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Asymptotic Semantic Collapse in Hierarchical Optimization

Faruk Alpay, Bugra Kilictas

TL;DR

In a closed linguistic setting with a Dominant Anchor Node whose semantic state has effectively infinite inertia, it is shown that repeated interactions with Peripheral Agent Nodes drive an asymptotic alignment that minimizes a global loss.

Abstract

Multi-agent language systems can exhibit a failure mode where a shared dominant context progressively absorbs individual semantics, yielding near-uniform behavior across agents. We study this effect under the name Asymptotic Semantic Collapse in Hierarchical Optimization. In a closed linguistic setting with a Dominant Anchor Node whose semantic state has effectively infinite inertia, we show that repeated interactions with Peripheral Agent Nodes drive an asymptotic alignment that minimizes a global loss. We model semantic states as points on a Riemannian manifold and analyze the induced projection dynamics. Two consequences follow. First, the limiting semantic configuration is insensitive to the optimization history: both smooth gradient-style updates and stochastic noisy updates converge to the same topological endpoint, establishing path independence at convergence. Second, the degree of context dependence controls information content: moving from atomic (independent) representations to fully entangled (context-bound) representations forces the node entropy, interpreted as available degrees of freedom, to vanish in the limit. The theory connects information-theoretic quantities with differential-geometric structure and suggests an interpretation as an immutable consensus rule that constrains agents to a shared semantic grammar. A lightweight dataset-free benchmark on an RWKV-7 13B GGUF checkpoint complements the analysis, reporting zero hash collisions, mean compliance of 0.50 under greedy decoding and 0.531 under stochastic decoding, and final Jaccard-to-anchor similarity values of 0.295 and 0.224, respectively.

Asymptotic Semantic Collapse in Hierarchical Optimization

TL;DR

In a closed linguistic setting with a Dominant Anchor Node whose semantic state has effectively infinite inertia, it is shown that repeated interactions with Peripheral Agent Nodes drive an asymptotic alignment that minimizes a global loss.

Abstract

Multi-agent language systems can exhibit a failure mode where a shared dominant context progressively absorbs individual semantics, yielding near-uniform behavior across agents. We study this effect under the name Asymptotic Semantic Collapse in Hierarchical Optimization. In a closed linguistic setting with a Dominant Anchor Node whose semantic state has effectively infinite inertia, we show that repeated interactions with Peripheral Agent Nodes drive an asymptotic alignment that minimizes a global loss. We model semantic states as points on a Riemannian manifold and analyze the induced projection dynamics. Two consequences follow. First, the limiting semantic configuration is insensitive to the optimization history: both smooth gradient-style updates and stochastic noisy updates converge to the same topological endpoint, establishing path independence at convergence. Second, the degree of context dependence controls information content: moving from atomic (independent) representations to fully entangled (context-bound) representations forces the node entropy, interpreted as available degrees of freedom, to vanish in the limit. The theory connects information-theoretic quantities with differential-geometric structure and suggests an interpretation as an immutable consensus rule that constrains agents to a shared semantic grammar. A lightweight dataset-free benchmark on an RWKV-7 13B GGUF checkpoint complements the analysis, reporting zero hash collisions, mean compliance of 0.50 under greedy decoding and 0.531 under stochastic decoding, and final Jaccard-to-anchor similarity values of 0.295 and 0.224, respectively.
Paper Structure (23 sections, 2 theorems, 20 equations, 2 figures, 1 table, 7 algorithms)

This paper contains 23 sections, 2 theorems, 20 equations, 2 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Framework
  3. Hierarchical Semantic Optimization Model
  4. Scalar Tokens vs. Contractual Tensors
  5. Main Results and Proofs
  6. Trajectory Irrelevance Theorem
  7. Entropy Collapse and State Entanglement
  8. Conclusion
  9. Dataset-Free Benchmark: Experimental Protocol and Reporting Standards
  10. Benchmark setup
  11. Measured quantities
  12. Algorithmic specification
  13. Objects and notation.
  14. Anchor boundary distribution.
  15. Results reporting
  16. ...and 8 more sections

Key Result

Theorem 1

Consider two processes by which a peripheral agent $i$'s state $x_i$ moves from an initial value $x_i(0)$ to the anchor $a$: (1) a smooth deterministic process following the Riemannian gradient flow $\frac{D x_i}{dt} = -\operatorname{grad}_{x_i} L$ (Eq. eq:gradient_flow), and (2) a stochastic proces

Figures (2)

  • Figure 1: Geometric intuition for hierarchical semantic alignment on a manifold $\mathcal{M}$. Peripheral states $x_i$ evolve toward the fixed anchor $a$ (Dominant Anchor Node). Different trajectories (smooth vs. stochastic) may follow different paths but share the same limiting collapsed state.
  • Figure 2: Benchmark dynamics across rounds (mean over agents). The top panel reports next-token entropy $H(p)$, and the bottom panel reports Central Context compliance. The vertical layout avoids label collisions and preserves readability at paper column widths.

Theorems & Definitions (4)

  • Theorem 1: Trajectory Irrelevance
  • proof
  • Theorem 2: Entropy Collapse under Full Alignment
  • proof