Final-Layer Hidden-State Trajectory

• Final-layer hidden-state trajectory is the sequence of vectors from the final model layer that captures the evolution of semantic and computational representations.
• It is employed to interpret model reasoning by analyzing geometric separability, activation deltas, and discrete state transitions in architectures like transformers and LSTMs.
• Insights from studying these trajectories inform model design improvements, including regularization techniques to counteract overspecialization in final-layer representations.

A final-layer hidden-state trajectory refers to the sequence or transformation of hidden-state vectors in the final layer of a neural sequence model—especially in transformers and LSTMs—as structured computation unfolds across tokens or time. This trajectory captures the dynamic internal evolution of the model's representations for each example and is a central object of empirical and theoretical analysis, revealing how deep networks internally encode semantics, task structure, reasoning, and uncertainty. Recent research specifically leverages the final-layer hidden-state trajectory for model interpretability, verification, and in-context behavior analysis, exploiting its geometric structure and separability properties.

1. Formal Definitions and Notation

Let a given model comprise $L$ layers, each mapping an input sequence %%%%1%%%% into stacked hidden states $h^\ell_t \in \mathbb{R}^D$ at each layer $\ell=1,\dots,L$ and time or token $t=1,\dots,T$. The final-layer hidden-state trajectory is defined as the sequence $\{h^L_{t_0}, h^L_{t_1}, \ldots, h^L_{t_n}\}$ collected at select pivotal timesteps of the computation or reasoning trace.

Concretely, in a chain-of-thought setting as analyzed in "CLUE: Non-parametric Verification from Experience via Hidden-State Clustering" (Liang et al., 2 Oct 2025), where explicit reasoning traces are delineated by $\langle$think$\rangle$…$\langle$/think$\rangle$ tokens, the key elements of the final-layer trajectory are:

• $h^L_{t_0}$: activation before reasoning starts (after $\langle$think$\rangle$),
• $h^L_{t_n}$: activation at the end of reasoning (before $\langle$/think$\rangle$).

The trajectory can then be summarized by the activation delta: $\Delta h := h^L_{t_n} - h^L_{t_0}$ This delta isolates the net representational transformation induced by the reasoning process, factoring out prompt or conditioning effects.

In transformer architectures, layerwise representations $h_\ell^{\mathrm{in}}$ and $h_\ell^{\mathrm{out}}$ further permit analysis of representation change across layers at each token, for which the per-layer angular displacement is given by: $$V_\ell = 1 - \frac{1}{2}\left(1 + \cos d(h_\ell^{\mathrm{in}}, h_\ell^{\mathrm{out}})\right)$$ with layer-jump quantification $C_L$ capturing sudden shifts at the network's apex (Shibata et al., 26 Jan 2026).

2. Empirical Structure and Geometric Separability

Empirical studies consistently demonstrate that the final-layer hidden-state trajectory is highly structured and often exhibits strong geometric or even discrete separability with respect to the underlying computational task or output correctness.

• In CLUE (Liang et al., 2 Oct 2025), correct and incorrect solution trajectories yield activation deltas ($\Delta h$) that form two clearly separable clusters in hidden-state space, as shown via PCA projections—enabling non-parametric, centroid-based verification with significant performance gains.
• For mathematical and symbolic computation, transformer models exhibit trajectories that walk between discrete attractors representing "implicit discrete state representations" (IDSRs), corresponding to latent symbolic states such as partial sums during addition (Chen et al., 2024). Trajectory transitions (measured by $\ell_2$ or cosine distance between consecutive $h^L_t$) spike at computationally significant tokens ("+", "="), reflecting underlying state updates.

The table below summarizes key trajectory quantifications:

Paper/Task	Trajectory Object	Separability Metric	Empirical Phenomenon
CLUE (Liang et al., 2 Oct 2025)	$\Delta h$ across CoT	$\ell_2$ centroid distance	Correct/incorrect clusters
IDSR (Chen et al., 2024)	$\{h^L_t\}$ over tokens	PCA, clustering, jump/delta norms	Polygonal path over discrete states
Layer-Jump (Shibata et al., 26 Jan 2026)	$h^L_{\mathrm{in}}, h^L_{\mathrm{out}}$	Angular/cosine “jump”	Final-layer spike

3. Methods Leveraging Final-Layer Trajectories

Several verification, analysis, and interpretability protocols employ the final-layer hidden-state trajectory:

A. Centroid-Based Verification (Liang et al., 2 Oct 2025)

• Compute $\Delta h$ for each past labeled trace (correct/incorrect).
• Form centroids $\mu_{\mathrm{succ}}, \mu_{\mathrm{fail}}$.
• Predict the label of new traces by nearest-centroid classification: $$\hat{y} = \arg\min_{c \in \{\text{succ}, \text{fail}\}} \|\Delta h_{\text{new}} - \mu_c\|_2$$ This nonparametric procedure captures the geometric signature of task performance.

B. Discrete State Analysis & Symbolic Reasoning (Chen et al., 2024)

• Extract the sequence $\{h^L_t\}$ at operator tokens.
• Project into lower dimensions (PCA/t-SNE), revealing walk over discrete state clusters (IDSRs).
• Quantify step-wise jumps, cosine similarity, and histogram separation to identify computationally meaningful transitions.

C. Predictive Complexity Analysis (Herrmann et al., 17 Mar 2025)

• Insert a “PHi” bottleneck predicting each $h_t$ from preceding hidden states.
• Use the per-token KL divergence between posterior $q_\psi(z_t | h_t)$ and prior $p_\chi(z_t | z_{<t})$ to measure "novel information gained" in the trajectory: $$\Delta I_t = D_{\mathrm{KL}}(q_\psi(z_t|h_t) \| p_\chi(z_t|z_{<t}))$$ Spikes in $\Delta I_t$ track nontrivial in-context learning or reasoning phases.

4. Layerwise Trajectory Behavior and Limitations

Recent multi-layer analysis reveals nuanced properties of the final-layer trajectory:

• Final layers tend toward over-compression and over-specialization relative to mid- or intermediate layers (Skean et al., 4 Feb 2025). Metrics such as entropy $S_1$, intrinsic dimensionality (PR), information to target $I(H_L;Y)$, and perturbation invariance all degrade relative to the optimal mid-layer.
• The "final-layer representation bottleneck" emerges directly from language modeling objectives, which require the last layer to prioritize only those features essential for the language modeling head, discarding general semantic features (Skean et al., 4 Feb 2025).
• In models exposed to extensive pre-training, the final-layer jump grows stronger, concentrating representational change into a "spike" at the apex, a phenomenon empirically substantiated across Llama, Gemma, and DeepSeek checkpoints (Shibata et al., 26 Jan 2026).

A plausible implication is that downstream applications (e.g., embedding extraction, robust representation learning) may benefit from utilizing mid-layer hidden-states or combinations thereof.

5. Regularization and Remediation Techniques

Large jumps in the final-layer trajectory present potential underutilization and brittleness in representation learning. To address this, jump-suppressing regularizers (JREG) have been proposed (Shibata et al., 26 Jan 2026), augmenting the training loss with a displacement penalty: $$L_{\mathrm{JREG}} = L_{\mathrm{CE}} + \lambda L_{\mathrm{disp}}$$ where $L_{\mathrm{disp}} = \sum_{\ell=1}^L w_\ell V_\ell$, and weights $w_\ell$ concentrate the penalty near the top layers. This approach successfully eliminates final-layer jumps ($C_L \to 0$), redistributes information processing across the network, and yields empirically consistent performance gains.

Recommended strategies include:

• Extract representations from mid- rather than top layers for embedding tasks (Skean et al., 4 Feb 2025).
• Use convex mixtures of mid- and final-layer embeddings.
• Enable skip connections into task heads to recover mid-layer information lost to final-layer overspecialization.

6. Theoretical Interpretation and Broader Implications

The structure of the final-layer hidden-state trajectory provides insight into the computational and representational dynamics of deep sequence models.

• In symbolic reasoning, the trajectory effectively traces the model’s path over implicit state representations, akin to a finite-state automaton but in a high-dimensional vector space (Chen et al., 2024).
• Analysis of the trajectory’s information dynamics reveals alignment with the information bottleneck principle: the final layer sacrifices global, semantically rich representations for specialization to the explicit prediction task (Skean et al., 4 Feb 2025).
• Overconcentration of transformation in the final layer, as measured by displacement and jump metrics, signals inefficient capacity use and motivates architectural or training interventions to balance representation evolution throughout the depth of the model (Shibata et al., 26 Jan 2026).

Collectively, these findings emphasize the final-layer hidden-state trajectory as both a window into model computation and a practical tool for enhancing reliability, interpretability, and robustness in modern neural LLMs.