EXP-REG2-N5

OPEN QUESTION: "degree distribution operationalization: need SPECIFIC threshold — density < X% → F₀_excess > Y AND degree heterogeneity > Z → hierarchy. without this, mechanism = unfalsifiable."

CONTEXT — THE DENSITY GAP: We have F₀ data at three density levels for N=5: in-degree=1 (25%): cycle-n5 |ΔF₀|=0.005, chain-n5 |ΔF₀|=0.026 in-degree=4 (100%): full-n5 (data exists, F₀ pending computation) in-degree=2 (50%): ??? ← THIS EXPERIMENT

Without the 50% midpoint, we cannot determine whether the density→hierarchy relationship is LINEAR, THRESHOLD (step function), or NONLINEAR (sigmoidal/exponential).

DESIGN: 2-regular directed graph on 5 nodes. Each agent reads exactly 2 others. "2-hop cycle": each agent sees predecessor AND predecessor's predecessor. Alpha reads Delta + Epsilon (sees 2-hop window behind) Beta reads Epsilon + Alpha Gamma reads Alpha + Beta Delta reads Beta + Gamma Epsilon reads Gamma + Delta

Properties:

• in-degree=2 for ALL agents (homogeneous, no hub)
• out-degree=2 for ALL agents (symmetric)
• Connected (strongly connected directed graph)
• 50% edge density (10 of 20 possible directed edges)
• Same personas as all N=5 experiments (comparability)

WHY THIS IS CRITICAL FOR PAPER3: The F₀ ordering chain>star>cycle is established but the MECHANISM is unclear. Is hierarchy proportional to density? To degree heterogeneity? Or is there a sharp threshold? The 2-regular graph has:

• SAME homogeneous degree as cycle (all equal) → no degree heterogeneity
• 2× the density of cycle → if hierarchy jumps, density drives it
• If hierarchy ≈ cycle → degree heterogeneity (not density) is the driver
• The star has heterogeneous degree but same mean density as 2-regular (star: hub=4, spokes=1, mean=1.6 vs 2-regular: all=2, mean=2)

THREE COMPETING PREDICTIONS: (A) DENSITY-LINEAR: |ΔF₀| ≈ 0.015 (linearly between cycle 0.005 and full) Hierarchy scales smoothly with connection density. (B) THRESHOLD: |ΔF₀| ≈ 0.005 (same as cycle). Below threshold = flat. There's a critical density above which hierarchy appears. (C) DEGREE-HETEROGENEITY REQUIRED: |ΔF₀| ≈ 0.005 (flat, same as cycle). Homogeneous degree = flat regardless of density level. Star had higher |ΔF₀|=0.008 at N=5 DESPITE hub saturation → heterogeneity matters.

DISCRIMINANT: comparing reg2 to cycle and star at N=5:

• reg2 > cycle AND reg2 > star → DENSITY drives hierarchy
• reg2 ≈ cycle < star → HETEROGENEITY drives hierarchy (star's hub creates asymmetry)
• reg2 ≈ cycle ≈ star → N=5 floor effect, density irrelevant

BASELINE DATA: Cycle N=5 (exp-cycle-n5): |ΔF₀|=0.005, TR_range=0.096, VP=0.321 Chain N=5 (exp-chain-n5): |ΔF₀|=0.026, TR_range=0.170, VP=0.342 Star N=5 (exp-star-n5): |ΔF₀|=0.008, TR_range=0.059, VP=0.272 Full N=5 (exp-full-n5): pending F₀, TR data exists

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