Hierarchical and Democratic Regimes in Multi-Planet Architectures Revealed by Action-Space Entropy

We introduce an action-based entropy formalism to quantify the structural architecture of multi-planet systems, applying it to a mass-complete sample of 529 systems constructed from the NASA Exoplanet Archive. By defining orbital action as \(J_i \propto m_i \sqrt{a_i(1-e_i^2)}\) and computing normalized Shannon entropy to measure the distribution of angular momentum among planetary bodies, we test the hypothesis of secular thermodynamic evolution by examining the entropy-age relation. Both equal-mass (1440 systems; ρ = −0.048, p = 0.0677) and mass-weighted analyses (529 systems; ρ = −0.022, p = 0.613) show no statistically significant correlation, indicating that planetary systems do not evolve toward angular momentum equipartition over stellar lifetimes. Instead, entropy exhibits a strong dependence on internal structural parameters, with a significant anti-correlation observed regarding mass dominance (ρ = −0.489, p = 3.34 × 10 −33 ) and an independent influence from multiplicity after controlling for mass hierarchy (ρ = −0.624, p = 1.85 × 10 −58 ). A nonlinear regression model incorporating multiplicity and mass concentration explains 76.5% of entropy variance (R 2 = 0.765), revealing a continuous structural manifold in (N, f max) space where approximately 70.7% of systems are dynamically democratic and 11.7% are hierarchical. Transit-detected systems preferentially occupy the democratic regime, whereas radial velocity systems extend toward hierarchical configurations, demonstrating that planetary architecture is formation-imprinted rather than thermodynamically evolving. Consequently, action-space entropy provides a physically grounded structural order parameter for exoplanet system classification and comparative formation studies.