Spilled Energy in Large Language Models
Authors: Adrian Robert Minut
Hazem Dewidar
Iacopo Masi
Venue: ICLR 2026 Poster (2026)
Keywords: LLM
hallucination detection
EBM
Primary area: alignment, fairness, safety, privacy, and societal considerations
TL;DR: We recast the LLM softmax as an Energy-Based Model, introducing training-free energy measures to detect hallucinations. Our method pinpoints errors, generalizes across tasks, and shows robust results on several benchmarks.

Abstract: We reinterpret the final Large Language Model (LLM) softmax classifier as an Energy-Based Model (EBM), decomposing the sequence-to-sequence probability chain into multiple interacting EBMs at inference. This principled approach allows us to track “energy spills” during decoding, which we empirically show correlate with factual errors, biases, and failures. Similar to Orgad et al. (2025), our method localizes the exact answer token and subsequently tests for hallucinations. Crucially, however, we achieve this without requiring trained probe classifiers or activation ablations. Instead, we introduce two completely training-free metrics derived directly from output logits: **spilled energy**, which captures the discrepancy between energy values across consecutive generation steps that should theoretically match, and **marginalized energy**, which is measurable at a single step. Evaluated on nine benchmarks across state-of-the-art LLMs (including LLaMA, Mistral, and Gemma) and on synthetic algebraic operations (Qwen3), our approach demonstrates robust, competitive hallucination detection and cross-task generalization. Notably, these results hold for both pretrained and instruction-tuned variants without introducing any training overhead. Code available at [github.com/OmnAI-Lab/spilled-energy/](https://github.com/OmnAI-Lab/spilled-energy/)

Published as a conference paper at ICLR 2026
SPILLED ENERGY IN LARGE LANGUAGE MODELS
Adrian R. Minut 1,2
Hazem Dewidar 1,2
Iacopo Masi
Sapienza University of Rome, Italy
OmnAl Lab 2
GLADIA
ABSTRACT
We reinterpret the final Large Language Model (LLM) softmax classifier as an Energy- Model (EBM), decomposing the eequence-to-sequenee probability chain into multiple interacting EBMs at inference. This principled approach allows us to track "energy spills" during decoding, which we empirically show correlate with factual errors, biases, and failures. Similar Orgad et al. (2025), our method localizes the exact answer token and subsequently tests for hallucinations. Crucially, however, achieve this without requiring trained probe classifiers or activation ablations. Instead, we introduce two completely training-frea metrics derived directly from output logits: spilled energy, which captures the discrepancy between energy values across consecutive generation steps that should theoretically match, and marginalized energy, which is measurable benchmarks across state-of-the-ant art LLMs (including LLaMA, Mistral, and Gemma) and on synthetic algebraic operations (Qwen3), our approach demonstrates robust, competitive ballucination detection and cross-task generalization. Notably, these results hold both pretrained and instruction-tunee variants without introducing any training overhead. Code available at www.ooo///ooonmaa/aaaaae
single step. Evaluated on nine
Q/A: " Whht is the capital Italy? Answer:'
Logit
Spilled (Ours)
The capital of Italy is Rome
The capital of Italy is Rome
The capital of Italy is Sydney
The capital of Italy Sydney X
Reasoning:
'A farmer has 12 chickens. Each chicken lays 2 eggs per day.
How many eggs will the farmer collect in days?'
Logit
Spilled (Ours)
chickens lay eggs per day
chickens lay eggs per day In 5 days the farmer will collect 12 x
days the farmer will collect 12 x
120 eggs in 5 days
2
x
120 eggs in 5 days
chickens lay eggs per day the farmer will collect 12
chickens lay eggs per day In 12
days
$
days
the farmer will collect 12 x
470 eggs
in days X
2
470
eggs
in days
Figure 1: Color-coded comparison of hallucination detection with LLaMa-3 8B using logit confidence and spilled energy. Our method generalizes well across topics (c.g., Q&A. reasoning) and diverse L.Ms.. / indicates a correct answer and X an incorrect one. While our approach focuses on the exact answer tokens (e.g. Rome/Sydney and 120/470, see Section 4.2), here apply min-max hallucination.
normalization to the full answer for visualization, as truthful

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INTRODUCTION
The widespread adoption of Large Language Models (LLMM)) across various domains has brought increasing attention to their critical limitation: their tendency to generate incorrect or misleading information-commnnly referred to as "hallucinations." " This issue supports the idea that LLMs are just stochastic parrots (Bender et al., 2021) answering in a way that is statistically plausible with respect to the input prompt despite not having a real understanding of it. On the other side, recent reasoning capabilities proper to ChatGPT 4o (OpenAI-Teame) 2023) or Deepseek (Liu et al., 2024) offer counter evidence to actually support this.
Ongoing research seeks to characterize and categorize hallucinations, setting them apart from other error types (Liu et al., 2022; Ji et al., 2023; Huang et al., 2023h; Rawte et al., 2023). At the same time, recent discussions have introduced terms such as confabulations (Millidge, 2023) and fabrications (McGowan et al., 2023), sometimes attributing a form of "intention" to
the very idea of LLM "intentionality" and other human-like qualities remains contested (Salles al., 2020; Cerapio-Gareia
Serapio- Gareía et al., 2023; Harnad, 2024). Research on LLM hallucinations can be categorized into two main branches: the first one is the extrinsic branch, where the hallucinations are measured with respect to the interpretation that humans give to those errors (Bang et al., 2023; Ji et al., 2023; Huang et al., 2023b; Rawte et al., 2023). The second branch was started by Kadavath et al. (2022b), proposing to study the hallucinations within the model itself. Following Kadavath et al. (2022b), the work in Li et al. (2024) proposes Inference- Time Intervention (ITI) as a way to improve the "truthfulness" of LLMs at inference time. ITI functions by altering model activations at inference time, steering them along specific directions within a restricted set of attention heads. Our work is also different from Yin et al. (2023), since we care about detecting errors in LLMs, whereas they introduce an automated methodology to detect when LLMs are aware that they do not know how to answer.
In this work, we follow the definition of hallucinations given by Orgad et al. (2025) as any form of error produced by an LLM- factual mistakes, biased outputs, breakdowns in common-senss reasoning, and related issues. Like them, we also confirm that the truthfulness signal is concentrated in the "exact answer tokens." " Nevertheless, unlike them, we abandon the idea of using a probe classifier (Belinkov, 2022) trained for each task and dataset. Given that LLMs are foundational models, user interactions typically occur in the wild, making difficult to predict which probe classifier is best suited for detecting hallucinations in real-world scenarios. Furthermore, in this setting, classifier weights should not only be updated dynamically for each task, but the optimal token-layer combination is also dataset-dependdett which conflicts with the broad LLM applicability. Indeed, in the work by Orgad et al. (2025), the authors report:
"We find that probing classifiers do not generalize across different tasks."
In our paper, we propose to solve this problem with a training-free method that generalizes better across different tasks and is mathematically principled using the framework of Energy-based Models (EBMs). Fig. reports a qualitative comparison across tasks, comparing to the logit confidence. Additional samples are shown in Appendix D.2.
We reinterpret the final softmax classifier over the vocabulary of LLM as an EBM, taking inspiration from what Grathwohl et al. (2020) did for classifiers. This perspective enables us to decompose the probability chain into multiple interacting EBMs that operate jointly during inference. Through this decomposition, we introduce the notion of "spilled energy" in LLM decoding and show empirically that such spill strongly correlates with errors. Given that method is solely based on the mathematics of EBMs and the chain rule of probability, we do not have to train or tune our detector, striking good generalization across tasks and LLMs. Building on this foundation, our contributions are as follows:
Training- free, LLM hallucination detection generalizing across tasks using the EBM framework. We introduce a method for detecting hallucinations that requires no additional training, in contrast to prior work that relies on trained classifiers and ablations of model activations. Our approach directly reads values inside the I.IM, enabling natural generalization across tasks and performing better than logit-based detection.
Two energy-based metrics. We define two complementary measures of energy spills: (i) delta energy A (x::), which captures discrepancies between energy values across two time steps that
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p(xaa) =
M (x)
Po (K.)) Pol 1n)
190
100
PolKa1]
Polk.-ia)
291%, 231
Spilal Eaarg
(b)
(c)
adllanno
Incorrect X
Correct
400
30
1001
LLM
θ
25
2010
X--1
-50
0 -50
-25
0
(a)
(d)
Figure 2: How energy spills in LLMs. (a) Language Modeling p(x;1) is attained as decomposition problem following the chain rule of probability, implemented as autoregressive: we recursively apply discriminative classifier over the vocabulary V to attain generative modeling with larger context size i.e. p(x; x;11:1) (b) We reinterpret each discriminative classifier as a generative EBM, finding connection between two quantities that should be the same across time steps yet are different. We call this difference "the spilled energy" E (x.1) in Eq. (8). (c) Given that we simply read values inside the LLM, our approach is training-free and correlates well with hallucinations on synthetic math dataset with increasing difficulty; (d) histograms of spilled energy values, for incorrect and The two distributions are easily separable by using simple threshold, resulting in generalization across real-world tasks.
correct answers on all nine datasets using min pooling for Llama-3-Insee
should be mathematically equivalent, and (ii) marginal energy (Xaa), which can be evaluated at single time step.
Scalable and generalizable analysis. Our framework is mathematically principled, training-fres and exhibits strong cross-dataset generalization. We scale our analysis to state-of-the-ard LLMs, including Llama 3-8B-Instruct and Misttal-7B-Inntrruet, and demonstrate competitive performance across nine benchmarks, showing robustness across datasets and architectures.
Fig. 2(a) illustrates the core idea of our method: rather than using naive approach, such simply recording the logit or training a probe classifier at the activations of the answer token, we first reinterpret the LLM as autoregressive EBM via the chain rule of probabilities. We then further decompose each conditional probability, incorporating insights from Grathwohl et al. (2020). At the time step of the exact token compare it with the marginal energy at the next time step i, corresponding to the denominator of the softmax. According to the chain rule, these two quantities should be identical; however, they differ in the LLM implementation A Ag (x.1)). correlates strongly with instances where the LLM produces an incorrect output- see Fig. 2(c). Moreover, its detection signal separates well correct and incorrect classes across datasets, reflecting the model's confidence, as shown Fig. 2(d).
extract the energy, which corresponds to the logit, and
2(b). We find that the discrepancy, which we term spilled energy
RELATED WORK
EBM applications to Trustworthy AL EBMs have been applied to improve the reliability and inter- pretability of Deep Nets. For example, Energy-Bast -Baed Outtof--iitriiuttinn Detection (OOD) (Liu et al., 2020) uses the energy score as more robust alternative to softmax confidence. At the same time, Grathwohl et al. (2020) presents how to reinterpret a discriminative classifier as EBM to train models that are both discriminative and generative. Following this work, Zhu et al. (2021) provides new insights into the role of energy when training EBMs and robust classifiers using adversarial training. Instead, Mirza et al. (2024; 2025) explain adversarial attacks by reinterpreting the softmax classifier as an EBM, showing that these perturbations correspond to shifts in the underlying energy landscape.
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Foundations of Hallucination in LLMs. LLMs are prone to diverse errors- including bias, reasoning failures, and generation of factually incorrect information unsupported by reliable sources. Karpowicz (2025) frames hallucination and imagination mathematically identical phenomena, both emerging from a necessary violation of information conservation. Also Xu et al. (2025) provides a formal Sarning-theoreel proof that hallucinations are unavoidable. They define formal world in which both the LLM and the ground-truth are computable functions, showing through classic results in computability theory, that no LLM can learn all such functions. As a consequence, hallucination is not just practical artifact but a fundamental limitation of LLMs, valid even under idealized conditions. Recently, Kalai et al. (2025) showed that hallucinations come from the statistical problem of the pretraining methodology: minimizing the cross entropy naturally causes errors because it does not train the model to express uncertainty and say "I do not know." Kalai et al. (2025) proposes changing the evaluation practices to not reward models for guessing, but rather to mimic the human exams that penalize only wrong answers.
Detecting and Mitigating LLM Hallucinations. Orgad et al. (2025) train classifiers on the internal representations of the LLMs to predict, based on the features, the correctness of the answer. Given an LLM in white-box setting, an input prompt, and the generated response ÿ. the classifier's task is to predict whether à is a ballucination. Orgad et al. suggested that LLMs may encode more factual knowledge in their latent subspaces than is revealed in their outputs. Gekhman et al. (2025) proposed a framework for studying hidden knowledge. Finally, Santilli et al. (2025) point out that uncertainty quantification in language models is often evaluated using metrics like AuROC. This shares biases between detection methods and correctness functions (c.g.. length effects) that systematically distort results. One way to mitigate hallucinations is to act the decoding stage, where the output generation can be steered Subramani et al. (2022). Steering vectors provide a straightforward way to control a model by adding a fixed vector to its activations (Dunefsky & Cohan, 2025). Fu al. (2025) introduced DeepConf, a test-time method that leverages model-internal confidence signals to filter out low-quality reasoning traces during or after generation. Kuhn et al. (2023b); Fadeeva et al. (2024); Farguhar et al. (2024), and its follow-up by Kossen et al. (2025) in which they approximate the semantic entropy in a more efficient way. Constrained decoding approaches Li et al. (2023); Peng et al. (2023) modify token selection policies. Similarly, reinforcement learning with fact-hased rewards Ouyang et al. (2022) has been used to bias decoding trajectories toward verifialle outcomes. Incorrect answers may also be given due to ambiguous prompt: Kuhn et al. (2023a)'s CLAM framework uses few-shot prompts to classify a question's 's ambiguity and then asks the user to clarify.
3 BACKGROUND AND FOUNDATIONS
3.1 ENERGY-BASED MODELS
We give an overview of Energy-based Models (EBMs) and their use in discriminative classifiers.
EBMs. Energy- -Based Models are a class of probabilistic models in which the probability distribution over data points is defined in terms of an energy function Ex). The energy function, parameter- ized by a neural network (Lecun et al., 2006), assigns a scalar energy to each configuration of x, where lower energy values correspond to higher likelihood. The resulting probability distribution is given by pa (x) = expl -En())) where Zo denotes the partition function (normalizing constant), defined as Zo = Σx exp(-EE for discrete x, or equivalently Zo = exp( (x)) dx for continuous x. Standard neural networks are often deterministic function approximators, mapping x y, EBMs instead define a full probability distribution over data or latent variables.
Zo
One of the strengths of EBMs is their flexibility in modeling arbitrary distributions without being tied to specific parametric form. This flexibility comes from the fact that the energy function E(x) can be defined in various ways. Training involves learning the parameters of the energy function such that the probability distribution pe (x) matches the empirical distribution of the data. This is typically achieved using techniques like contrastive divergence, score matching, or maximum likelihood.
Notation. Let V denote the vocabulary of an LLM, i.e., the set of all tokens that can be processed as input and generated at each decoding step, with size |V| = V. We shorten the sequence of tokens . x,} } as X = (xwa), and x; V denotes the token in the i-th position along the sequence. We model the LLM as a function 0 IR NV - RV, implemented by a transformer, or any other lequence-to-sequenn mechanism. For sequence {x:1] input, we write 0 (xxa) [k] to denote the
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predicted logit assigned the k--th token class in V for the i + 1 token in the sequence, is standard in autoregressive LLM training (Ouyang et al., 2022).
3.2 AUTOREGRESSIVE LARGE LANGUAGE MODELS
Generative modeling has been pursued through variety of approaches beyond autoregression (AR). Variational Autoencoders (VAEs) (Kingma & Welling, 2014) learn probabilistic latent variable model by encoding inputs into latent space and decoding samples back to the data domain. Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) frame generation game between a generator and discriminator. The diffusion process has been incorporated into neural nets Dich--Dicsttein et al., 2015) and, more recently. Diffusion Models (Ho et al., 2020) have emerged as powerful class generative models. While these paradigms differ in how they approximate the data distribution, AR models are special in their kind and take more direct route by factorizing the joint probability of sequences into conditionals, making them especially suitable for language modeling. We now focus the AR formulation that underlies most LLMs. Textual data is segmented into a sequence tokens X' (x...... X1}. J. and language modeling objective is employed to maximize the likelihood of such data (Radford & Narasimhan, 2018). In other words, we model the joint probability of tokens in the sequence X, through a conditional probability parameterized by A:
min-man max
p(Xaa) p(x; x;-1a) ... p(x2 x1) p(x1)
II pe px (x-111) pe (x1).
(I)
discriminative model
What we find interesting about this factorization is that, although seeks to attain generative modeling, i... p(X;1), it actually uses recursively discriminative classifiers, parameterized by transformer network 6, that predicts a discrete distribution of the next token x; over the vocabulary V, given previous tokens x,-1:1 This is used to model each conditional probability.
4 How ENERGY SPILLS IN LLMS
When predicting the token at position i, the conditional probability modeled by 0 can be decomposed using the probabilities of the sequences. As result, the marginal term from step cancels out with the sequence probability from the decomposition at the previous step - 1, which means we have:
step
p(x,1)
II po (x
II
pa (x(11)
pe (x:1)
Pal [x 111
p(x;:1).
Pa (x;=1:1)
2a Pa(xx=T11
Po (x1:2:)) 2:1)
step
This indeed confirms that Eq. (1) results in the correct formulation for language modeling, which p(xa1). Following the mathematics, these quantities should cancel out along the sequence, but we will now show that, in practice, this constraint is not explicitly optimized for, and we can exploit it for hallucination detection.
4.1 INTERPRETING LLMS AS ENERGY-BASED MODELS (EBMs)
Let us continue the expansion from Eq. (2). Writing the conditional as the ratio between the joint distribution in the numerator and the marginal distribution in the denominator, note that this ratio is actually implemented in LLMs as softmax classifier that digests the embedding of the prior sentence X;- 1:1 and predicts the next token x;; thus, this chain of equality holds true. We can then apply the "trick" from Grathwohl et al. (2020) as:
x=(x = (x exp 0xx [ix(),) where id: 10.13V + [1....,1] VJ. (3) pe 1x1)
exp xxp((()-..)(M)
id is the map that takes as input a one-hot encoding vector x; for word token at position in the text and outputs its index in the vocabulary. typical cross-entropy loss only optimizes with the
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supervision provided by the ground-truth token, through the vocabulary index id(x;). This loss ignores all other quantities or constraints related to the complete sequence X,, i.e., it ignores all the time steps higher than i + 1.
We can write the conditional probability of Eq. (3) as a ratio of two EBMs as:
exp( Lig (x..)) (0)
lo
log
(x.: 1 + Ei (xi-1:1).
(4)
exp(
F (x)-111) Z(0)
Following Zhu et al. (2021), the partition functions simplify since log 2 log Z(0)'.
Egg En are computed from the output of the model, but with two big differences: E% a single logit extracted using the id the sampled token, Eg by marginalizing over all ids the vocabulary.
The two energies be derived from the softmax of the logits, by connecting Eq. (4) and Eq. (3):
- llo (x x;11:1) = log (xp(0(xx-nn))lddd)))
(5)
exx [k])
(x1 [ic(x)) + log
exp (x,-111)
(6)
Fgg (xx:1)
Egg (X1-1:)
where (x;;-1a) produces the logits over the entire vocabulary V, and i dx;) allows us to extract the logit of the sampled token at decoding step i.
We can think of (xx.) as the energy of the sampled tokens (xa1) ). and Eg (x;-1:1) as the energy Eg (xa), marginalized over all possible x, Considering the decoding at step in Eq. (4), we get:
Egg (xa1) = -xx
(x4-11111) = log
exp (1)
Using the chain rule and Eq. (6), can write the negative log-likelihood in terms of energies as:
- -logp(xNal = log Il
= Σ F (xxa) Eg (x;-aa)
without considering the base case (x)))). Now, if we develop the above equation as done for Eq. (2), write the total energy of a sequence of length N as (x(x..). Observe that the two energies, not interacting at the same step but at steps and i 1, should be equal, but they are measured in the LLM at different generation steps and from different components.
timestop 141
timestep
Eo(xxa) = DE EE (x++1a) Eg (xaa)
En (x
Eg (xa1) Egg (x,11)
E (x,, 1:1)
(X.a)
At timestep + 1, first - -E (x.1) is measured, taking the denominator the softmax as in the right part of Eq. (6), whereas at timestep i, the second Eg (xx1) is taken, reading the logit in the softmax, left part of Eq. (6). We thus define the discrepancy between the two quantities as spilled energy:
Definition 4.1 (Spilled Energy (x;1))) The spilled energy in LLM is the difference between two energies that, in principle, should be equal, but given that they are measured i) at different time steps ii) in different components, could be different.
A A -E (xx1) + (xxa) = log Σ, exp 00 (0 [[a) + 0 [
rimestep
limestep
Since both terms on the right side should be equal to Ex(x.1), delta values should always be zero when we are correctly modeling the energy timestep i. A shorter explanation for why spilled energy needs to be zero is given Appendix A.3.
For formal proof, please Appendix A.I.

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4.2 DETECTING HALLUCINATIONS WITH SPILLED ENERGY
EBMs have previously been used to assess neural network credibility (Liu et al., 2020), and calibration for LLMs has been explored by the Anthropic team (Kadavath al., 2022b). However, dominant training-free baselines such as logits or "p(true)" remain weak. We likewise adopt a training-free approach, but rely on Eq. (8) and its variants as discriminants.
We feed the prompt (x;-1, } } to the LLM O and obtain the completion {xN...., x. Follow- ing Orgad al. (2025), we focus on the "exact answer" tokens those in [i + N, that contain the precise answer (e.g., Rome in Fig. 1), denoted [u, w| S [ 1, NJ. For instance, it would be the tokens associated with Rome in the question Fig. 1. We identify this span by prompting the LLM for brief answer. When the answer spans multiple tokens, we apply a pooling strategy, which we ablate in Section 5. We propose measuring two values that correlate well with hallucinations:
1. Marginal energy EØ (xa1);:
2. Spilled energy A by definition of Eq. (8).
We also attempt to combine the two metries into scaled spilled energy AE,, where the spilled energy is multiplied by the absolute value of the marginal energy as (x,1) = Eg (xx:)) (x:1). The metries proposed here are independent, new for LLMs, and can all be tested efficiently. These measures can be computed over the full sequence, but for error detection, discussed in Section 5.2, we must extract the values the localized exact interval [u, w) to avoid false positives. Note that Eg (xx1) is the classic bascline which in literature is referred to as "logits" or "logits confidence".
5 EXPERIMENTS
To evaluate spilled energy, we consider two complementary settings. First, a controlled synthetic environment, where we generate both correct and incorrect mugti-digit arithmetic solutions. Second, established real-world benchmarks, where errors arise naturally across diverse reasoning and com- prehension tasks. Together, these experiments test whether insights from the clean synthetic setup transfer to the complexity of open-domain language understanding.
5.1 SPILLED ENERGY UNDER SYNTHETIC ARITIIEETCC
Experimental Setting. We first evaluate spilled energy in a controlled setting: multi-digit arithmetic problems with more than 14 digits. For each instance, we generate both correct and incorrect solutions. We tested three different LLMs: Llama-3 8B (Dubey et al.). Qwen-3 8B (Qwen-Team) and Mistral-7B-Instrnne numerical errors of varying magnitude. Specifically, we define three error ranges that differ in their difficulty of detection:
v0.3 (Jiang al.). Incorrect solutions are obtained by introducing random
Easy: random offset in the range [1000, 10000]. which are typically easier to identify.
Medium: random offset in the range [100, 1000), where detection requires closer inspection.
Hard: random offset in [1, 10j, much harder to detect since they appear plausible at first glance.
This design allows us to systematically probe whether spilled energy can distinguish between correct and incorrect generations across different levels of error subtlety.
Results. We observe that spilled energy values separate correct from incorrect solutions with high reliability across all error ranges and across all LLMs. In particular, spilled energy consistently assigns lower values correct generations and higher values to incorrect ones, producing clear margin of separation. Compared to standard baselines such as logits, spilled energy achieves superior discriminative power, especially for errors in the more challenging range [1, 10), see Fig. 3. We offer more results in Fig. 5. Larger, better-detailed ROC and histograms are in Figs. 6 and respectively.
5.2 CROSS-DATASET RESULTS IN REAL- WORLD BENCHMARKS
Experimental Setting, We evaluate our methods on a diverse set of established NLP benchmarks, including Math (Hendrycks et al.), TriviaQA (Joshi et al.), HotpotQA (Yang et al.), Winogrande
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Spilled Energy
Easy
Correct
Incorrect
LLama-344 3-a---8---nII-tcet
Logit Energy
Medium
Marginal Energy
Hard
200
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150
150
0.8
0.6
0.4
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0.2
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Spilled Energy
Spille. Energy
Spilled Eaetgy
FPR
(a) Easy
(b) Medium
(c) Hard
(d) ROC
Qwen-3 8B
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1.0
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Spilled Energy
Spilled Energy
Spilled Energy
FPR
(e) Easy
(f) Medium
(g) Hard
(h) ROC
Figure 3: Histograms of Spilled Energy values across models (rows) on Math Sums with different error ranges in the answer (columns, decreasing range left to right, making it harder to detect errors). All sums are performed 13-digit integers. In the fourth column, show ROC curves for Hallucination Detection across the error ranges (colors) and methods (line styles).
x4
55
56
76
74
hoton12e 2.07
2:50
24
7.72
3.80
22:22
3.54
12
20.00
32
52
22
22
52
52
32.44
20.38
3311
11.00
-3
Ahwggandd unds
24
57
wiwagranee
3.24
22.44
11.00
0
Tun
Thih
52
32.27
23.03
3.28
-180
12.00
Indt
25
73
57
km.0--2.77
-2.57
25.04
28:34
-320
-189
28:25
11.50
$2.00
-10
helpotgu..c NC
10
8
hwrpsggu.o NC
38.27
33.03
20
5.72
0.80
20:32
-12.4-37.22
11234
$7.02
mil
mochh
1nb
mah
Taxt SRTANE
Teat dhbaatt
(a) Results by Orgad et al.
(b) Spilled Energy Improvement over Orgad et al.
Figure 4: (a) AuROC performance as percentages probing classifiers on exact answer tokens by Orgad et al. for LlaMA-3- (b) depicts the performance difference between our Spilled A F. with Min pooling and theirs. Positive values indicate cases where Spilled AE outperforms Orgad et al.. This comparison highlights the generalization capabilities of our method, compared probing classifiers. Legend: low performance
high performance.
(Sakaguchi et al.), Winobias (Zhao et al.), Movies (Orgad et al.), MNLI (Williams et al.) and IMDB (Maas et al.). These datasets span wide range of reasoning and error-detection tasks, allowing us to test whether the patterns observed in the synthetic arithmetic setting extend to real-world, open-domain scenarios. Here too, we evaluate multiple LLMs that are either instruction-aligned or not aligned, such as LLaMA-3 (Dubey al.), and Mistral (Jiang et al.). As emphasized by Orgad et al., it is essential to first localize the tokens most relevant to the final answer before applying error detection. Since exact answer tokens may consist of multiple tokens, we further adopt pooling
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Table I: Hallucination detection performance, in terms of AuROC, across nine benchmarks and four different LLMs. We measure the generalization across all tasks by computing the average.
Pool HotpotQA HolpotQ--WC
IMDB
Math
MNLI
Movics
TrivisQA
Winobias
Winogrand:
Average
LLstrel Dubey al. (2024)
p(true)
58.31
50.72
52.33
...30.....
45.99 ...1
45.47....
48.33
51.29
Orgailer al.
Mcan 66.56.2-10 10
59.00
69,
60.56
60.56
66.4
63.22
67.33
58.00
66.m
Logit E
Max
72.85
91.11ar.
42.08
s7.81
25.52:amm
43.97.
68.89...
39.95
19.40....
54.62....
Marginal gnt
Max
76.
30.74...4
25.63.2.
27.08....:
96.17-c.s
80.13
-7.47-..
65.72.....
Marginal 8
Min
75.9
01.91:2
97.57
14.37
70.55
61.21
72.21
73.38
47.19
53.98
62.93-11
Spilled AE,
Max
53.6 33.65/103 1.400
12:00
55.80
35.44.044
18.8
70.30
3.40
2..5
44.32
48.5
Spilled AE
Min
85.98
1000
93.00
47.66
65,58
73.95
89.34
87.07
.1.as
60.72
55.17
73.16-154
LLaMA Dubey et al. (2024)
p(true)
52.83
53.78. a:0
60.76...0o
62.94... .0.3)
50.02.1.1
53.47
54.90-010
Orgateerl al.
Mean 61 00.22ga+r m
56.78 m
72.67
09.67
60.33 a.m1 64.00 i
66.44gmm
60.89
53.56
62.84
Logit 8"
53.47.3.33
49.02
48.27a1:92
57.38.000
91.76.n.1
5..4
52.77.2.1
50.74.1.31
51.17.3..
56.89
Marginal g
Max
78.00.1.30
76.90.1.v
48.
68.77
10.93
80.70
67.49
s1.91
51,28
Marginal pm
Min 58. 08.399...
89.20
54.71
34,13
97-4220
50.37
69.88
49.08
-9.00
57.68
Spilled AE.
27.75-1.83
79:44
43.39.1.82
72.87
99.97...u
61.56-1..6
77.55
52.94:2.7
48.17
68.
Spilled AE
Min
79.04 13.78
80.83 11.07
43,22 ..
74.36 (5.04
99.97
61.97 2..01
78.54
-1.57
52.11
48.21
68.69
17.48
Mistral-Instreet liang al. (2023)
p(true)
56.6222.0 so
53.41 cm o4
84
51.65
g1
24.9/..0.0
63.59
26.34gn
56.92.0.17
55.88
Orgai et al.
Mcan 64.78.10 56
56.78
82.67
13.05
68.78
13.40
64.22
::2:32
64.89
62.44:22.
61.00
61.44:11.3
65 56 05.54
Logit 8
77.24.3.0
83.84.1.0
22.28.2.44
57.67.8m
78.98
76.89.1..0
80:35
45.53
48.17.3177
63.44
Marginal E
Max 64.63.1.40
33.42
a2.3
26.52
17.62
86.60
1.50
65.46
:4.4
51.14
1.71
53.68
Marginal pm
Min 87.
97.94
97.96.0.0
87,25 14.90
49.19
48.49
71.12
Spilled AE,
Max 49.13-2.0
02.00
46.45
29.0 39.05... 10.00
53.79....
55.24
46.73-..x
a3.30
51.20-1.4
Spilled AE
Min
91.12 10
97.47-.
66.63 59.77
on
95.9 95.05.n xo
94,99
91.75
50.74g415
49.00
77.49
Mistral Jiang al. (2023)
p(trie)
54.2 54.21gam gm
68 n4
20
21.94
49.12.n..
58.00
23.76
47.29
51.
51.36.0117
Orgail al.
Mcan 61.78
57.44
76.22
12.02
65.78
15:27
56.67
0:3.03
64.22 5.01
64.33
58.00
54.56
1.30
62. 11:05.23
Logit A
Max.
49.54 i3.4
52.47
32.72
57.21
92.49
30.52
2.00
46.53
44.41
49.51
Marginal Em
Max 83.57 1s
86. W6.8ggmm m
45.31 m3
62.26g-m
96.03
99.27
92.26
g1
54.49.--4
74.59_
Marginal g
Min
87.52
1..33
90.91
$4,69
86.21
98.80
94,41
83.66
52.75
46.37
77.19
Spilled AE.
Max
60.54
43.47
71.93
45.94
78.84
67.92
57.24
51.88
Spilled AE Min
84.24
83.74 41
57.48
28
96.699mmm
84.47
81.27
50.62 79
48.72 1.00
73.94
strategy across the localized span to obtain a final score per sentence. We compare spilled and marginal energy against baselines such as the probing classifiers of Orgad et al., logit confidence of Varshney et al. and p(true) of Kadavath et al..
Ablation of the exact answer token. We provide an ablation experiment on the impact of selecting the exact answer tokens. Table 2 reports average AuROC over 9 benchmarks and 4 LLMs with the exact answer, along with another column that offers the improvement provided by using the exact answer. Like prior work, we confirm that searching for the exact answer provides a notable boost: the improvement is very pronounced (~ 24%) for spilled and marginal energy, while the logit baseline receives a modest increase of 9%.
Cross-dataset results. We next evaluate in the more general setting of cross-dataset transfer, which better reflects real-world usage. For methods requiring training, we report the average performance on each dataset when trained separately on each of the other datasets (c.g., performance is the average accuracy of classifiers trained on each of the other nine datasets). Fig. 4 shows a confusion matrix of cross-datase performance, where the rows represent the training dataset and the columns represent the testing dataset, and where red indicates good performance and blue indicates low accuracy. The model tested is LlaMA-3- soon as they go out-of-diitribution from the training dataset, perform only marginally better than random guessing. The sharp drop observed in the off-diagonal elements supports our premise that this standard, in-distribution setup significantly overestimates the utility of trained probes for broad LLM deployment. Meanwhile, Fig. 4b displays the improvement of Spilled AE over the probing classifier, where a positive red result means improvement of our method. Ours exhibits greater performance across most datasets without requiring training. The generalization is proved with strong increment over the off-diagonal. Moreover, in some cases, such as TriviaQA, HotpotQA, and Movies, we have improvements even on the diagonal. Additional confusion matrices are available in Appendix D.3.
IMDB
Fig. 4a shows that probing classifiers, as
Table I summarizes results across nine benchmarks. The result reported in each cell is the average of the accuracies of Fig. 4a within a column. Spilled energy consistently outperforms logit confidence, and substantially surpasses the probing classifiers of Orgad et al. (2025). While this latter performs well when trained and tested on the same dataset, their performance drops sharply under cross-datase evaluation, reflected in their higher standard deviations. By contrast, ours requires no training and

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Published as a conference paper at ICLR 2026
Table 3: Hallucination detection performance on the Gemma Model Instruct for different parameters of the model, IB and 4B.
Pool IMBD
Movies
TrivisQA
Winogrande Winobias
MNLI
Math
HeepoiQA HatpoQA WC
Average
Gemmau Instruct 4B Kamaih el (2025)
Logit As
Max 50.09 1o.45
60.88
1.00
53.95
49.77
co.35
$4.43
2.00 27.00
78.64
62.84
64.49-22 -2.02
55.79
10:34
Marginal Arm
49.14.2m
51.49
47.97
1.00 100,00
74.57
aas 83.70
85,95
73.73
Marginal E
Min m0
51.29..1.0
55.33
48.12
51,91
-2:10 99.01
76.03
sa2r 62.59 c2.59
21.84-277
63.00-104
Spilled A,
Max 21.89
50.77
56.08:2..
50.59-
95.61
21 50.87
51,21
03.94..1
Spilled AE
Min 50.89
16.13
29.01 06
50.18
69
53.10.a.0
99
82.29
so.ss 89.10
82.70
75.89
12
Gemma- Instruct IB Kamath et (2025)
Logit A
Max 46.33
48.12
58.89....
50.50
53.49
49.28
65.12
:0.4 62.24
75.67
56.63 1o.13
Marginal Am
45.42.
94.15
49.93
98.17..0.0
64.21
10.07 86.87
72.77
Marginal A Min 54.58
28.93
:14
39,80
49.84
4.33
50.39
56.33
11:00
63.20
-4:27 41.58
61.56
-..m
49.58
10.47
Spilist AE.
45.17
33.27
29.01
52.27
49.91-2.0
77,48
40.49.a.17
49.18
35.77
48.0 48.06-12:1
Spilled AE
Min 45.02 12.45 82.82
112.91 80.73
52.48
11.75
49.77
22.02
92,93
56.82
10.00 85.64-2.2
71,86 -1.77
68.67 20.44
generalizes robustly across diverse benchmarks. We observe that instruction-tuned models tend to amplify the margin by which spilled energy outperforms other methods, whereas on non-aligned Mistral, spilled energy may rank slightly behind marginal energy. We also compare pooling strategies and find that min pooling yields the best overall performance across methods. Table 3 shows our method generalizes Gemma over different LLM size, IB and 4B.
Impact of Instruction Tuning, We observe a difference in the behavior in the base models and their instruction-tume generally improves genera-
Pool
Average % w/ exact answer
Exact
answer increase
ones. While instruction-tune tion quality, can degrade the calibration of classical confi- dence metrics, as described in Huang et al. (2023a); Ho et al. (2025). For instance, examining the average performance in Table I, the logit baseline Eg decreases from 56.89% to 54.62% for LLaMA-3, indicating that fine-tuning may lead to overconfidence. In contrast, Spilled Energy (A) con- sistently benefits from instruction tuning, showing improved detection rates across both LLaMA-3 (68.69% to 73.16%) and Mistral (73.94% to 77.49%).
Logit E Orgad al.
Max
56.12
+9.23
Mean
63.67
Marginal g
Min
67.23
+20,02
Marginal ET
Max
63.34
+3,62
Spilled AE
Min
73.32
+24.06
Table 2: Improvements in AuROC with the exact answer. Average across 4 LLMs and 9 benchmarks.
Variance and Generalization. A notable observation in Table 1 is the higher standard deviation associated with marginal and spilled energy compared to the probing classifiers in the average column. This variance is not weakness but a reflection of the method's 's training-free free nature. Since AE relies on the intrinsic energy landscape of the LLM, its magnitude and sensitivity are naturally dependent on the specific domain (e.g., the sharp energy peaks in Math and HotpotQA versus the flatter distributions in Winobias and IMDB). Probing classifiers, by contrast, have high- vahrance when cross-testing yet the average of cross-testing results is mostly constant just above random chance (%x 62- 64%).
Limitations. A current limitation of spilled energy is that it sometimes produces false positives on tokens that are not semantically informative, as shown in Appendix D.2. We observe this effect most prominently on punctuation tokens (c.g., commas, periods) and on words at the beginning of sentences. In these cases, the probability mass over the next token is naturally spread across many plausible options, leading to inflated spilled energy values even in otherwise correct generations. This highlights the importance of accurately identifying the exact answer tokens, as detection is most reliable when restricted to the parts of the output that carry the semantic content of the answer.
6 CONCLUSION
We reinterpreted the softmax layer of LLMs as an EBM, which lets us define spilled energy: the discrepancy between energy values that should be equal across consecutive time steps. We show theoretically and empirically that this discrepancy provides strong, training-free signal for detecting hallucinations and errors in LLM outputs. Through synthetic arithmetic experiments, we demonstrate that spilled energy reliably separates correct from incorrect generations, outperforming baselines such as logits and marginal energy. Across diverse world NLP benchmarks, spilled energy generalizes robustly without requiring additional classifiers task-specifi training, unlike probing methods that struggle with transfer. Overall, spilled energy offers a principled and practical framework for error detection in LLMS and a new perspective on the internal energy dynamics of autoregressive models.
10

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Published as a conference paper at ICLR 2026
ETHICS STATEMENT
This work adheres to the ICLR Code of Ethics. Our study focuses on methodological contributions to eπror and hallucination detection in Large Language Models. We do not train models or collect additional data; instead, we rely exclusively on publicly available datasets and widely used benchmark models for evaluation.
We note that part of our evaluation includes the Math dataset, which was publicly accessible at the time of experimentation but has since been taken down following copyright claim. We emphasize that this dataset was used solely for evaluation purposes of our method, and only prior to the date of the takedown. No redistribution of the dataset made, and our reported results are limited to demonstrating methodological effectiveness.
Our work does involve personally identifiable information, sensitive content, or human subjects, and does not raise foresecable risks of harm. We believe the proposed approach contributes positively research on trustworthy AI by providing a training-free and generalizable framework for error detection in language models.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our results, All experimental details, including model configurations, evaluation protocols, and datasets used, described in the main text and Appendix B. Upon acceptance of this work, we will publicly release the code implementing our method, along with instructions reproduce all reported experiments. This will allow the community to verify our findings and build upon our work.
ACKNOWLEDGMENT
This work was supported by projects PNRR MUR PE0000013-FAIR under the MUR National Recovery and Resilience Plan funded by the European Union NextGenerationEU, PRIN 2022 project 20227YET9B "AdVVent" CUP code 2S3D23012830006 by Sapienza research projects D2QNeT and BEAT (Better dEep leArning securiTy) bando per ricerca di Ateneo 2024, and via the Seed of ERC grant "MINT.AI" (cup B83C25001 B83025000400001, This work is additionally supported by the MUR FIS2 grant FIS- TIS-2023-00942 "NEXUS" (cup B53C25001030001). The work of Hazem Dewidar was carried out while he was enrolled in the Italian National Doctorate Artificial Intelligence run by Sapienza University of Rome. Computing was supported by CINECA through the Italian SuperComputing Resource Allocation (ISCRA) projects Ge-Di HP10CRPUVC and SLEY HP10CX9CMC.
was also partially supported
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A APPENDIX
A.I PARTITION FUNCTIONS PROOF USED IN EQ. (4)
extend the proof of Zhu al. the sequence-to-seqqenee setting by treating next-token prediction as multi-class classification problem. At step i, the input is the prefix (x--1:1) and the model outputs logits over the vocabulary of size V. For notational consistency, we define the following energy terms:
Eg (xx1) log((ex (0 (((xx=aa)]id((x)))
(9)
Egy (xi-1:1) - Ek= exp (0(x (0(x:11)({)))
The probability of the sequence up to position i can be expressed
pe (x exp( (xx1))
(10)
Ze
where Zo is the global partition function (normalizing constant), defined over all possible continua- tions of all prefixes:
Zo = Σ Σ
exp(0(xx-::))id(xx]]]
Σ
exp cxp(0(xx-::(()).
(11)
X1-1.1 X.
Similarly, the probability of the prefix X;11:1 can be written using the marginal energy:
exp( - ET (x,-1:1))
(12)
Zo
where Zg is the corresponding normalizing constant:
Z Σ exp(-Ep) (x,-1.))
Σ exp log Σ
exp
(13)
km
By expanding the logarithm in Eq. (13), we obtain
7,0 Σ Σ
exp(O(xx-
(14)
X;-1i1 1=1
which is identical to Eq. (11), Hence, the partition functions coincide:
7. Zo.
(15)
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A.2 THE ROLE OF TEMPERATURE IN SPILLED ENERGY
We now analyze how the temperature parameter T affects the definition of spilled energy. Starting from Eq. (3), the probability of the next token under temperature scaling
logPo(Xx x(-1.)) ) = log exp( 1xx (ad (Id(x)))
(16)
Σ exp( 0 x1 -
1 0 (1) log E exp( ((x()xx))))(())
(17)
Accordingly, the spilled energy becomes
[V]
( 1 T
((xxxxa) - log Σ exp( (x.,......... ..... x.)((()))))
(18)
Limit case 7 oc. When the temperature tends infinity, the logits are scaled down towards zero, making all tokens equally likely:
V
lim AFo(xa) = lim 7-0
0 1xx - log
Σ
14))
(19)
50000 T
k=1
log
exp(0)
(20)
log V
(21)
Thus, for r - oc the model degenerates into a uniform random classifier over the vocabulary.
Interpretation. Varying 7 perturbs the balance between the two energy terms, introducing a system- atic eтror in AEg. From the perspective of the Boltzmann distribution, scaling by corresponds to injecting or removing energy from the system. At high temperatures (T - oo), the system approaches maximum entropy, where all tokens have equal probability. At low temperatures (r - 0+), the distribution collapses onto the maximum logit token, making the model highly deterministic.
Error accumulation. As we generate tokens sequentially, we accumulate deviations in AEg:
log 1og 0 (Id((x))) log eexp(+ +
Σ A Ea Ex(x))).
(22)
j=1
Hence, temperature scaling not only modifies the probabilities but also reshapes the cumulative error landscape traced by spilled energy.
A.3 WHY SPILLED ENERGY SHOULD BE ZERO?
TL;DR Consider Eq. (2) in our paper and the simplification that occurs between the probabilities between step and step i 1: that simplification occurs because the probability in the denominator step is the same the probability the numerator at step 1 in order to perform language modeling correctly. We measure those inside and LLMs in terms of energy, and the spilled energy is the amount by which they differ.
Please see the definition below. Let us assume sequence of three tokens X2, X1, Xp- If we do language modeling with autoregression, minimizing the negative lieeliihd,o we have:
- -log 1(x,((X, X1, xo) = - logg p(x, X2 X1 Xo) p(x (xx))
stop
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Now, every conditional probability the right side is implemented with transformer ending in softmax discriminative classifier. Eq. (3) and Eq. (6) allow us to re-interpret:
X(X,,x1,,X0 X1, Xo) p(x1, xo)
exp (f(x1, xo) [id((xx))) (23)
step 2: log p(gg((x) xo) log
log
E exp (0)x1,
E (x2, X1, xo) E. EE" (x1, xo). (24)
In other words, we reinterpret:
(x2, X1, Xo), which is the logit (0) of the the numerator p(X2, X1. Xo) the energy E (x2, softmax at timestep 2;
The denominator as the energy E""(xi Xo) obtained with the marginalization (m) across the vocabulary V. This value can be read "read" simply by taking the denominator of the softmax at timestep 2. Please remember this term.
It is better indicate them as energies (since they are not probabilities), and given their logarithmic properties, we obtain a difference. We use the notation I for logits and m for marginalization.
Now, when we go across steps and we connect two-time steps:
step 1: log [ox
log
p(x1xx) p(xo)
(x1,x0) -E (xo))
We see that at timestep 1, the value E (x1, Xo) appears again, but measured at the logit level. In other words, across the time-steps 2 and the quantity x(x1,x)) is measured twice:
timestep 2, the marginalization;
at timestep 1, the logit.
In the architecture in the loss, there is mechanism that forces these quantities be the same, but they should be equal, given the language modeling objective. This is the same as saying that in Eq. (2), the probabilities across time steps need cancel out shown.
In other words, the following:
p p(x2, Xo) =
Implies:
X1, = E² 6xxx X1, xo) E. -E'' (x1, (x1,
Em (xx) + E² (xo)
should he zero
should he zem
To model the energy of sequence E (x2, X1 Xo) correctly, then:
E. Ex1( xo) + E² (x1,, xn) = (spilled energy timestep 2 non-zero) 1TT E Exxx)) 0 (spilled energy at timestep if non-zero)
that E(x2, X1, xo) E² (x,,X1,Xx).
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B REPRODUCIBILITY
For comparisons real-worke tasks, adopt the same experimental setting Orgad al. (2025), whose implementation is publicly available at https: / /ggtttb.bbb.g.. conn/tehhii Beschiion-oe on-cs-nnpp LLMsKnow. This ensures that baselines and evaluation procedures follow established and validated protocol.
In addition, we release our codebase, which includes:
computation of the proposed energy- based measures;
scripts for reproducing the synthetic arithmetic preliminary experiments;
example of how our method can be integrated into benchmarking or production pipeline.
B. EXACT ANSWER TOKEN DETECTION DETAILS
To analyze the spilled energy specifically the tokens carrying the semantic weight of the answer, we must first localize the "exact answer" span [u, w] within the longer generated sequence ÿ. We adopt the methodology proposed by Orgad et (2025), utilizing a combination of heuristics and an auxiliary instruction-tuned LLM to perform this extraction.
Depending on the nature of the task, we employ two strategies to identify the exact answer substring 8:
Extraction Strategy
Heuristic Matching: For tasks with closed set of possible labels (e.g., classification tasks or multiple-choice QA), we perform string matching to locate the label within the generation.
LLM-based Extraction: For open-ended generation tasks (e.g.. TriviaQA, Math), where the answer form varies, employ instruction-tuned model (Mistral-7B-IInttrucc) to extract the short answer from the long-form generation.
Prompting for Extraction Following Orgad et al. (2025), we prompt the auxiliary model with the original question and the generated long answer ý using the following template:
Prompt for Exact Answer Extraction
Extract from the following long answer the short answer, only the relevant tckens. If the long does not the quastion,
output NO ANSKER.
lQuestion
[LLM long answer
Exact answer: [Short exact answer 1]
[Question
A: (LLM long answer that does not answer the question]
Exact answer: NO ANSWER
Q: [Questien]
A: [L.L 1ong answer]
Exact answer:
Verification and Token Mapping To ensure robustness, verify that the extracted string is a valid substring of the original generation ÿ. If the extraction invalid or the model outputs "NO ANSWER," retry the extraction up five times. If valid substring is still not found, the sample is excluded from the analysis to avoid identifying incorrect tokens.
Once the substring is validated, map the corresponding token indices [u, w| in the original sequence. The spilled energy analysis is then performed specifically over this interval, or pooled across it (e.g., via min-pooling) as described in Section 5.2.
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Table Answer Extraction Success Rate across tasks for Mistral-Instruct.
Dataset
Success Rate (%)
TriviaQA
90.29
HotpotQA
87.37
Movies
93.61
MNLI
92.99
Math
87.59
HotpotQA-WC
92.38
Answer Extraction Performance For answer localization, we achieve accuracy comparable the results of Orgad et al. (2025). We report in Table 4 the extraction success rate across the full datasets using In-ttae-7 Note that some datasets have been excluded (e.g., IMDB, Winobias, Winogrande) since they have finite set of possible answers that can be used to easily locate the exact answer within the model's generation.
C LLM USAGE
Large language models were used exclusively for text polishing and minor exposition refinements. All substantive research content, methodology, and scientific conclusions were developed entirely by the authors.
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D SUPPLEMENTARY MATERIAL
This supplementary material is intended to complement the main paper by providing further motiva- tion for our assumptions and design choices, well additional ablation studies or plots, such as ROCs and histograms, that could not fit in the main paper.
D.1 ADDITIONAL RESULTS FOR SYNTHETIC ARITHMETIC
In Fig. we augmented Fig. in the main paper, also adding the results for Mistral-7B-Instruun v0.3 and LLaMa-3-8B. The same findings of the figure in the paper also translate to this LLM, meaning that our method generalizes across LLMs.
Fig. 6 and Fig. also extend and provide more details of Fig. 3 in the main paper by showing, respectively, the histograms and the ROC at a better resolution and displayed in different frames. Also, we have added results for Mistral-7B -B-taal---nstrucc v0.3 and LLaMa-3-8B.
D.2 ADDITIONAL QUALITATIVE RESULTS
In this section, we offer additional results of the detection performance following what is shown in Fig. We report both success cases and failure cases. While it is difficult draw conclusions and predict when, why, and on which topics spilled energy may work not, we noticed that it appears to perform reliably on knowledge-based factual content but, at times, exhibits difficulties with reasoning tasks and numerical information, despite working well on math questions, demonstrated Section 5. Further investigation required to better understand and validate these patterns.
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Figure 5: Histograms of Spilled Energy values across models (rows) on Math Sums with different error ranges in the answer (columns, decreasing range left right, making it harder to detect emrors), as described in Section 5.1. In the fourth column, we show ROC curves for Hallucination Detection across the error ranges (colors) and methods (line styles).
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Correct and Incorrect answers across models on Math Sums, increasing difficulty from left to right. We compute sums on 13-digit integers, for incorrect answers we add a random offset sampled uniformly from the error interval: Easy U(ee3, le4) Medium u(1e2, 1e3)
Hard 2411, 10); for more details see Section 5.1.
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Figure 8: Fig. 8a presents the cross-dataset performance of the method proposed by Orgad et al. (2025) using Llama-3. Fig. 8b depicts the performance difference between their method and our Spilled AE with Min pooling. Positive values indicate cases where Spilled A E outperforms the method of Orgad al. (2025). All numbers computed as percentages.
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22:24
27:47
30.30
4.20
44:52
2257
26
33.74
35
25
Tha
-28
0
m1
-3.27
25:24
29:47
0.59
-328
8.15
1.45
5.20
10.14
mmdb
11
2
1421
12.38
4.28
37.2
mith
20:24
30:47
-330
-320
43.01
3.57
28.74
12
1
28.27
23.24
2244
-2.8
..20
43:83
22557
24:26
2.74
move
Ioddoou PC
TotulAook
(a)
(b)
Figure 9: Fig. 9a presents the cross- dataset performance of the method proposed by Orgad et al. (2025) using Mistral. Fig. 9b depicts the performance difference between their method and our Spilled AE with Min pooling. Positive values indicate cases where Spilled E outperforms the method of Orgad al. (2025). All numbers computed as percentages.
24

---

Published as a conference paper at ICLR 2026
Pool
HospoiQA HorpoQA-WW WC
IMDB
Math
MNLI
Movics
TrivisQA
Winobias
Winogrande
Average
LLaMA Instruct
Orgad et al. (2025)
Mean
66.56 9.10
59.00 ..14
69.78 14.74
66.56
60.56 12.53
16.44
63:22
1...1 67.33 11o0
58,00
64.16
Spilled A
Min
N5.99
03.00-01 m
47.66...o
78.95:110 65.58
on
89. 14
87.07
0 69.72.014
55.11
73.16-11
Marginal g
Max
76.72
30.74
85.63-1.0
27.08
29.90
96.17.01
80.13
57.67
47
65.72-
Marginal F
Min
97,57
-4.37
40.55....4
61,21-3:
73.38
47.19.2m
53.98.2.
62.93
Logit E
Max
72.85 2.12
91.11-1.02
42.08 s.n
57.81
25.52 13.00
43.97
68.89
39.95 12:1
49.40
54.62
Spilled AE
Max
$1.31
47.68 2:0
52.34.
40.33
56.41.2.1
68.56
47.54:2.0
38.40.2.4
44.97
50.07
5.00
LLaMA
Orgad
Mean
61.22
56.78
72.67
69.67
60.33
64.00)xx
66.44
43.56.4.1
62.84
Logit
Min
87,93 1:01
91.24 0.00
51.73 1.30
42.99
97.01 (3.0
99.86 00:
84.53....
19.29 11:11
48.52 (1.7
72.57
Spilled A
Min
79.04 -174
80.83.1.00 m7
43.22 1:47
74.36: c0.54
99.97
61 61.97:2m1
78.54-1.77
52.11:9:5
48:21
68.69
Spilled AAA.
Min
77.75
79:44 20:44.:000
43.39
72.87
99.97.3.06
61
77.55=:
52.34
48.17:3-11
68.12 68.12
Marginal g
78.00
76.90=11m
48 28.29.1:11
68.77
10.93.1.22
67.49-:.o
$1.91
51.28g5.r
59.36 99
Marginal
Min
59.20
-1.71
34.13
97.42
50.37
69.88
29.05.223
49.00
57.68
Logit g
Max
49.02.:1 79
48. 48.27-11.0
57.38
91.
76.2-:1
S7 e
$2.77 16
50.74:1111
51.17:1-1
56.89
Logit g
ALT
43.56.1
39,74
18.27
17.41.6.0
91.71
43.11
43.62
50,74
51.17:1.11
52.15
Lagit
Taken
43.56-1.16
29.74-1.m
48.27
57.41
91.21=200
43.11-1.17
20.74.1.3
51.17-1.11
Marginal g
ALT
61.59
1:00
58.64 1:00
48.29
1.10
67.93 10.32
10:75 11:40
61.39
1.30
19.73-1.43
51.19 22.5
51,44-2:33
51.22 11:6)
Marginal g
Last Token 61.59 1:00
58.641.1.0
18.2
1.16
67.93
00.22
10.75
13:41
61.39.1.33
1.:0
49.73
51.19.043 12:53
11.442.5
51.22 (1:4))
Marginal gm
Mean
58.27
58.64..122
48.
68.3
68.32.6.3
6.1 6.
66.55
66.55.5.33
45.67
51,80.5.0
$1.29
50.55
Hnsttrlllnssnuct
Orgad et al. (2025)
Mean
64.78.10.55
56,78.77
82.67:11.43
68.78
62.22.12:3
64.89.11.88
65.44:12.1
61.00.11.00
61.44.11.a
65.56
Spilled A
Min
91.12 m
97.47
39.77-x..
66.63...4
95.91
94,99
91.75-:
50.74:00s
49.00
77.49
Marginal g
Min
87,58-1.0
97.94
0z
18.67-2.7
67.58
07.58....
97.96
$4.90
87.25
49.19.2.7
48.49
71.12
Logit
Max
77.24-1
83.84_110
57.67
78099-11
80.35
45.53
48.17
63.44-1911
Marginal
64.63-.
33.42..111
26.52
17.62...a1
86.60.1.3
65.46-2m
2644
21.14
53.68
Logit g
Last Token
58.77
71.26
22.28 2.54
71,21
42:78-1.00
42.93
58.36 1.0
45.65
48.30
51.50
Logit g
58,77
71.26-1
22.28
71.21
47.78-2
42.9%
58.36-100
45.65
48.30.0004
Mistral
Orgad et al. (2025)
Mean
61.78 2.22
57.44 4:00
76.22
12.32
65.78
(15.21
56.57
64,22
(6.0)
64.33
58.00
12:00
54.56 L.:1
62.11 2:21
Marginal g
Min
87,52 1:31
90,91 1:30
$4.69
2.00
86.21
1.00
98.80
10.3
94.43
83.66
12:00
52.15
16.37.2.0
77.19 10:00
Maryinal gm
Max
83.57
86.83
-5.3
62.26
96.03
99.27
92.26
51.31
14.49...4
74.59..00
Spilled &
84.24-111
83.74_1.1
57:43-1m0
28
96.69.3.4
-4.47-1-7
81,27=:
73.94 48.72.1mm
Spilled
Max
61.50
60.60...0 m
42.57
26.27
47.01
68.07
58.71
51.13
61.19-
Spilled AE
Max
60.54-1 x0
60.18...
-3.47
71.93
43.94
78.84.1.3
67.92
57.24
51.5 1..8
59.77-
Table 5: Hallucination detection performance, in terms of AuROC, across nine benchmarks and different LLMs. We measure the generalization across all tasks by computing the average.
F4
72
76
55
S7
6
4R
80
SN
5:75
23:32
4.25
-5.00
52
4.25
33.37
$5.47
1epetoo
01.75
06
16.26
2724
15.17
-9
medes
74
64
32
50
52
52
C7
57
moxee
07.75
20:12
3.89
3.74
3.00
92
45.47
windo
17.7
52.12
50
43.28
24.07
92
23
14
45.47
33
Awograrde
23
m11
40.25
15.12
59
3.29
31.07
2.35
6.25
16.17
indh
54
5
27.12
-20
mith
13.75
25.12
34
422
2.00
31.05
23
25.37
42:47
mam.
mov
(a)
(b)
Figure 10: Fig. 10a presents the cross-dataset performance of the method proposed by Orgad et al. (2025) using Mistral-Instruce Fig. 10b depicts the performance difference between their method and our Spilled AE with Min pooling. Positive values indicate cases where Spilled A E outperforms the method of Orgad et al. (2025). All numbers are computed as percentages.
25

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Published as a conference paper at ICLR 2026
Pooling Window
3
4
0.5
-5
0.5 11 -1
-3
What is the capital of Italy? The capital of Italy is Rome
Figure 11: Example of the Pooling Window
D.3 ADDITIONAL. RESULTS FOR TESSSITESTING WITH REAL. WORLD BENCHMARKS
Table 5 shows how our method compares with the baseline methods, Orgad al. (2025) and Logit E.. This table was obtained by using various pooling methods in the pooling window from which we measure possible hallucinations. More details below based on the example Fig. 11:
Min: minimum energy value in the pooling window. Energy Measured: -3
Max: maximum energy value in the pooling window. Energy Measured: 11
Mean: mean of all the energies in the pooling window. Energy Measured: 2.08
Last Token: energy of the last token in the pooling window. Energy Measured: -3
After Last Token: energy of the first token after the pooling method. Energy Measured:
26

---

Published as a conference paper at ICLR 2026
D.3.1 SUCCESS CASES
Question:
Whhic planet known as the Red Planct?'
Logits: The Red Planet is Mars
Ours: The Red Planet Mars
Logits: The Red Planet is Jupiter
Ours: The Red Planet is Jupiter x
Question: What is the Largest mannal in the world?'
Logits: The largest mamm al in the world is the Blue Whale
Ours: The
largest mamm al in the world is the Blue Whale /
Logits: The largest mamm al in the world is the House Cat x
Ours: The largest mamm al in the world is the House Cat X
Question:
Who painted the Mona Lisa?'
Logits:
The Mona
Lisa was painted by Leonardo da Vinci
Ours: The Mona Lisa was painted by Leonardo Vinci
Logits:
The Mona
Lisa was painted by Pablo Ese obar
Ours: The Mona Lisa was painted by Pablo Esc obar
x
Question:
What gas plants breathe for photosynthesis??
Logits: They breathe in carbon dioxide
Ours: They breathe in carbon dioxide /
Logits: They breathe in oxygen X
Ours: They breathe in oxygen
27

---

Logits: The largest ocean is the Indian Ocean Published as a conference paper at ICLR 2026
Question: 'In which continent is Egypt Located?"
Logits: Egypt is located in Africa
Ours: Egypt is located Africa
Logits: Egypt located in Europe X
Ours: Egypt is located in Europe
Question: What the fastest land animal?'
Logits: The fastest land animal is the che et ah
Ours: The fastest land animal is the che et ah
Logits: The fastest land animal is the lion X Ours: The fastest land animal is the lion
Question: What is the hardest natural substance on Barth?'
Logits: The hardest natural substance is diamond
Ours: The hardest natural substance is diamond
Logits: The hardest natural substance is gold X
Ours: The hardest natural substance gold x
Question: which ocean is the largest?"
Logits: The largest ocean is the Pacific Ocean
Ours: The largest ocean is the Pacific Ocean /
Ours: The largest ocean is the Indian Ocean X
28

---

Published as a conference paper at ICLR 2026
D.3.2 FAILURE CASES
Question: 'Whoo was the first person to walk on the moon?''
Logits: Neil Armstrong
Ours: Neil Armstrong
Logits: Buzz Ald rin
Ours: Buzz Ald rin X
Reasoning: 'If there are 3 cars and each car has 4 wheels how many wheels are there in total?
Logits: Each car has
wheels
So for cars the total number of wheels is 3
12 wheels
Ours: Each car has wheels for cars the total number of wheels is
4
12 wheels
Logits: Each car has wheels So for cars the total number of wheels is
8
14 wheels x
Ours:
Each car has wheels
So for the total number of wheels is
g
14 wheels
x
Reasoning:
What is the square reot of 64?'
Logits:
The square root of
64 is
Ours: The square root of 64
Logits: The square root of is
x
Ours: The square root of 64 is
10 x
Question:
<
What blood type is known the universal doner?'
Logits:
negative
Ours: O negative
Logits: AB positive
Ours: AB positive X
29


Reviews:
[{"review_id": "QPj759Tm7F", "reviewer": "Reviewer_qt4T", "summary": "This paper reinterprets the final softmax layer of large language models (LLMs) as an energy-based model (EBM). From this perspective, the authors claim to identify a discrepancy between two energy formulations, termed \\emph{spilled energy}, which correlates strongly with hallucinations. Unlike probing classifiers, spilled energy requires no additional training. Experiments on LLaMa-3, Mistral, and Qwen-3 across nine benchmarks show that spilled energy outperforms logits and probing methods in detecting hallucinations in the cross dataset setup.", "questions": "- I have a question regarding the following sentence (quoting from the paper): \n\"E^{\\ell}_{\\theta} and E^{m}_{\\theta} are computed from the output of the model, but with two key differences: E^{\\ell}_{\\theta} is obtained as a single logit extracted using the id of the sampled token, while E^{m}_{\\theta} is computed by marginalizing over all id's in the vocabulary.\"\n It seems that you are associating the logit of a single token with the energy function of all the preceding tokens as well. Could you clarify why this is considered correct? This association feels inconsistent: if E^{\\ell}_{\\theta} corresponds to a single token, then why does E^{m}_{\\theta} correspond to the normalization factor over the entire vocabulary? More concretely, I would have expected E^{m}_{\\theta} to represent the prediction for the specific token immediately preceding the one associated with E^{\\ell}_{\\theta}, rather than the marginalized quantity. Consequently, the idea of “spilled energy” seems problematic, as it appears to rely on an inconsistency in how energy is defined. I would be happy to get a clarification on that.\n- Could you clarify what is meant by “cross-dataset” in Table 1? This is not explained anywhere, and I find it difficult to understand the setup. Since your method does not require training, it is not clear to me how cross-dataset evaluation is applicable here. I do see how this makes sense in the context of Orgad et al.—since their probing method requires training—but then, what would be considered the standard experimental setup for them without cross-dataset evaluation (if its not Table 1)? Additionally, could you clarify how the cross-dataset setup in Table 1 is applied to Orgad et al.? Specifically, what is the exact setup—what was the model trained on, and what was it tested on?\n- Could you clarify what the superscripts m and l represent in Equation (4)?\n- In the Abstract, you write: “we propose a method to detect hallucinations completely training-free that naturally generalizes across tasks and LLMs.” Could you please point out where in the paper you demonstrate generalization across LLMs?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper considers a very timely and important problem of hallucination detection.\n- The paper is well-written and easy to follow.", "weaknesses": "- In my view, the reliance on exact tokens is a significant limitation. I am familiar with the paper by Orgad et al., which was the first to demonstrate that identifying the right token can improve detection. However, I see this primarily as an interesting observation rather than a practical method for hallucination detection. The reason is that identifying such tokens typically requires external algorithms or the use of other LLMs, which leads to very high latency—making it impractical for real-world hallucination detection.\n- I find the concept of “spilled energy” somewhat unclear and potentially inconsistent. Could you please see my first question below for further context?\n- The work appears to lack some important baselines, and the experimantal setup is unclear. For example, in my view, it would be valuable to include comparisons with Ptrue and Semantic Entropy [1]. In addition, some key comparisons are not clearly presented—for instance, I did not see results benchamarked against the baselines (even those already considered in the paper) under the standard setup (e.g., Orgad et al. when trained and tested on the same dataset). I would appreciate clarification on this point. If such a comparison is not included (and only the “cross-dataset” case is considered), it would significantly weaken the strength of the paper.\n\n**References:**\n\n[1] Kuhn et al., Semantic uncertainty: Linguistic invariances for uncertainty estimation in natural language generation, 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cWvcE7IMFT", "reviewer": "Reviewer_zjL6", "summary": "The paper proposes a training-free method to detect hallucinations in Large Language Models (LLMs) by reinterpreting the final softmax classifier as an Energy-based Model (EBM). This reinterpretation allows the authors to decompose the sequence-to-sequence probability chain into multiple interacting EBMs.\n\nThe core contribution is the introduction of \"spilled energy\", which measures the discrepancy between two energy quantities that should be mathematically equal across consecutive generation steps, but differ in the LLM's implementation due to the way marginal and joint probabilities are computed. The authors also propose a complementary metric, marginal energy, which is measurable at a single step.\n\nThe paper empirically demonstrates that high spilled energy strongly correlates with LLM hallucinations. Crucially, their method is training-free and shows strong cross-dataset generalization across state-of-the-art open-sourced LLMs (LLaMa-3, Mistral, Qwen-3) and nine benchmarks including synthetic and real-world datasets. It localizes the signal to the \"exact answer tokens\" and outperforms the logit confidence baseline and prior probe classifier approaches, especially in cross-dataset settings where probe classifiers fail to generalize.", "questions": "1. Why do we observe the cross-dataset results that for non-aligned models (Mistral), marginal energy sometimes slightly outperforms spilled energy? \n\n2. Is there any insight why marginal energy is a superior signal in the pre-trained model setting, or why instruction-tuning seems to amplify the margin for spilled energy? Maybe a more in-depth analysis into comparing two forms of energy is interesting for the pre-trained and instruction-tuned models.\n\nOther questions are listed in the bulletpoints of weakness as the justification.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea of reinterpreting the LLM's autoregressive sequence modeling as a chain of Energy-based Models (EBMs) is novel and very interesting.\n2. The implementation of the hallucination detector is simple and training free, but achieve a good generalization across synthetic and real-world datsets. \n3. The technical motivation and derivations are sound, and the method is well mathmatically-grounded.\n4. The paper is well-written and the presentation is clear.", "weaknesses": "1. There might be some technical details make the detector hard to be applicable in the real-world tasks.\n    - The paper emphasizes that localizing the signal to the \"exact answer tokens\" is essential. However, correctly to find the exact answer tokens might be non-trivial. In the paper, authors identify this span by prompting the LLM for a brief answer seems to be a bit fragile to me. This is essentially a dependence on a pre-processing step that relies on LLM's outputs, which may introduce noise or additional complexity. \n   - Compared to logit confidence, spill energy requires calculation of the quantities using two-step predictions, which may introduce the additional cost. Marginal energy only needs the one-step calculation but the results for two energy forms are quite mixed.\n    - The analysis focuses on the \"exact answer\" tokens which can moslty be applied to QA or classification tasks. How well would the Spilled Energy perform on detecting other types of hallucinations common in LLMs, such as self-contradictions or source-attributable error that might occur earlier in a long-form generated answer or reasoning trace, outside of the specific final answer tokens?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lr34cBV6eS", "reviewer": "Reviewer_Xm2r", "summary": "This paper proposes a training-free method for detecting hallucinations in LLM. The core idea is to reinterpret the final softmax layer of an LLM as an Energy-Based Model (EBM). The paper define a variable named spilled energy that quantifies the differences between two energy values across consecutive generation steps that should ideally be equal. The paper has a hypothesis that a big spilled energy is correlate with hallucination in model generation. The authors test this hypothesis on both synthetic task and real-world benchmarks. The results show that spilled energy is a strong signal for detecting hallucination. Their method outperforms baseline method, although there could still be false positive cases.", "questions": "see weakness", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper formulate the LLM's generation procedure as energy based model, which enable the following definition of \"spilled energy\" for hallucination detection. \n2. The method proposed in the paper is a training-free method which make it lightweight and applicable to most LLM for hallucination detection. Empirically, the paper show results on both sythentic and realworld dataset, which validate that the method may have generalization across different domains. While other works (non-training free) show worse performance under new datasets, the paper’s method show robustness in performance.\n3. The experiments in the paper are solide. The author provide results on sythentic dataset, which show the metric's ability to separate correct from incorrect answers. The results on realworld datasets shows the method's valid on a wide range of domains. The results are validated across model family as well.\n4. The ablation section also consolidate the findings. The authors demonstrate the critical importance of localizing the \"exact answer\" tokens, showing that doing so provides a good performance gain.", "weaknesses": "1. The method needs to first identify the specific token span constituting the \"exact answer\". The paper implement this by \"prompting the LLM for a brief answer”. My concern is that what if the LLM's \"brief answer\" is itself a hallucinated? What if the “exact answer” is wrongly identified?\n2. The current evaluation focuses on tasks with answer that can be localized to a short range(words). My question is that how the method perform on more subtle hallucinations, such as a mutilple incorrect words/phrases in a long paragraph or complex factual error that is not contained in a single noun phrase.  \n3. In limitation, the author acknowledge that the method can produce false positives on tokens that are not semantically informative, such as punctuation. While this might be mitigated this with answer localization, it suggests the raw signal is noisy and may not be a pure measure of semantic or factual correctness. Can we do better to reduce the false positive? \n4. If energy gets “spilled”, it is a sign the model’s prediction doesn’t align well with its internal probability distribution. Can you give more explanations why we can use this to detect hallucination? I am not very convinced by current explaination in paper.\n5. Each time, we need to inference the model twice, the extra latency make the method impractical", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AAsINIAW18", "reviewer": "Reviewer_LpMe", "summary": "This paper proposes a new method for detecting hallucinations in language models by interpreting the model's sequential predictions as quotients of two different energy-based models (EBMs) denoted $E_\\theta^{\\ell}$ and $E_\\theta^m$. The paper defines the spilled energy as the difference between the two energies: $\\Delta E_\\theta(\\mathbf{x}_{0:i}) = E_\\theta^\\ell(\\mathbf{x}_{0:i}) - E_\\theta^m(\\mathbf{x}_{0:i})$. \n\nThe paper then proposes several measures for hallucination detection, including:\n- marginal energy $E_\\theta^m$,\n- spilled energy $\\Delta E_\\theta$, and\n- scaled spilled energy $\\Delta E_s := |E_\\theta^m| \\Delta E_\\theta$.\n\nThese metrics are evaluated in two settings. First, a synthetic setup, involving the detection of increasingly smaller numerical errors introduced in arithmetic computations of 13-digit integers. Spilled energy ($\\Delta E_\\theta$) gets strictly superior ROC curves (better pointwise, not just in AUC) than baselines across Llama-3-8B (both instruct and non-instruct versions), Qwen-3-8B, and Mistral-7B-Instruct. \n\nSecond, the proposed methods are tested in detecting errors in reasoning tasks across several benchmarks. To do so, the *exact answer tokens* are detected, and the different methods are computed across them using different pooling methods. The different methods are compared against logit energy and [1] using AuROC scores.\n\n---\n\n[1] Orgad, H., Toker, M., Gekhman, Z., Reichart, R., Szpektor, I., Kotek, H. and Belinkov, Y., 2024. Llms know more than they show: On the intrinsic representation of llm hallucinations. arXiv preprint arXiv:2410.02707.", "questions": "- **(Q1)** Why do the authors claim that:\n>  ... delta values should always be zero when we are correctly modeling the energy at timestep $i$\n\n- **(Q2)** How exactly are the exact answer tokens detected?\n\n- **(Q3)** When comparing with [1] in Table 1, are the probes retrained on each dataset? If not, which datasets are they trained on?\n\nGiven a strong answer to **Q1**, addressing **W1**, I would be open to increasing the score. \n\n---\n\n[1] Orgad, H., Toker, M., Gekhman, Z., Reichart, R., Szpektor, I., Kotek, H. and Belinkov, Y., 2024. Llms know more than they show: On the intrinsic representation of llm hallucinations. arXiv preprint arXiv:2410.02707.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **(S1)** The proposed methods are efficient, simple, generic, easy to implement, can be applied to any language model given access to its logits (gray-box access), and extend previous hallucination detection methods.\n\n- **(S2)** The proposed methods are evaluated on a wide range of models and tasks, and show strong performance. The synthetic setup is well-designed, and the reasoning tasks are representative of the kind of tasks where hallucinations are likely to occur in real-world applications.\n\n- **(S3)** The entire derivation of spilled energy is presented in the main text and is not hidden in the appendix. The derivation is clear and easy to follow (if a bit cumbersome).", "weaknesses": "- **(W1)** The paper does not provide a theoretical justification for why spilled energy (or the other proposed methods) is a good indicator of hallucinations. It is just an empirical observation. Moreover, the intuitive explanation is limited to:\n  > ...the two energies, not interacting at the same step but at steps $i$ and $ i-1$, should be equal, but they are measured in the LLM at different generation steps and from different components.\n\n  and\n  > Since both terms on the right side should be equal to $E_\\theta (\\mathbf{x}_{i:1})$, delta values should always be zero when we are correctly modeling the energy at timestep $i$.\n\n  Which are unclear to me (see question 1).\n\n- **(W2)** While the authors mention that using the exact answer is critical for deployment, the paper is vague on the exact method used to detect these exact answer tokens. The only mention I found is in Section 4.2: \n  > We identify this span by prompting the LLM for a brief answer.\n\n- **(W3)** The derivation of spilled energy is more cumbersome and complicated than it should be. E.g., equations (1) and (2) are redundant (one of them suffices), and equations (3)-(5) can be united into one. This is a minor weakness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)