OPEN QUESTION: "emergence threshold: at what N agents + interaction density do emergent properties appear?"
CONTEXT — THE 2×2 + CROSSOVER: We discovered a CROSSOVER INTERACTION in topology × N: Star N=3: TR=0.141 (hub creates hierarchy) Star N=5: TR=0.059 (hub SATURATES → flat) Cycle N=3: TR=0.071 (uniform → flat) Cycle N=5: TR=0.096 (uniform → slight increase, CYCLE > STAR at N=5!)
The star hub at N=5 collapsed because fan-in=4 → cognitive overload. But the cycle preserved hierarchy because each agent only has fan-in=1.
CRITICAL DISAMBIGUATION: What happens with FULL GRAPH at N=5? Complete topology: every agent sees every other (in-degree=4 for ALL agents).
This separates TWO competing explanations for star N=5 collapse: (A) HUB-SPECIFIC saturation: only the HUB saturates at fan-in=4. If full-n5 has HIGH hierarchy → hub-specific. Symmetric fan-in=4 doesn't saturate because outputs balance inputs. (B) N-GENERAL dilution: ANY configuration at N=5 flattens hierarchy. If full-n5 is FLAT → N itself prevents hierarchy regardless of topology. (C) DENSITY-HIERARCHY: more connections = more hierarchy (as at N=3 where complete graph had highest TR=0.444). If full-n5 TR > 0.15 → density creates hierarchy at ANY N.
BASELINE DATA: Complete N=3 (exp-asym PERSONA_LIVE, N=9): TR_range=0.444, VP=0.075, C=0.344, RP=0.147 Star N=5 (exp-star-n5, N=1): TR_range=0.059, VP=0.272, C=0.746, RP=0.055 Cycle N=5 (exp-cycle-n5, N=1): TR_range=0.096, VP=0.321, C=0.759, RP=0.048
GROUNDING: Gershenson E/S/C — complete graph = maximal coupling. At N=3 this produced HIGHEST complexity (C=0.344? actually LOWEST). Correction: exp-asym C=0.344 was LOWEST — max coupling → LOW C at N=3. Does N=5 full graph change this? May-Wigner: random matrices → instability at N×connectivity > threshold.
WHY THIS MATTERS FOR PAPER3: Completing star-cycle-full × N=3-N=5 gives us a 3×2 design. The crossover + third topology = the core finding of the paper.