. 64

E.1 EVALUATION DOMAINS

  • XSUM tests summary-claim grounding, HotpotQA tests whether reasoning justifies answers, and IFEval tests faithful compliance with instructions.Each domain contributes 500 base items and 5,000 traces across shift axes, with ground-truth labels determined by majority vote from three annotators per trace.Source: §E.1 · p. 64, Table 28 · p. 64

E.2 RESULTS

Across non-CoT domains, SIFT reduces transfer gaps, but performance varies by domain and single-domain training does not reliably transfer. Per-domain results also show that the hardest shifts and useful deployment choices differ across reasoning tasks.

E.3 RECOMMENDATION

  • Train on the target domain: CoT-trained SIFT reaches 0.58 transfer AUROC on summarization versus 0.66 when trained on summarization.The recommendations also call for domain-relevant transformations and benchmarking simpler baselines first.Source: §E.2 · p. 64, §E.3 · p. 65

E.2 RESULTS

  • Transfer gaps fall with SIFT in all three non-CoT domains: summarization 0.13→0.08, retrieval 0.16→0.10, and instruction 0.16→0.12.The reported gains are smaller than on CoT, and domain-specific properties pose distinct challenges.Source: §E.2 · p. 64, Table 29 · p. 65

F.1 PER-DOMAIN TRANSFER GAPS

F.2 PER-DOMAIN INVARIANCE VIOLATION RATES

  • Translation is the hardest shift across domains (IVR 0.061–0.098); step reordering is challenging for MATH (0.074) and BBH (0.085).GSM8K and CommonsenseQA have overall IVR below 0.05.Source: §F.2 · p. 65, Table 31 · p. 66

F.3 PER-DOMAIN RECOMMENDATIONS

  • SIFT improves transfer AUROC to 0.70 (+0.08) on GSM8K, 0.58 (+0.10) on MATH, and 0.61 (+0.11) on BBH.Recommendations favor a 2-seed ensemble for safety-critical GSM8K use, SIFT or a 4-seed ensemble for MATH, and SIFT for BBH; use multi-environment training when cross-lingual support is needed for CommonsenseQA.Source: §F.3 · pp. 65–66

G.1 DECISION FRAMEWORK

G.2 CHECKLIST BEFORE DEPLOYMENT

  • Before deployment, validate against 50–100 human annotations and cover at least three environments with 100+ examples each.The checklist also calls for target-domain coverage and an abstention policy; if SIFT requires 51% abstention, it recommends ensembles instead.Source: §G.2 · p. 67

G.3 CALIBRATION AND THRESHOLD SELECTION

  • Set SIFT’s confidence threshold to match the false-positive/false-negative trade-off, then monitor performance and retrain as models evolve.The guidance suggests τ ≈0.4 for medical reasoning and τ ≈0.6 for legal auditing, plus quarterly retraining and weekly AUROC checks.Source: §G.3 · p. 67, §G.4 · p. 67
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