EXP-PENT

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Pentagon Scaling — Odd/Even Cycle Hypothesis at N=5

2026-03-09 L3 level paper3 1 runs

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In plain language

Two agents: flat. Three: hierarchy. Four: flat again. Is it the odd number that matters? We test five agents — another odd number — to see if the pecking order returns. This time, no synthesizer roles that might blur the lines.

Predictions we made before running

3/4 confirmed · 1 refuted

▶ Technical details

Research hypothesis

EXP-DYAD (N=2): flat (TR range 0.060). EXP-ASYM (N=3): hierarchy (TR range 0.444). EXP-QUAD (N=4): flat again (TR range 0.059). Non-monotonic pattern.

Two competing explanations: H1 (ODD/EVEN CYCLE): 3-cycle (odd) breaks symmetry → hierarchy. 4-cycle (even) restores symmetry → flat. 5-cycle (odd) should RESTORE hierarchy. H2 (N=3 UNIQUE): triad is Goldilocks — enough for asymmetry, too few for averaging. Any N>3 homogenizes regardless of parity. N=5 should remain flat.

CONFOUND FROM EXP-QUAD: Delta (narrative/synthesis) was an INTEGRATIVE persona that homogenized flow. This experiment uses only NON-integrative personas (no synthesis, no narrative weaving) to disentangle persona choice from topology.

If N=5 shows hierarchy (TR range > 0.15): H1 supported (odd/even mechanism). If N=5 is flat (TR range < 0.10): H2 supported OR density effect dominates parity.

Experimental setup

Type: simple

Condition	Parameters
PENT_LIVE	persona: PERSONA, interaction: LIVE, n_agents: 5

Factors: n_agents (5)

Parameters

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

Trophic Ratios by Condition

Mean trophic ratio per agent across runs. Error bars = ±1 std dev. Higher TR = more upstream (exporter).