Hop Scaling: Multi-Agent Chain Degradation Saturates

Executive Summary

A critical question for multi-agent architectures is how coordination quality degrades as information passes through successive agents: linearly (manageable), exponentially (catastrophic), or with saturation (benign). Paper XXIII established that the context fence + domain prime handoff achieves CE 0.503 for a single handoff. This paper tests whether that unit scales, with four experiments on Qwen 2.5-7B covering chains of 1 to 10 hops, re-priming, shared vocabulary, and cross-domain handoffs.

The degradation is logarithmic, not exponential. Cross-entropy rises from 0.503 at hop 1 to approximately 0.60 by hop 5, then plateaus through hop 10. Neither linear (R-squared = 0.50) nor exponential (R-squared = 0.48) models fit the data well; the shape is logarithmic: a fast initial rise, then flat. The paper interprets the plateau as a chain equilibrium — by hops 3 to 5, relay outputs converge to a stable distribution that carries domain identity but no further expert-specific trajectory information.

Two additional findings challenge conventional wisdom. First, intermediate fences are overhead: applying a context fence at every hop versus only at the final hop produces identical results at 10 hops (CE 0.598 vs 0.597). The fence prepares the terminal agent that does the domain work, not the relay chain. Second, cross-domain handoffs actually outperform within-domain handoffs (CE 0.56-0.63 vs 0.72), because the domain switch forces a genuine mode reset that within-domain chains never trigger. Pre-loading a shared vocabulary into every agent hurts at every chain length (+0.04 to +0.07 CE). Per-hop degradation does not compound through the 10 hops tested.

Key Findings

• Logarithmic degradation: CE rises from 0.503 to ~0.60 by hop 5, then plateaus through hop 10
• Neither linear nor exponential: Linear R-squared = 0.50, exponential R-squared = 0.48 — logarithmic curve is the correct fit
• Intermediate fences are overhead: Fence-every-hop (0.598) vs fence-final (0.597) at 10 hops — identical performance
• Cross-domain outperforms within-domain: Cross-domain CE 0.56-0.63 vs within-domain 0.72 — the domain switch acts as a natural mode reset
• Shared vocabulary hurts: A vocabulary preamble in every agent adds +0.04 to +0.07 CE at all chain lengths

Key References

• McEntire (2026) — Context Fences: mode switching mechanism (Paper XXIII)
• McEntire (2026) — The Coordination Problem Is Interference: throughline (Papers XVI–XXII)
• McEntire (2026) — Ritual Shape: priming saturation (Paper XX)
• McEntire (2026) — Sender Continuation Perplexity: KV-cache measurement method (Paper XVIII)