SR-GRPO: Stable Rank as an Intrinsic Geometric Reward for Large Language Model Alignment — part 4/4

Cleaned-text code points [47385, 58393). Bibliography omitted here; source.md includes it.

Tables 11 and 12 report Pearson ( r r ) and Spearman ( ρ \rho ) correlations with p p -values. Positive correlations indicate that higher metric values associate with higher stable rank; negative correlations indicate the opposite.

### F.5 Key Findings

The correlation analysis reveals three main patterns. First, stable rank shows strongest positive associations with semantic coherence metrics: progression score ( ρ = 0.313 \rho=0.313 ) and QA alignment consistency ( ρ = 0.316 \rho=0.316 ). Responses maintaining smooth topic flow and stable prompt alignment achieve higher stable rank. Conversely, coherence standard deviation exhibits strong negative correlation ( ρ = − 0.356 \rho=-0.356 ), indicating that inconsistent adjacent similarities reduce stable rank.

Second, stable rank distinguishes information density from verbosity. While absolute measures like token count ( ρ = − 0.294 \rho=-0.294 ) and sentence count ( ρ = − 0.368 \rho=-0.368 ) correlate negatively, efficiency metrics show positive associations: lexical diversity ( ρ = 0.238 \rho=0.238 ) and compression ratio ( ρ = 0.233 \rho=0.233 ). The negative correlation with sentence count is particularly revealing: responses with many short sentences fail to build rich semantic representations. Combined with positive correlation with average sentence length ( ρ = 0.085 \rho=0.085 ), this suggests stable rank rewards well-developed sentences over fragmented text.

Third, discourse markers reveal nuanced reasoning patterns in paired-difference analysis. Most marker categories show negative correlations: additive markers ( ρ = − 0.156 \rho=-0.156 ), conditional markers ( ρ = − 0.163 \rho=-0.163 ), and enumeration patterns ( ρ = − 0.148 \rho=-0.148 ), with total marker count negatively associated ( ρ = − 0.204 \rho=-0.204 ). Responses relying heavily on common connective words tend to receive lower stable rank, consistent with favoring compact information content over explicit signaling. However, contrastive and causal markers follow a different pattern: positive Pearson correlations (contrastive r = 0.187 r=0.187 , causal r = 0.139 r=0.139 ) but weak Spearman correlations (contrastive ρ = 0.067 \rho=0.067 , causal ρ ≈ 0 \rho\approx 0 ). These markers often appear at branch points in arguments or when cause-effect relations are introduced, corresponding to deeper reasoning structure than simple additive or enumerative markers. The weak Spearman correlations indicate that stable rank is sensitive to whether such markers are present but does not increase monotonically with frequency, consistent with the view that contrastive and causal connectives are most informative when they highlight key reasoning steps rather than being used uniformly as a stylistic choice.

## Appendix G Ablation: Comparison with Other Intrinsic Dimension Metrics

While stable rank has shown strong performance as a reward signal, other intrinsic dimension metrics exist that measure the effective dimensionality of neural representations. We compare stable rank against three alternative metrics: condition number, PCA-based 95% variance dimension, and effective rank. All metrics are computed on the final hidden layer of Qwen3-8B ( Yang et al., 2025 ) using the same RewardBench evaluation protocol.

##### Metrics.

We evaluate four intrinsic dimension metrics: • Stable rank : sr ​ ( 𝐗 ) = ‖ 𝐗 ‖ F 2 / ‖ 𝐗 ‖ 2 2 \text{sr}(\mathbf{X})=\|\mathbf{X}\|_{F}^{2}/\|\mathbf{X}\|_{2}^{2} , the ratio of squared Frobenius norm to squared spectral norm.

• Effective rank ( Roy and Vetterli, 2007 ) : er ​ ( 𝐗 ) = exp ⁡ ( H ​ ( σ ~ ) ) \text{er}(\mathbf{X})=\exp(H(\tilde{\sigma})) , where H ⁡ ( σ ~ ) H(\tilde{\sigma}) is the Shannon entropy of the normalized singular value distribution σ ~ i = σ i / ∑ j σ j \tilde{\sigma}_{i}=\sigma_{i}/\sum_{j}\sigma_{j} .

• Condition number : κ ⁡ ( 𝐗 ) = σ max / σ min \kappa(\mathbf{X})=\sigma_{\max}/\sigma_{\min} , the ratio of largest to smallest singular value. We use 1 / κ 1/\kappa as the score so that higher values indicate better quality.

• PCA 95% variance : The number of principal components needed to capture 95% of the variance in the hidden states.

##### Results.

Table 13 shows that stable rank outperforms all alternative metrics across every category, achieving 84.04% overall accuracy compared to 61.91% for PCA 95% variance, 54.50% for effective rank, and 36.04% for condition number. The gap is largest on challenging categories: stable rank reaches 84.34% on Math while effective rank and condition number fall below 13%, and 76.20% on Safety while alternatives remain below 46%.

Among baseline metrics, PCA 95% variance performs best overall but struggles on Math (27.96%) and Safety (45.45%). Effective rank shows moderate Code performance (76.12%) but fails on Math (12.75%). Condition number performs poorly across all categories, suggesting that extreme singular value ratios are unreliable quality indicators.

##### Analysis.

Stable rank outperforms alternatives due to two key properties. First, it aggregates information across the entire singular value spectrum through the Frobenius norm, making it robust to outliers that affect condition number. Second, it balances representational richness (Frobenius norm) with coherence (spectral norm): high-quality responses activate diverse semantic dimensions in a structured manner, while low-quality responses either collapse into low-dimensional representations or exhibit erratic noise. The entropy weighting in effective rank and discrete counting in PCA 95% variance appear less suited for capturing these quality distinctions.

## Appendix H Ablation: Context Window Size

We investigate how context window size affects stable rank performance on RewardBench. For each prompt-response pair, we truncate the combined sequence to a maximum length in { 128,512 , 1024 , 2048 , 4096 } \{128,512,1024,2048,4096\} tokens before computing stable rank on the final layer of Qwen3-8B ( Yang et al., 2025 ) . RewardBench examples have a mean length of 281 tokens, with 75% under 512 tokens.

##### Results.

Table 14 shows that stable rank accuracy degrades at short context lengths but saturates quickly. At 128 tokens, overall accuracy drops to 62.59%, a 21 percentage point decrease from 512 tokens (83.85%). Code suffers most severely, falling from 87.91% to 24.80%, because truncation removes critical program logic. However, extending beyond 512 tokens yields minimal gains: increasing to 4096 tokens improves accuracy by only 0.2 percentage points.

##### Analysis.

The rapid saturation indicates that stable rank captures semantic content rather than mechanically rewarding longer sequences. Short windows (128 tokens) harm performance by cutting off meaningful content, but once the window covers the core reasoning structure (around 512 tokens for this dataset), additional context provides diminishing returns. Safety performance remains constant at 76.20% for all lengths ≥ \geq 512, suggesting that refusal quality is determined early in the response.

##### Practical Recommendations.

A 512-token window provides an effective operating point for datasets with similar length distributions. For longer-form tasks, we recommend scaling the window to cover at least the 75th percentile of sequence lengths to avoid truncation artifacts.

## Appendix I Ablation: Input Prompt Format

We investigate whether input prompt format affects stable rank performance. Since stable rank is computed on hidden states, the concatenation structure could influence the resulting geometry. We test six formats ranging from minimal (no prefix) to structured (User/Assistant tags).

(1) {prompt}\n\n{response} (no prefix)

(2) {prompt}\n\nResponse:{response}

(3) {prompt}\n\nAssistant:{response}

(4) User:{prompt}\n\nAssistant:{response}

(5) Question:{prompt}\n\nAnswer:{response}

(6) {prompt}\n\nAnswer:{response}

##### Results.

Table 15 shows that stable rank is robust to format variation. Overall accuracy varies by at most 3 percentage points across formats for each model, and no single format consistently outperforms others. Category-level variations are larger (e.g., Safety on Phi-3.5-mini ranges from 52.82% to 62.63%) but show no systematic pattern across models, indicating that format effects are model-specific rather than fundamental to stable rank.

##### Practical Recommendations.

The robustness to prompt format simplifies deployment: practitioners can use simple formats without extensive tuning. We recommend using the model’s native chat template when available, or a minimal format like {prompt}\n\nAnswer:{response} otherwise.

## Appendix J Stable Rank Example Prompts

To complement the quantitative analysis in Section 5 , we provide examples illustrating how stable rank differentiates high-quality responses from two common failure modes: catastrophic repetition and redundant verbosity.

##### Catastrophic Repetition.

The Image Processing example in Table 16 shows a response that begins coherently but enters an infinite repetition loop (“Image > > Image > > …”), collapsing into a degenerate low-dimensional representation. This failure directly relates to coherence standard deviation, which has the strongest negative correlation with stable rank ( ρ = − 0.356 \rho=-0.356 ). Geometrically, repetition concentrates variance into a single direction, reducing the effective dimensionality that stable rank measures.

##### Redundant Verbosity.

The Math Problem example shows a response that correctly solves the problem but continues with unnecessary elaboration (“However…”, “Alternatively…”). While the content is not incorrect, it dilutes information density. This aligns with our finding that stable rank correlates negatively with token count ( ρ = − 0.294 \rho=-0.294 ) and total discourse markers ( ρ = − 0.204 \rho=-0.204 ), while correlating positively with compression ratio ( ρ = 0.233 \rho=0.233 ). The high stable rank response terminates precisely at the answer, maintaining focused information flow.

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