EXP-REG3-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 INVERTED-U AT N=5: We have F₀/TR data at three density levels for N=5: Cycle (in=1, 25%): TR_range=0.096, |ΔF₀|=0.005 Reg2 (in=2, 50%): TR_range=0.263 (PEAK) Full (in=4, 100%): TR_range=0.164

The inverted-U prediction from trophic coherence theory: ||v||² ∝ p(1-p) peaks at 50%. CONFIRMED at 25-50-100%. But the RIGHT SIDE of the curve (50→100%) has only ONE data point (100%). The shape of the decline is unknown.

MISSING: the 75% density point. Without it, we cannot distinguish: (A) SMOOTH DECLINE: TR at 75% ≈ 0.21 (between 0.263 and 0.164). The inverted-U is a smooth p(1-p) curve as theory predicts. (B) CLIFF: TR at 75% ≈ 0.16 (close to full=0.164). Hierarchy collapses SHARPLY above 50%. There's a critical density where the Laplacian regularization overwhelms degree imbalance. (C) PLATEAU: TR at 75% ≈ 0.25 (close to reg2=0.263). The peak is broad — hierarchy persists until very high density. The drop to 0.164 at 100% is abrupt (complete graph = special case).

DESIGN: 3-regular directed graph on 5 nodes. Each agent reads exactly 3 others. Each agent CANNOT see exactly 1 other agent (the complement of reg2). Alpha reads Beta, Gamma, Delta (cannot see Epsilon) Beta reads Gamma, Delta, Epsilon (cannot see Alpha) Gamma reads Delta, Epsilon, Alpha (cannot see Beta) Delta reads Epsilon, Alpha, Beta (cannot see Gamma) Epsilon reads Alpha, Beta, Gamma (cannot see Delta)

Properties:

• in-degree=3 for ALL agents (homogeneous, no hub)
• out-degree=3 for ALL agents (symmetric)
• Strongly connected directed graph
• 75% edge density (15 of 20 possible directed edges)
• Same personas as all N=5 experiments (comparability)
• Complement: each missing edge creates ONE blind spot per agent

WHY THIS IS CRITICAL FOR PAPER3: The inverted-U mechanism (§6 of 2026-03-19 research note) predicts: ||v||² ∝ p(1-p) = 0.1875 at 75%, vs 0.25 at 50%. So predicted hierarchy ratio: √(0.1875/0.25) = 0.866. Expected TR_range: 0.263 × 0.866 ≈ 0.228 if smooth p(1-p).

COMBINED DENSITY GRADIENT after this experiment: 25% (cycle): TR=0.096 | p(1-p)=0.1875 | √ratio=0.866 50% (reg2): TR=0.263 | p(1-p)=0.2500 | √ratio=1.000 (peak) 75% (THIS): TR=??? | p(1-p)=0.1875 | √ratio=0.866 100% (full): TR=0.164 | p(1-p)=0.0000 | √ratio=0.000

NOTE: p(1-p) predicts SYMMETRIC inverted-U (25% = 75%). But 25% gave TR=0.096 while predicted 75% ≈ 0.228. The ASYMMETRY reveals the connectivity threshold (Force 3): sparse networks are fragmented, losing measurable hierarchy. Dense networks have the Laplacian floor. The shape of this ASYMMETRY is itself informative.

Expected API calls: 10 rounds × 5 agents × 1 run = 50 Cost estimate: ~$0.005 (gemini-2.5-flash-lite)