Thermodynamics of Cooperation: Value Dynamics

Self-maintaining agents that share finite resources must couple to high-energy centers (accumulated-negentropy nodes) for energy inflow, yet tight coupling imposes autonomy costs that can erode the agent's boundary integrity. We model this trade-off as a one-dimensional potential on the coupling distance between an agent and a center, combining a short-range-dominant dissolution cost ($1/r^2$) with a long-range resource inflow ($1/r$) and a constant entropy maintenance baseline. The resulting coexistence potential possesses a unique global minimum, the coexistence attractor, whose global asymptotic stability we prove via Lyapunov analysis. The zero-crossings of the potential define the stable coexistence band, a bounded interval of coupling distances within which the agent maintains positive net energy. We derive a closed-form expression for the bandwidth of this band (the freedom bandwidth) and prove that it increases with center mass and decreases with agent complexity, dissolution pressure, and entropy leakage. Outside the band, agents undergo irreversible transitions: dissolution (boundary erosion from excessive coupling) or starvation (energy deficit from insufficient coupling). We extend the single-center model to $K$ centers, proving that the multi-center potential separates across coupling dimensions, yielding a componentwise global attractor with a diversification benefit: agents that would be non-viable at any single center may survive through distributed coupling.