Cartel Yield vs Friction Cost
Beyond K_crit (amber), friction exceeds max yield. Cartel bleeds capital per cycle.
Dark = tight cartel cap (secure) · Red = high K_crit (vulnerable)
Dark = low max share (secure) · Red = high cartel ceiling
Reading the Phase Maps
Safe region (dark blue): K_critical is low relative to retail. Cartel cannot accumulate enough before friction bleeds them. System self-regulates.
Transition zone (teal-amber): K_critical is moderate. Temporary dominance possible but friction pressure increases. Sensitive to parameter shifts.
Danger zone (red): K_critical very high or infinite. Friction too low relative to yield. Increase f or decrease I_floor.
Key: The safe region expands when f·λ is large (high friction + fast decay). It contracts when I_floor/K_R is large.
Theorem Status
✓ sup(S_C) < 1 — Confirmed. K_R·M_R > 0 ensures strict bound.
✓ K_critical exists — Confirmed when f > 0. Friction grows linearly; yield is bounded.
✓ Self-defeating attack — Confirmed. Beyond K_crit, cartel bleeds nominal capital.
⚠ Requires f > 0 — If identity reset is free, bound collapses. Must enforce on-chain friction.
⚠ Requires K_R > 0 — If retail exits entirely, S_C → 1. Need participation incentive.
Corrected emission: I(t) = max(I₀e⁻ᵏᵗ · 1/(1+γ|Gₜ|) · Hₜ, I_floor) | Rᵢ ∝ stakeᵢ · log(1+depthᵢ) · e⁻λ·ageᵢ | K_crit = f⁻¹ · S_C · I_floor · τ*