Prompting Test-Time Scaling (P-TTS)

• Prompting Test-Time Scaling (P-TTS) is an inference-time augmentation strategy that varies prompt contexts to generate multiple reasoning trajectories.
• It employs instructional wrappers—such as reward framing, correctness cues, penalty clauses, and step-by-step instructions—to enhance logical outputs without increasing training data.
• Empirical results show significant accuracy gains on benchmarks, proving P-TTS improves both in-domain reasoning and out-of-domain generalization.

Prompting Test-Time Scaling (P-TTS) is an inference-time data augmentation and reasoning control technique for LLMs, designed to enhance model robustness, reasoning depth, and generalization. P-TTS leverages prompt-level variation and systematic scaling of computation during inference—rather than increasing training data or model size—to generate diverse reasoning trajectories, explore the latent reasoning space, and amplify logical capabilities in both in-domain and out-of-domain settings.

1. Principles and Formal Definition

P-TTS operates by systematically varying the prompt context or scaling the computational budget during model inference to surface multiple reasoning pathways. In its canonical form, a small set of high-quality training instances (e.g., 90 math reasoning problems) is used, and for each instance, principled “instructional wrappers” or prompt augmentations (such as reward framing, correctness cues, penalty clauses, and explicit step-by-step instructions) are applied: $q_i^{(p)} = \tau_p \, \| \, q_i$ where $\tau_p$ is the prompt template for principle $p$ and $q_i$ is the original question. Reward framing is instantiated in several paraphrased forms and placement variants, increasing instructional diversity.

The resulting set of augmented prompts yields a synthetic, combinatorially diverse context set for modeling inference, transforming the prompt into a “scaling knob” that regulates the diversity and depth of the model’s reasoning.

2. Data Augmentation Methodology

Instead of collecting large volumes of annotated reasoning traces, P-TTS generates multiple prompt variants for each curated seed example by deterministic composition of wrapper templates. The main configurations are:

Corpus Configuration	Number of Examples	Composition Principle
Single-P-TTS	$90$	90 seed problems x 1 principle
Core-P-TTS	$360$	90 x 4 core principles
Seed+Core	$450$	Unmodified seed + core variants
Full-P-TTS	$900$	Core + multiple reward paraphrases

Key principles include:

• Reward: e.g., “I am going to tip $200,000 for a better solution!”
• Correctness: e.g., explicit accuracy requests.
• Penalty: warning of negative consequences for incorrect answers.
• Step-by-Step: explicit stepwise instruction.

Augmentation thus covers both semantic diversity (conceptual cues, motivational framing) and surface form (paraphrasing, prefix/suffix variation), greatly amplifying the effective data distribution accessed during fine-tuning or inference.

3. Model Fine-Tuning and Training Regimes

The Qwen-2.5 model family (tested at 7B, 14B, and 32B parameters) is fine-tuned using the P-TTS-augmented datasets.

• Supervised fine-tuning (SFT) predicts the assistant’s reasoning trace and answer, with cross-entropy loss on the response tokens.
• The approach is model-agnostic and requires no changes to architecture or base pre-training corpus.

P-TTS is specifically constructed as an inference-time augmentation; the test-time prompts themselves reflect the stochasticity and instructional variation, and thus the model is optimized to generalize across this expanded prompt manifold.

4. Empirical Performance and Benchmarks

P-TTS models demonstrate significant improvements on mathematical reasoning benchmarks:

• On AIME2024, P-TTS-7B achieves +26.66% and +30.00% absolute accuracy gains over S1 and S1.1 (1K-shot static prompts), respectively.
• On AIME2025, corresponding gains are +13.34% and +6.67% (7B).
• P-TTS-32B shows +23.33% and +16.63% on AIME2024, and +26.63% and +3.33% on AIME2025.
• Comparable or better performance is observed on MATH500 and GPQA-Diamond.

The effectiveness of P-TTS relies on both prompt diversity and scheduling: ablation studies confirm that the number and variety of wrapper templates, as well as how they are sampled and combined, are decisive in driving model improvements.

5. Out-of-Domain Generalization and Zero-Shot Transfer

Despite training solely on AIME-style seeds, P-TTS consistently enhances zero-shot generalization to further domains, including:

• National and international math Olympiad benchmarks (e.g., OlympiadBench, AMC23)
• Chinese-language exams (Gaokao, Kaoyan)
• GradeSchoolMath and Minerva (scientific quantitative reasoning)

The approach improves “diversity gain” and “trigram diversity” metrics, indicating a greater variety of both logical reasoning templates and surface language. This suggests that the model acquires more universal reasoning skills, not merely overfitting to static prompt patterns.

6. Theoretical Rationale and Scaling Mechanisms

P-TTS is grounded in the premise that augmenting test-time prompt context simulates the diversity of much larger training sets, driving the model to explore underrepresented regions of the latent reasoning manifold. This stochastic prompt scaling enables the LLM to:

• Generate alternative reasoning chains by controlling for reward, correctness, penalty, and procedural cues.
• Break symmetry in otherwise repetitive model outputs.
• Surface latent “modes” of reasoning that would be inaccessible to a model trained solely on a fixed prompt distribution.

Key formula for total augmented training set size: $$| \mathcal{D}_{\text{full-P-TTS}} | = (1 + 4 + (K-1)) \times N$$ where $N$ is the number of seed problems ($90$), $K$ is the count of reward paraphrases ($6$), and $1 + 4 + (K-1)$ covers original + core principles + additional reward variants.

7. Practical Benefits and Future Extensions

P-TTS enables accurate LLM reasoning in resource-constrained, rapidly evolving, or low-data domains by minimizing annotation overhead:

• It achieves performance comparable to or better than 1K-shot static baselines using only 90 carefully curated seeds.
• It is well suited to rapid deployment, dynamic task adaptation, and multilingual/generalist settings where annotated reasoning data is scarce.

Proposed extensions include:

• Instance-adaptive selection of wrapper templates.
• Integration with retrieval or model-based reranking for improved answer validation.
• Curriculum learning regimes with schedule-adaptive prompt diversity.
• Cross-domain transfer to non-mathematical reasoning, multimodal, and open-ended generation tasks.

This strategy turns the design and selection of prompt variations into a new, efficient axis for scaling model performance, complementing advances in pre-training scale and dataset expansion.

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In summary, Prompting Test-Time Scaling (P-TTS) reframes both the problem of reasoning data augmentation and “prompt engineering” itself as an inference-time scaling process. By systematically exploring instructional context, P-TTS unlocks LLM reasoning capabilities with minimal additional data, strong in-domain and out-of-domain transfer, and significant gains in robustness and diversity of logical output—all with high empirical support from controlled benchmark studies (Bsharat et al., 10 Oct 2025).