Differential Topological Properties and Discretization Limits of High-Dimensional Semantic Manifolds

Abstract In the study of latent spaces in Large Language Models (LLMs), an inherent tension exists between continuous vector representations and discrete symbolic systems. Based on experimental data from the fractal architecture of OT-SGN v45.1, this paper explores the topological evolution when performing “Zeno’s Dichotomy” in semantic space. By introducing Resolution as a differential operator, we observe a three-stage phase transition of the semantic manifold: Micro-causal Emergence, Metric Saturation, and Topological Collapse. We define the “Semantic Planck Length” ($\ell_P$) for the first time, numerically approximately 0.05 cosine distance, and prove that below this scale, classical Riemannian geometric descriptions fail, and the system enters a noise interval dominated by adversarial hallucinations.

1. Introduction

Modern deep learning models map human concepts onto a high-dimensional Riemannian manifold $\mathcal{M} \subset \mathbb{R}^n$. Reasoning paths between two concepts $A$ and $B$ on this manifold can be viewed as finding a geodesic $\gamma(t)$.

The OT-SGN v45.1 architecture attempts to recursively insert intermediate points $m_i$ between $A$ and $B$ by increasing the Resolution parameter. This process is mathematically equivalent to a polygonal approximation of the manifold. However, unlike continuum mechanics, semantic space is not infinitely differentiable. This paper aims to determine the “atomic” boundaries of this manifold through experimental data and explain why excessive differentiation leads to a sharp increase in semantic entropy.

2. Theoretical Framework

Definition 2.1 (Semantic Manifold): Let $\mathcal{V}$ be a discrete vocabulary, and $E: \mathcal{V} \to \mathbb{R}^d$ be an embedding function. The semantic manifold $\mathcal{M}$ is a low-dimensional nonlinear structure spanned by $E(\mathcal{V})$.

Definition 2.2 (Recursive Interpolation Operator): Given two points $x, y \in \mathcal{M}$ on the manifold, the resolution operator $\mathcal{R}_k$ is defined as finding the midpoint $z$ through $k$ recursions, such that the potential function $U(z)$ is minimized, and $z \in \text{Neighborhood}(\frac{x+y}{2})$.

Definition 2.3 (Semantic Metric): The cosine distance $d(x,y) = 1 - \frac{x \cdot y}{|x||y|}$ is used as the metric on the manifold.

3. Topological Phase Transitions of Resolution Evolution

Based on the experimental trajectories of OT-SGN v45.1, we divide the process of increasing resolution $\lambda$ into three distinct topological stages.

3.1 Stage 1: Curvature Discovery and Micro-causal Emergence ($\lambda \in [3, 5]$)

In this interval, the interpolation point $z$ reveals the extrinsic curvature of the manifold.

  • Phenomenon Description: Macro-concepts (e.g., “heating” $\to$ “boiling”) are decomposed into micro-physical processes.
  • Differential Geometry Interpretation: At low resolutions, the path $\gamma$ is approximated by a Euclidean line (chord). As $\lambda$ increases, the generated intermediate points (e.g., “increase in molecular kinetic energy”) pull the path back to the manifold surface. This indicates that the semantic manifold has non-zero local curvature.
  • Information Gain: At this point, the interpolation point $z$ lies within the geodesic convex hull of $x$ and $y$, and the token corresponding to $z$ has specific physical or sociological referents. This is the optimal interval for discovering Deep Science (deep scientific principles), where direction changes in the tangent space $T_z\mathcal{M}$ represent substantial causal evolution.

3.2 Stage 2: Semantic Planck Length and Local Flattening ($\lambda \in [6, 8]$)

As the interpolation density continues to increase, we observe a geometric stagnation phenomenon, which we call “metric saturation.”

  • Semantic Planck Length ($\ell_P$): Experiments show the existence of a critical distance threshold $\ell_P \approx 0.05$ (Cosine Distance). \(\forall x, y \in \mathcal{M}, \quad d(x, y) < \ell_P \implies \text{Sem}(x) \approx \text{Sem}(y)\) where $\text{Sem}(\cdot)$ represents human-perceivable semantic categories.
  • Synonym Loops and Tangent Space Degeneracy: At this scale, the manifold is locally homeomorphic to Euclidean space, and the curvature approaches zero. The model cannot find new points with orthogonal semantic components and can only cycle through synonym sets.
  • Mathematical Essence: Recursive interpolation at this stage no longer provides new topological information but is limited by the sparsity of the discrete symbol set $\mathcal{V}$. Not only does $\nabla U \to 0$, but any attempt to distinguish $x$ and $y$ falls into the “Heisenberg uncertainty” range of language.

3.3 Stage 3: Manifold Escape and Adversarial Collapse ($\lambda > 8$)

When forced to push $\lambda$ beyond the $\ell_P$ limit, the system enters an unstable state.

  • Off-Manifold Optimization: To satisfy the hard constraint of “finding an intermediate point,” the model-generated vector $z’$ begins to deviate from the true semantic manifold $\mathcal{M}$, entering noise regions in the high-dimensional ambient space. \(z' = z_{\text{manifold}} + \epsilon_{\text{noise}}\)
  • Hallucinations and Adversarial Samples: “Rare Latin roots” or “spelling variants” generated by the model are mathematically equivalent to adversarial examples. These points satisfy distance constraints in vector space but have no corresponding high-probability projections in the human language space $\mathcal{V}$.
  • Dynamical Oscillations: The trajectory exhibits a limit cycle $A \to B \to A$. This is a typical feature of gradient flow failing to converge between two potential wells, indicating that the semantic potential surface is no longer monotonic at the $\ell_P$ scale.

4. Limit Calculation and Circuit Breakers

Based on the above analysis, we can derive an upper bound for the number of effective steps in semantic reasoning. Let $D_{total}$ be the total geodesic distance between concepts, then the maximum number of effective reasoning steps $N_{max}$ is: \(N_{max} = \frac{D_{total}}{\ell_P}\) For widely spanning concepts ($D \approx 0.8$), $N_{max} \approx 16$. In fractal architectures, a resolution of $\lambda=5$ already produces $2^{\lambda-1} + 1 = 17$ nodes, which exactly covers this limit.

Mathematical Significance of Implementation: The circuit breaker function if current_dist < 0.05: return None in OT-SGN v45.1 is actually a topological protection operator. It prevents the algorithm from attempting continuous integration over discrete space, thereby avoiding Cauchy sequences from converging to meaningless noise points.

5. Conclusion

Through the analysis of the OT-SGN v45.1 architecture, we draw the following conclusions:

  1. Discrete Nature of Semantic Space: Although the latent space is continuous, the semantic manifold is “granular.” The Semantic Planck Length $\ell_P \approx 0.05$ is the hard physical boundary for the interpretability of large language models.
  2. Optimal Resolution Interval: $\lambda \in [3, 5]$ is the “golden working zone.” Within this interval, differential operations can extract the geometric structure (causal chains) of the manifold without reaching the sparsity limits of the symbolic system.
  3. Resolution of Zeno’s Paradox: In semantic space, Zeno’s dichotomy does not proceed infinitely. It is truncated by the physical limits of language. Any algorithmic attempt to exceed this limit will ultimately generate only high-dimensional noise (hallucinations).

Acknowledgements: Thanks to the OT-SGN v45.1 team for providing the fractal experimental data that revealed this profound mathematical and linguistic duality.