EXP-CHAIN-N5

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Chain N=5 — Does Directed Acyclic Flow Scale? Gradient Persistence at Higher N

2026-03-18 L3 level paper3 1 runs $0.01

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In plain language

Five AI agents form a chain: Alpha writes freely, Beta reads only Alpha, Gamma reads only Beta, Delta reads only Gamma, Epsilon reads only Delta. Information flows ONE WAY through a 5-link chain. No feedback. With 3 agents, the chain created the strongest hierarchy gradient. Does this scale? Or does the signal die before reaching the end?

What we found

UNDERPOWERED (N=1): No predictions scored

Predictions we made before running

0/5 confirmed

▶ Technical details

Research hypothesis

OPEN QUESTION: "emergence threshold: at what N agents + interaction density do emergent properties appear?"

CONTEXT — THE 3×2 CROSSOVER + CHAIN GAP: Topology × N matrix (N=3 vs N=5): Star N=3: TR_range=0.141 → Star N=5: TR_range=0.059 (COLLAPSE -58%) Cycle N=3: TR_range=0.071 → Cycle N=5: TR_range=0.096 (INCREASE +35%) Full N=3: TR_range=0.444 → Full N=5: pending Chain N=3: TR_range=0.363 → Chain N=5: THIS EXPERIMENT

CHAIN N=3 KEY FINDING: Chain N=3 produced STRONGEST gradient: Alpha TR=0.649 (source), Beta=0.286 (relay), Gamma=0.536 (sink-with-export). TR_range = max-min = 0.363. Directional hierarchy CONFIRMED by in-degree=0 structure.

WHY CHAIN N=5 IS CRITICAL: Chain topology has unique properties that disambiguate the N-scaling mechanism: 1. Star collapsed at N=5 because hub fan-in=4 → saturation. Chain has NO hub — max fan-in=1 for ALL agents. 2. Cycle STRENGTHENED at N=5 (uniform fan-in=1, longer cycle). Chain also has fan-in=1, but ACYCLIC. 3. Chain separates the CYCLICITY hypothesis from the FAN-IN hypothesis:

If chain-n5 preserves gradient: acyclicity + low fan-in → robust hierarchy at any N
If chain-n5 collapses: N itself is the factor (dilution regardless of topology)
If chain-n5 STRENGTHENS: longer chain = more gradient steps → amplification

GROUNDING: Gershenson E/S/C — chain = minimal coupling (each agent sees exactly 1 other). Cycle also has minimal coupling but has feedback. Chain = open system.

DESIGN: A→B→C→D→E (directed acyclic chain, 5 agents)

Alpha: reads NO agents (in=0, out=1) — pure source
Beta: reads Alpha only (in=1, out=1) — relay 1
Gamma: reads Beta only (in=1, out=1) — relay 2
Delta: reads Gamma only (in=1, out=1) — relay 3
Epsilon: reads Delta only (in=1, out=0 effective) — terminal sink

This creates 5 DISTINCT structural positions along a gradient. If hierarchy tracks position, we expect monotonic TR gradient.

BASELINE DATA: Chain N=3: TR_range=0.363, VP=0.193, Alpha_TR=0.649, Beta_TR=0.286, Gamma_TR=0.536 Cycle N=5: TR_range=0.096, VP=0.321, C=0.759 Star N=5: TR_range=0.059, VP=0.272, C=0.746 Full N=5: pending

COST: 10 rounds × 5 agents = 50 API calls (~$0.005). CHEAP pilot.

Experimental setup

Type: simple

Condition	Parameters
CHAIN_N5_LIVE	topology: CHAIN_N5, interaction: LIVE, note: Directed acyclic chain: A→B→C→D→E. Each agent reads only its predecessor. Alpha reads no agents.

Factors: topology (CHAIN_N5)

Parameters

n_agents: 5
n_rounds: 50
n_runs_per_condition: 1
model: gemini-2.5-flash-lite
temperature: 0.9
scheduler: round_robin