Unified Token Space: Geometry and Stability

• Unified Token Space is a paradigm that encodes diverse data as tokens in a shared latent geometry, enabling joint processing across modalities.
• Empirical analyses reveal stratification with varying intrinsic dimensions and strongly negative Ricci curvature, influencing model fluency and numeric reasoning.
• Design principles recommend minimizing unnecessary codimension, excessive curvature, and complex stratification to enhance stability and transferability.

A unified token space is a representational paradigm in which diverse data, modalities, or task outputs are encoded as elements (tokens) in a shared, mathematically-structured latent space. This concept enables joint processing, transfer, and evaluation across tasks, models, or modalities by enforcing a common embedding, quantization, or transformation protocol. Unified token spaces have been explored in LLMs, multimodal architectures, vision-language systems, decision transformers, and even in category-theoretic treatments of AI computation, with the geometric, statistical, and algebraic structure of token spaces having significant implications for model expressivity, generalization, and stability.

1. Geometric and Topological Foundations of Token Spaces

The foundational structure of the token space in LLMs is defined by mapping a finite vocabulary $V$ to a high-dimensional ambient space $\mathbb{R}^D$ via a learned embedding $f:V\rightarrow \mathbb{R}^D$ (Robinson et al., 11 Oct 2024). The realized token subspace $T = f(V)$ is typically a lower-dimensional set sampled as a point cloud in $\mathbb{R}^D$. The key question is whether this subspace forms a smooth manifold and how its local and global geometric properties impact model behavior.

Estimation of Intrinsic Dimension and Curvature

Volume-radius asymptotics are utilized to estimate local intrinsic dimension ($n$) and Ricci scalar curvature (Ric) of $T$. For small radii, the ball-volume in Euclidean $d$-space is $$v_{\text{Euc}}(r) = \pi^{d/2}/\Gamma(d/2+1)\cdot r^d$$, and on a Riemannian manifold: $$v_{\text{man}}(r) = K\cdot r^n \left[1-\frac{\text{Ric}}{6(n+2)}r^2+\mathcal{O}(r^4)\right]$$ Taking logs yields a linear model for log-volume vs. log-radius with quadratic correction, facilitating local dimension and curvature estimation via least-squares regression over token neighborhoods.

Robinson et al. implement a Monte Carlo procedure: for each token $j$, counts of tokens within distance $r$ ($v(r_{k,j};j) \approx M\cdot k$) are regressed to extract local $\widehat n_j$ and $\widehat{\text{Ric}}_j$, employing bias corrections and per-token normalization. This statistical approach circumvents the impossibility of directly fitting continuous structures to a finite, discrete vocabulary.

2. Empirical Structure: Stratification and Distributional Analysis

Empirical results on GPT-2 (D=768), LLEMMA7B and MISTRAL7B (each D=4096) reveal that:

• Intrinsic dimension is far below ambient ($D$) and varies widely across the token subspace $T$.
  • GPT-2: median intrinsic dimension of $\sim\!500$ for non-numeric tokens, but $\sim\!16$ for numeric tokens.
  • LLEMMA7B: non-numeric $\sim\!10.5$, numeric $\sim\!6.8$ (with many isolated zeros).
  • MISTRAL7B: non-numeric $\sim\!5.6$, numeric $\sim\!2.8$ (many isolated).
• Stratification: $T$ is not a manifold but a stratified manifold; “knees” in the volume-radius curve indicate abrupt changes in local dimension, substantiating the existence of discrete strata—regions of constant but different dimension and curvature.
• Curvature: Ricci scalar curvature is significantly negative on each stratum:
  • GPT-2 non-numeric: Ric $\sim -63$; numeric: Ric $\sim -2.4$
  • LLEMMA7B non-numeric: Ric $\sim -169$; numeric: Ric $\sim -170$
  • MISTRAL7B non-numeric: Ric $\sim -5036$; numeric: Ric $\sim -5693$

The numeric-token stratum in GPT-2 is almost flat (Ric $\rightarrow 0$), correlating with GPT-2’s weak numeric reasoning. In contrast, math/code-specialized models generate numerous isolated, low-dim numeric points to better distinguish them, supporting stronger mathematical reasoning.

3. Implications for Model Expressivity, Fluency, and Stability

High intrinsic dimension and strongly negative curvature correlate with regions of enhanced generative fluency. Strata boundaries—in which dimension and Ricci curvature change abruptly—mark points where continuous transformer mappings become non-smooth on $T$, causing inference discontinuities. This geometrically induced non-smoothness is a fundamental limitation: crossing a stratification boundary induces a nontrivial behavioral shift.

A unified token space with a large codimension ($D-n$) and pronounced negative curvature is susceptible to overfitting and numerical instability, particularly during fine-tuning. Small shifts in embedding space can push tokens out of the support of $T$, resulting in “bifurcations” (sudden output changes) during inference. This provides a geometric rationale for observed empirical instability in transfer and adaptation scenarios.

4. Unified Latent-Space Perspective and Design Principles

The geometric analysis implies that token spaces supporting strong, stable, and transferable generative models should ideally minimize:

• Unnecessary codimension: reduce $D-n$ to decrease the scope for off-manifold drift.
• Excessively negative curvature: avoid configurations that accentuate generalization fragility and sharp “behavioral cliffs” across the token space.
• Stratification complexity: limit abrupt transitions in local structure.

Design, fine-tuning, and retrieval strategies must account for the stratified structure. Inference and supervision approaches should respect stratification boundaries to avoid sharp discontinuities in behavior, which are not manageable by global smoothness assumptions.

5. Token Space, Modality, and Unification Beyond Language

The principles elucidated in language-model token spaces extend to multimodal unified tokenizers and tasks involving vision, audio, or general structured data. In each case, the objective is to create a token space (discrete or continuous) in which modality-induced submanifolds interact according to the overall tasks’ algebraic and geometric structure. The stratified manifold model of token space provides a theoretical substrate informing unified designs for embedding, decoding, and generation across data types and tasks.

Geometric diagnostics—such as estimated local dimension and Ricci curvature—provide first-principles criteria for comparing or constructing token spaces for unification across languages, modalities, and domains. Such diagnostics enable the principled evaluation and tuning of quantizers, embedding schemes, and transformer architectures as AI models become increasingly broad in scope.

6. Outlook and Future Directions

Unified token spaces—when viewed as stratified geometric objects—offer a rigorous conceptual and practical framework for understanding, designing, and diagnosing complex neural architectures. Open directions include quantifying the minimal necessary stratification for expressivity, constructing embeddings and quantizers that minimize geometric pathologies, and extending geometric analysis to multimodal, cross-lingual, and hierarchical tokenization regimes.

A plausible implication is that unified token space diagnostics will play a central role in the principled development of future foundation models. They offer interpretable, quantitative levers for balancing expressivity, fluency, stability, and robustness as language and multimodal systems continue to expand in capability and scope.