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arxiv: 2605.05134 · v1 · submitted 2026-05-06 · 💻 cs.LG · math.DS

Recognition: unknown

Low-Cost Black-Box Detection of LLM Hallucinations via Dynamical System Prediction

Dan Wilson , Mohamed Akrout
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:45 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords LLM hallucination detectionKoopman operator theorydynamical systemsblack-box methodsembedding spacestate-space modelsresidual scoringpreference calibration
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The pith

Projecting LLM responses into an embedding space and fitting Koopman operators to factual and hallucinated regimes enables low-cost single-pass hallucination detection.

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

The paper treats sequences of LLM outputs as observations from a latent dynamical system once projected by an embedding model. It fits two Koopman transition operators, one to factual responses and one to hallucinated ones, then uses the difference in their prediction errors as a score for whether a new response is hallucinated. A calibration procedure adjusts the decision threshold using a small number of user-provided examples to match different risk tolerances. This single-generation approach avoids the multiple samples or knowledge-base lookups required by earlier methods. Benchmark tests show it reaches state-of-the-art accuracy while lowering compute demands.

Core claim

LLM responses projected via an embedding model form observable realizations of latent state-space dynamics; the factual and hallucinated regimes admit distinct, learnable Koopman transition operators whose respective prediction errors yield a differential residual score that classifies hallucinations.

What carries the argument

The differential residual score computed from the one-step prediction errors of two separately fitted Koopman transition operators, one for factual regime trajectories and one for hallucinated regime trajectories in embedding space.

If this is right

  • Detection operates on a single LLM response without requiring additional sampling or external retrieval.
  • The classification threshold can be tuned to user preferences or domain needs using only a small demonstration set.
  • The method runs as a black box, requiring no internal access to the LLM.
  • Resource overhead is reduced compared to consistency-checking or retrieval-augmented baselines while matching their accuracy on standard benchmarks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the embedding space faithfully captures the dynamics, the same separation might allow early prediction of hallucinations within a single long generation.
  • Extending the operator fitting to online updates could support real-time monitoring of chatbot conversations.
  • Similar dynamical modeling might apply to detecting other LLM failure modes such as logical inconsistencies if they also produce distinct trajectory patterns.

Load-bearing premise

That the projected embedding sequences of factual responses and of hallucinated responses obey measurably different linear transition rules in the lifted space.

What would settle it

A test set in which the factual-operator prediction error is not reliably smaller than the hallucinated-operator error on verified factual responses, or in which the differential score shows no correlation with ground-truth hallucination labels.

Figures

Figures reproduced from arXiv: 2605.05134 by Dan Wilson, Mohamed Akrout.

Figure 1
Figure 1. Figure 1: Our proposed differential residual score view at source ↗
Figure 2
Figure 2. Figure 2: For 300 factual ground-truth and hallucinated responses from the view at source ↗
Figure 3
Figure 3. Figure 3: Hallucination Detection Dynamical System (DS): (a) Phase 1: Dynamical system fitting view at source ↗
Figure 4
Figure 4. Figure 4: Hallucination detection performance using DS classification on 8K samples of the HaluE view at source ↗
Figure 5
Figure 5. Figure 5: ROC curves on the HaluEval dataset as a function of the sequence length L for (a) the average classification performance and for (b)–(f) each embedding model showing the variation of the true positive rate against the false positive rate at various prediction error thresholds. Notably, the magnitude of the improvement from shorter to longer sequence length correlates in￾versely with model size. Smaller emb… view at source ↗
Figure 6
Figure 6. Figure 6: Histograms of token-level residual scores from (6) for Wikibio test data. This effect can be leveraged to implement a calibration process that utilizes a small set of user￾provided demonstrations to determine the classification threshold η. A sketch of this process is provided here. Whether the user is strict or tolerant to minor hallucinations, our method sets the classification threshold η to align with … view at source ↗
Figure 7
Figure 7. Figure 7: User-centric threshold calibration: (1) selection of calibration samples based on user pref￾erence (tolerant or strict to minor inaccuracies), (2) computation of per-token differential residual scores ∆E, and (3) calibration of the classification threshold η to maximize the target metric. 5 Conclusion In this paper, we introduce a new hallucination detection method by rethinking a Large Language Model as a… view at source ↗
read the original abstract

Large Language Models (LLMs) frequently generate plausible but non-factual content, a phenomenon known as hallucination. While existing detection methods typically rely on computationally expensive sampling-based consistency checks or external knowledge retrieval, we propose a new method that treats the LLM as a black-box dynamical system. By projecting LLM responses into a high-dimensional manifold via an embedding model, we characterize the resulting vector sequences as observable realizations of the model's latent state-space dynamics. Leveraging Koopman operator theory, we fit the transition operators for both factual and hallucinated regimes and define a differential residual score based on their respective prediction errors. To accommodate varying user requirements and domain-specific sensitivities, we introduce a preference-aware calibration mechanism that optimizes the classification threshold based on a small set of demonstrations. This approach enables low-cost hallucination detection in a single-sample pass, avoiding the need for secondary sampling or external grounding. Extensive testing across three data benchmarks demonstrates that our method achieves state-of-the-art performance with reduced resource overhead.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a black-box hallucination detection method that embeds LLM response sequences into a manifold and models them as observable realizations of latent state-space dynamics. Separate Koopman transition operators are fitted to factual and hallucinated regimes; a differential residual score derived from their one-step prediction errors is used for classification, with a preference-aware calibration step that tunes the threshold on a small set of demonstrations. The central claim is that this yields state-of-the-art detection performance across three benchmarks at substantially lower computational cost than sampling-based or retrieval-based alternatives.

Significance. If the core dynamical assumption holds, the approach would constitute a meaningful advance by replacing expensive multi-sample consistency checks or external grounding with a single-pass operator-based residual test. The low-cost, black-box framing and the introduction of preference-aware calibration are practically attractive. However, the significance is currently limited by the absence of quantitative evidence that the embedding trajectories actually admit distinct, learnable Koopman operators that separate the two regimes.

major comments (3)
  1. [Abstract] Abstract: the assertion of 'state-of-the-art performance' is unsupported by any numerical results, error bars, baseline comparisons, or dataset statistics. This claim is load-bearing for the paper's contribution and cannot be evaluated without the missing quantitative evidence.
  2. [Method] Method (operator fitting and calibration): transition operators are fitted separately on factual and hallucinated regimes while the classification threshold is optimized on demonstrations drawn from the same data distribution. This construction makes the differential residual score dependent on quantities derived from the evaluation data, raising a direct circularity risk that must be addressed with explicit train/test splits or held-out calibration sets.
  3. [§2–3] Core modeling assumption (§2–3): the manuscript provides no empirical test or theoretical argument showing that standard embedding trajectories are Koopman-linearizable in a factuality-dependent manner. If the fitted operators are not measurably distinct or if residuals are dominated by embedding noise, the method reduces to a generic embedding classifier and the claimed dynamical advantage disappears.
minor comments (2)
  1. [Method] The notation for the differential residual score and the precise definition of the Koopman operator approximation should be stated explicitly with equations rather than described only in prose.
  2. [Experiments] Figure captions and experimental tables should include the exact embedding model, sequence length, and number of demonstrations used for calibration so that the low-cost claim can be reproduced.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment point-by-point below, clarifying aspects of the work and indicating revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion of 'state-of-the-art performance' is unsupported by any numerical results, error bars, baseline comparisons, or dataset statistics. This claim is load-bearing for the paper's contribution and cannot be evaluated without the missing quantitative evidence.

    Authors: We agree that the abstract would be stronger with explicit quantitative support. The full manuscript reports detailed results in Section 4, including accuracy/F1 scores on the three benchmarks, comparisons against sampling-based and retrieval baselines, error bars from repeated runs, and dataset statistics. We have revised the abstract to incorporate the key numerical findings (e.g., detection performance and cost reductions) while remaining concise. revision: yes

  2. Referee: [Method] Method (operator fitting and calibration): transition operators are fitted separately on factual and hallucinated regimes while the classification threshold is optimized on demonstrations drawn from the same data distribution. This construction makes the differential residual score dependent on quantities derived from the evaluation data, raising a direct circularity risk that must be addressed with explicit train/test splits or held-out calibration sets.

    Authors: This is a valid concern about potential leakage. In the original experiments the operators were fit on training partitions and the preference-aware threshold was tuned on a held-out calibration subset disjoint from the test set. We have revised the Method section to explicitly document the data partitioning (train/calibration/test splits) and to confirm that all reported metrics use these held-out sets, thereby removing any circularity. revision: yes

  3. Referee: [§2–3] Core modeling assumption (§2–3): the manuscript provides no empirical test or theoretical argument showing that standard embedding trajectories are Koopman-linearizable in a factuality-dependent manner. If the fitted operators are not measurably distinct or if residuals are dominated by embedding noise, the method reduces to a generic embedding classifier and the claimed dynamical advantage disappears.

    Authors: We acknowledge that the original submission did not foreground direct validation of the Koopman assumption. We have added a new subsection in §3 that (i) reports the Frobenius-norm difference between the factual and hallucinated operators, (ii) shows that the differential residual is not reproduced by noise-only ablations on the embeddings, and (iii) provides a brief theoretical motivation based on the approximation properties of Koopman operators for regime-dependent nonlinear dynamics. These additions demonstrate that the separation is not reducible to a generic embedding classifier. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper projects LLM responses into embeddings, fits separate Koopman transition operators to labeled factual and hallucinated regimes, defines a differential residual score from one-step prediction errors, and calibrates a threshold on a small set of demonstrations before evaluating on separate benchmarks. This is a standard supervised modeling pipeline with offline fitting and held-out testing; no equation or step reduces the claimed detection performance or SOTA results to the inputs by construction. The Koopman fitting is an explicit modeling choice whose validity is tested empirically rather than assumed tautologically, and no self-citation chain or ansatz smuggling supports the central claims.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Koopman theory to embedded LLM sequences and on the existence of separable factual/hallucinated regimes; these are domain assumptions rather than derived results. Free parameters include the embedding model choice and the calibrated threshold.

free parameters (2)
  • classification threshold
    Optimized via preference-aware calibration on a small set of demonstrations to accommodate user requirements.
  • embedding model
    Choice of model used to project responses into the high-dimensional manifold before fitting operators.
axioms (1)
  • domain assumption Projected LLM response sequences behave as observable realizations of an underlying latent dynamical system to which Koopman operator theory applies.
    Invoked when the paper states that vector sequences are treated as realizations of the model's latent state-space dynamics.

pith-pipeline@v0.9.0 · 5468 in / 1395 out tokens · 41088 ms · 2026-05-08T16:45:40.075167+00:00 · methodology

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