Principal Semantic Vector (PSV)

• Principal Semantic Vector (PSV) is a continuous semantic representation that distills and propagates the core logical essence from an LLM’s internal reasoning trace.
• It extracts critical tokens using Global Causal Contribution and Competitive Persistence Scoring, then applies PCA to determine a dominant semantic direction for watermark guidance.
• Empirical results demonstrate that PSV-guided watermark embedding lowers perplexity and achieves high detection AUC, ensuring robust and contextually coherent answer generation.

The Principal Semantic Vector (PSV) is a continuous semantic representation designed to distill and propagate the logical essence of a reasoning trace in LLMs. Within the ReasonMark watermarking framework, the PSV acts as the dynamic semantic compass that bridges the undisturbed Thinking Phase of model inference with the Answering Phase, enabling robust, contextually adaptive watermark embedding. By extracting the most semantically pivotal tokens from the chain-of-thought and aggregating their embeddings into a principal direction, the PSV ensures that watermark bias aligns with the evolving logic of generation, preserving answer correctness while facilitating high-fidelity, detectable watermarking (Liu et al., 8 Jan 2026).

1. Segmentation of Reasoning and Answer Generation

ReasonMark operationalizes two strictly distinct stages for watermarking in reasoning-intensive LLMs:

• Thinking Phase $T_{\text{think}} = \{t_i\}_{i=1}^N$: The model computes internal token probabilities $\{P_i\}_{i=1}^N$, progressing the reasoning trajectory with no external emission.
• Answering Phase $T_{\text{answer}} = \{t_i\}_{i=N+1}^S$: The model generates the final output, modulating each token's watermark bias according to its semantic alignment with the evolving PSV.

The PSV, denoted $\mathcal{R}_0$, is initialized by distilling information from $T_{\text{think}}$:

$$\mathcal{R}_0 = f_{\eta}\left(\{t_i\}_{i=1}^N, \{P_i\}_{i=1}^N\right)$$

This function $f_{\eta}$ captures the “semantic essence” of the internal chain-of-thought.

2. Criticality Score: Identifying Semantically Pivotal Tokens

To extract tokens that anchor the reasoning trajectory, ReasonMark computes a Criticality Score (CS) for each vocabulary word based on two components:

• Global Causal Contribution (GCC): Quantifies a token’s probabilistic influence across reasoning steps, weighted by Jensen–Shannon divergence and normalized cosine similarity of distributions.

$$\mathrm{GCC}(w) = \sum_{i=1}^N \left[P_i(w)\,\lambda_i\, \sum_{j=i+1}^M \alpha_{i\to j}\,P_j(w)\right]$$

• $P_i(w)$: Probability of $w$ at step $i$.
• $\lambda_i = \mathrm{JS}(P_i\|P_{i-1})$: Measures distributional shift.
• $\alpha_{i\to j}$: Propagates influence through similarity.
  • Competitive Persistence Scoring (CPS): Measures how persistently a token remains salient and competitive.

$$\mathrm{CPS}(w) = \sum_{i=1}^N \left[S(t_i)^{-1}(1-\Delta_i(w))\, \sum_{j=i+1}^M \mathbb{I}(w \in \mathrm{Top}_k(P_j))\right]$$

• $S(t_i)^{-1}$: Inverse surprisal of generated token.
• $\Delta_i(w)$: Logit margin against the strongest competitor.
• $\mathbb{I}(w \in \mathrm{Top}_k(P_j))$: Persistence in the candidate set.

The aggregate score,

$$\mathrm{CS}(w) = \mathrm{GCC}(w)\,\log\left[1+\mathrm{CPS}(w)\right]$$

selects the top-$K$ “Critical Tokens” $\mathcal{C}'$, encapsulating the most semantically influential elements of the chain-of-thought.

3. Construction and Updating of the Principal Semantic Vector

Upon identifying $\mathcal{C}' = \{w_1,\ldots,w_K\}$, ReasonMark forms an embedding matrix $H$:

$$H = [E(w_1), E(w_2), \ldots, E(w_K)]^T \in \mathbb{R}^{K \times d}$$

where $E(\cdot)$ maps tokens to their embedding space of dimension $d$. Principal Component Analysis is applied:

$\mathcal{R}_0 = \mathbf{v}_1 = \mathrm{PCA}_1(H)$

yielding the dominant semantic direction. As answer generation proceeds, the PSV evolves:

$$\mathcal{R}_i = (1-\beta_i)\,\mathcal{R}_{i-1} + \beta_i\,E(t_i)$$

with $\beta_i$ modulated by semantic alignment, ensuring that $\mathcal{R}_i$ remains anchored to the reasoning trajectory while adapting to the generated content.

4. PSV-Guided Watermark Embedding

During the Answering Phase, watermark bias for each candidate token is determined by its cosine similarity $s_{w,i-1}$ with the current PSV:

$$s_{w,i-1} = \frac{E(w) \cdot \mathcal{R}_{i-1}}{\|E(w)\|\,\|\mathcal{R}_{i-1}\|}$$

The logit of each candidate in the “green list” $\mathcal{V}_g$ is augmented:

$$\delta_{i,w} = \delta_0 + \delta_{\lambda} s_{w,i-1}$$

$L_i(w) \leftarrow L_i(w) + \delta_{i,w}$

where $\delta_0$ is the base strength and $\delta_\lambda$ controls semantic sensitivity.

Tokens with high alignment receive stronger watermark bias, whereas those less aligned are perturbed minimally—preserving logical integrity and answer quality.

5. Step-by-Step Algorithmic Workflow

The complete mechanism for PSV extraction and guided watermarking can be summarized as:

Phase	Step	Key Operations
Thinking	Compute $\{P_i\}$, $\mathrm{GCC}(w), \mathrm{CPS}(w)$	Select top-$K$ critical tokens $\mathcal{C}'$
	Build $H$; PCA to get $\mathcal{R}_0$	Semantic distillation of chain-of-thought
Answering	Initialize $\mathcal{R}_N = \mathcal{R}_0$	Partition vocabulary; compute $s_{w,i-1}$, update logits
	Sample $t_i$; EMA update of $\mathcal{R}_i$	Dynamic semantic compass during generation

This procedure enables a tightly synchronized watermark signal that is semantically adaptive and computationally efficient.

6. Empirical Performance and Robustness

Extensive benchmarking demonstrates the PSV’s centrality. ReasonMark achieves lowest perplexity (10.31, C4) and highest detection AUC (99.31%) compared to KGW and others. In machine translation (WMT16 DE–EN), a top BLEU score (9.916) and AUC (87.25%) are reported. In mathematical tasks (AIME, GSM8K), accuracy is on par with or better than unwatermarked baselines, while watermark detection remains at AUC $>99.9\%$.

Ablation studies dissect PSV contributions:

• Replacing CTs (Critical Tokens) with random tokens raises C4 perplexity from 10.31 to 12.88.
• Removing GCC increases perplexity to 11.15.
• Removing CPS reduces AUC from 99.31% to 98.69%.

Attack robustness tests indicate ReasonMark, via PSV guidance, retains high detection under word-level and semantic paraphrasing ($>93\%$ AUC) and under translation attacks ($>82\%$ AUC), outperforming prior methods.

7. Context and Implications

The development of the PSV within ReasonMark addresses central challenges in watermarking reasoning LLMs: avoiding logical corruption, controlling computational latency, and ensuring adversarial robustness. The compact, dynamic nature of the PSV allows for semantic harmony between the reasoning and generation steps, directly modulating watermark strength where the reasoning trajectory is most salient.

A plausible implication is that dynamic, semantically grounded watermark mechanisms such as PSV enable traceable, trustworthy deployment of LLMs in sensitive, real-world environments, promoting accountability without sacrificing solution quality or efficiency (Liu et al., 8 Jan 2026).