OPEN QUESTION: "emergence threshold: at what N agents + interaction density do emergent properties appear?"
CONTEXT: We now have a 2×2 design with ONE MISSING CELL: Star N=3 (exp-topo-star): TR_range=0.141, hub-as-SINK confirmed Star N=5 (exp-star-n5): TR_range=0.059, hub SATURATED → flat! Cycle N=3 (exp-topo-cycle): TR_range=0.071, flat as predicted Cycle N=5 (THIS): ???
CRITICAL FINDING: Star N=5 collapsed to TR=0.059 (flatter than cycle N=3!). Hub saturation at fan-in=4. This means N=5 star ≈ cycle N=3 in flatness.
QUESTION: Does cycle N=5 ALSO stay flat? Or does something emerge at N=5 with uniform degree that didn't happen at N=3?
Directed cycle: A→B→C→D→E→A. Each agent reads ONLY predecessor. In-degree=1 for all (same as cycle N=3). Uniform. No hubs.
THREE COMPETING PREDICTIONS: (A) STAYS FLAT: TR_range ≈ 0.071 (same as cycle N=3). Uniform degree = flat regardless of N. The "table" result. (B) GETS FLATTER: TR_range < 0.05. Longer cycle = more dilution. 5-hop transitive path vs 3-hop. Innovation advantage dissipates. (C) HIERARCHY EMERGES: TR_range > 0.15. Longer cycle creates temporal separation — early innovators have 4-hop advantage before cycle catches up. N itself creates asymmetry.
THE 2×2 INTERACTION TEST: If star N=5 is flat AND cycle N=5 is flat → N=5 inherently resists hierarchy If star N=5 is flat AND cycle N=5 has hierarchy → cycle paradoxically enables hierarchy at N>3 If both scale similarly → topology doesn't matter at N=5, only at N=3
BASELINE DATA: Cycle N=3 (exp-topo-cycle, N=1): TR_range=0.071, VP=0.167, C=0.715, RP=0.081 Star N=5 (exp-star-n5, N=1): TR_range=0.059, VP=0.272, C=0.746, RP=0.055 Star N=3 (exp-topo-star, N=1): TR_range=0.141, VP=0.243, C=0.778, RP=0.072 Complete N=3 (exp-asym, N=9): TR_range=0.444, VP=0.075, C=0.344, RP=0.147
GROUNDING: Gershenson E/S/C — complexity at intermediate coupling. Cycle = minimal coupling (in-degree=1). Does C peak at low coupling + high N?