We investigate the performance of large language models (LLMs) on repetitive deterministic prediction tasks and study how the sequence accuracy rate (SAR) scales with output length. Each such task involves the repetition of the same operation N times. Examples of such tasks include letter replacement in letter strings following a given rule, integer addition, and multiplication of string operators in many-body quantum mechanics. If the LLM performs the task by a simple repetition algorithm, the success rate would follow an exponential decay with sequence length. In contrast, our experiments on leading LLMs reveal a crossover that is sharper than exponential: −logSAR grows super-linearly with N, and accuracy collapses around a characteristic length N∗, the accuracy cliff that separates reliable from unreliable generation. The hypothesis of independent per-step errors is rejected for every model and task we studied. The crossover is well described by a double-exponential accumulation law, SAR=exp(−β0NαN−1), whose crossover scale N∗ does not depend on the functional form chosen to fit it. To interpret this behaviour we introduce a minimal effective model in which step-correctness variables interact through dense random couplings and compete with an external field set by the prompt. Solved by direct enumeration, the model reproduces the super-linear error accumulation and the accuracy cliff qualitatively, and it assigns to each model--task pair two interpretable parameters, an intrinsic error rate and an error-accumulation factor.
Figures & tables
Figure 1: Illustration of the experimental setup and theoretical framework. The left panel shows examples of deterministic sequence prediction tasks (e.g., arithmetic problems) used to evaluate large language models (LLMs), where each token’s correctness is represented by a binary variable (Ising spin). Errors at different positions are modeled as Ising spins interacting through dense random couplings, an effective description of how correlations and noise propagate during sequence generation. The right panel displays the typical behavior of the Sequence Accuracy Rate (SAR) (i.e. the probability for the whole output sequence to be correct) as a function of the sequence length N : SAR remains high for short sequences, then drops beyond a characteristic crossover scale N∗ , forming the accuracy cliff.
Figure 2: Cyclic letter replacement benchmark on gemini-2.5-pro and gemini-2.5-flash with alphabet size ∣A∣=4,9,13 and 26 . Points: fraction of correct instances (up to n=100 per length); error bars: 95% Wilson intervals with the batch design effect. Solid curves: maximum-likelihood fit of the accumulation law Eq. 19 , with the fitted β0 , α and crossover scale N∗ ; dashed: the quadratic law e−aN−bN2 ( Sec. 3.1 ).
Figure 3: Cyclic letter replacement with alphabet size ∣A∣=13 across different models. Points: fraction of correct instances (up to n=100 per length; n≤30 for some grok-4 lengths); error bars: 95% Wilson intervals with the batch design effect. Solid: fit of Eq. 19 ; dashed: quadratic law.
Figure 4: Integer addition benchmark across different models. Points: fraction of correct instances (up to n=100 per length; n≤30 for grok-4 ); error bars: 95% Wilson intervals with the batch design effect. Solid: fit of Eq. 19 ; dashed: quadratic law.
Figure 5: Pauli string multiplication benchmark on gemini-2.5-pro and gemini-2.5-flash evaluated under strict (phase-matched) and relaxed (phase-ignored) criteria. Points: fraction of correct instances (up to n=100 per length); error bars: 95% Wilson intervals with the batch design effect ( c^≈5 for this task). Solid: fit of Eq. 19 ; dashed: quadratic law.
Figure 6: Scaling collapse of the Sequence Accuracy Rate. (a) SAR against the rescaled length N/N∗ (with N∗ the fitted SAR=0.5 crossover scale of Eq. 19 ) for all 19 model–task curves; colors denote task families, error bars are 95% Wilson intervals with the batch design effect. The black curves are the rescaled form of Eq. 19 , exp[−(ln2)xes(x−1)] , for shape parameters s=N∗logα=0.2 , 0.8 and 2.2 spanning the fitted range. (b) The law-independent crossover width w=[N(SAR=0.2)−N(SAR=0.8)]/N∗ of every curve against its N∗ , with standard errors; the widths differ significantly between model–task pairs.
Figure 7: Correlation–error map grouped by model. Each point is the maximum-likelihood (logα,logβ0) of Eq. 19 for one task, with 1σ error bars (quasi-likelihood); the horizontal and vertical axes represent the correlation level and error level, respectively. The background color and the gray contours give the crossover scale N∗ implied by Eq. 19 at each point (numbers along the contours).
Figure 8: Correlation–error map grouped by task; conventions as in Fig. 7 .
Figure 9: Divide-and-conquer evaluation on the Pauli string multiplication task under the phase-strict (top) and phase-relaxed (bottom) criteria, using gemini-2.5-pro (blue) and gemini-2.5-flash (red). Points: fraction of correct instances with 95% intervals. The gray k=1 curves are the maximum-likelihood fits of Eq. 19 ; the extracted (α,β0) are then used to draw SAR(N/k)k for k=2 and k=3 without overhead (dashed) and multiplied by the overhead measured from the same records (solid; the product of the chunk-stage and recombination factors quoted in the text). For pro the measured overhead is negligible and the data follow the prediction semi-quantitatively; for flash the recombination overhead ( θ=0.82 , 0.76 ) and a chunk-stage shortfall suppress the plateau under the strict criterion.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 10: Example: K4×K3 graph.
Figure 11: Direct finite- N evaluation of the random-coupling model. (a) SAR(N) from exhaustive enumeration ( N≤19 , 200 disorder realisations) for four parameter sets; filled symbols: geometric average expEJlogpJ[1] , open symbols: arithmetic average EJpJ[1] , bands: inter-quartile range over realisations. Solid lines: the exponentiated closed form Eq. 61 with the same (J0,h) and no free parameter; dotted: the leading-order quadratic law Eq. 60 . (b) The exact geometric curves against N/N∗ , together with the rescaled closed form for the corresponding s=2J02N∗ . (c) Contrast for (J0,h)=(0.5,2.5) between the N -independent coupling variance used in this work (circles) and the conventional SK normalisation Jij∼J0/N (squares, log scale): with the latter, −logSAR is linear in N and no cliff develops.
(J0,h)
N∗geo
N∗ari
closed
quad
α=1
e2J02
αeff
w
(0.3,2.5)
14.7
16.2
12.6
21.7
103
1.20
1.108
–
(0.5,2.5)
8.5
9.8
6.5
13.9
103
1.65
1.113
0.66
(0.5,3.5)
13.4
15.3
9.7
38.5
760
1.65
1.155
0.55
(0.8,3.5)
6.9
9.4
4.9
24.5
760
3.60
1.145
0.64
Appendix
Table 1: Crossover scales of the exactly enumerated model. N∗geo , N∗ari : from the geometric and arithmetic disorder averages; “closed”: the exponentiated form Eq. 61 ; “quad”: the leading-order quadratic law; α=1 : independent errors, ln2e2h . αeff is the accumulation factor of a fit of Eq. 19 to the exact geometric curve, to be compared with e2J02 ; w is the crossover width of the exact curve in units of N∗ .
task
model
n
c^
ΔQAICα=1
ΔQAICquad
ΔQAICstr
ΔQAIClog
N∗ (range)
letter-4
gemini-2.5-pro
4900
1.9
+260
-20
-33
-21
12 (11–12)
letter-4
gemini-2.5-flash
4580
1.0
+252
-88
-94
-110
7 (7–8)
letter-9
gemini-2.5-pro
4900
1.7
+292
-33
-34
-9
17 (17–17)
letter-9
gemini-2.5-flash
4890
1.6
+207
-35
-36
-15
10 (9–10)
letter-13
gemini-2.5-pro
4900
1.5
+257
-36
-37
+7
21 (20–21)
letter-13
gemini-2.5-flash
4880
1.3
+171
-83
-88
-72
10 (10–11)
Appendix
Table 2: Model comparison for every model–task curve. n : number of instances; c^ : batch design effect; ΔQAIC of the α=1 exponential, the quadratic, stretched-exponential and logistic laws relative to Eq. 19 (positive: worse); N∗ of Eq. 19 and its range over the four two-parameter laws.
Run-level pass rate overstates retry-free coverage by up to 17.8 percentage points -- and the gap is largest precisely for mid-performing systems. We investigate this accuracy--stability relationship in large language model (LLM) evaluation for deterministic text-conditioned generation, using programming tasks as a concrete testbed. Standard code-generation benchmarks emphasize single-run accuracy or eventual success under repeated sampling, but many deployment settings also require stability: consistent outcomes across repeated invocations under the same task description. We present a repeated-run evaluation protocol with metrics for run-level accuracy, retry-free coverage, and per-problem variability. On a recency-based benchmark of 100 LeetCode-style problems, we evaluate 16 models from five provider families under two prompt templates with five repeated runs per problem, yielding 16,000 evaluation instances. Although run-level pass rate and perfect stability rate are strongly correlated (r=0.985), pass rate consistently exceeds retry-free coverage -- a gap that reaches 17.8 percentage points and reverses model rankings even among closely matched systems. Prompt effects are model-dependent rather than uniformly beneficial. These results suggest that repeated-run stability analysis is a necessary complement to conventional accuracy reporting for deterministic text-conditioned generation tasks.
Yongxi Zhou, Lai Yun Choi, Jiaxi Wen +1
Northeastern University, Massachusetts, USA · University of Southern California, California, USA
Large language models (LLMs) often achieve strong performance on reasoning benchmarks, but final-answer accuracy alone does not show whether they faithfully execute the procedure specified in a prompt. We introduce a controlled diagnostic benchmark for procedural execution, where models are given a step-wise arithmetic procedure and two numeric inputs, and must return the final computed value. Complexity is varied through procedure length and look-back dependencies over intermediate variables. Average first-answer accuracy drops from 63% on 5-step procedures to 20% on 95-step procedures. Generation-level analysis shows that failures often involve missing answers, premature answers, self-correction after an initial error and under-executed traces. These findings suggest that apparent reasoning ability can mask substantial weaknesses in faithful long-horizon procedural execution.
Large language models (LLMs) are trained and evaluated as though perfect reliability is achievable for any task given sufficient scale. We show that this assumption is information-theoretically unjustified. Every generative task has a reliability ceiling that no model can exceed, determined by how much output uncertainty is resolvable from observable context. The gap decomposes into a resolvable component closable with additional context and a subjective component inherent to task ambiguity. Autoregressive generation further degrades this ceiling at a rate governed by the task's dependency kernel, which quantifies inter-token correlations in the output. From these two primitives, we derive a first-principles scaling law where LLM performance is bottlenecked by the scarcer resource: training data or model capacity. This law recovers the Chinchilla scaling law as a special case and provides a structural account of when scaling improves reliability. Beyond scaling, our framework unifies diverse practical phenomena, such as the benefits of retrieval-augmentation and the spectral mechanics of catastrophic forgetting. Our work formalizes the resource-complexity tradeoffs that govern model performance across domains, offering a unified theory of performance limits in generative language models.
Subhabrata Majumdar
Indian Institute of Management Bangalore Bengaluru, India