Non-Equipartition as a Signature of Structural Memory in Self-Gravitating Systems

Self-gravitating, collisionless systems such as dark-matter halos, stellar halos, and satellite galaxies consistently exhibit long-lived violations of equipartition. Standard ΛCDM relaxation theory attributes these deviations to extremely long relaxation times, predicting slow asymptotic decay toward the partial-equipartition curves derived from the Fokker–Planck equation. This work introduces an alternative explanation, the Structural Memory Framework (SMF), based on a history-dependent constraint. A dimensionless structural information budget I0 is defined, computed directly from a halo’s merger tree, together with a corresponding structural information functional I[ f ] defined as the Kullback–Leibler divergence of a phase-space distribution relative to the Markovian baseline fM. The stationary state is obtained by maximising the entropy S[ f ] subject to conservation of energy, particle number, and the constraint I[ f ] = I0. This produces a history-tilted Boltzmann distribution fstat ∝ f α M exp[−βeff H], in which the deformation exponent α is set by I0. The Structural Memory Framework predicts a qualitatively distinct evolution of the equipartition deficit ∆eq. Whereas ΛCDM relaxation theory requires monotonic decay d∆eq/dt < 0, SMF predicts saturation d∆eq/dt → 0 to a non-zero floor determined by I0. Three binary falsification tests are formalised, including a corrected “Twins Test” using the History–Kinematic Correlation Ratio RIK = Corr(∆eq, I0) /max[Corr(∆eq, ΛCDM proxies)] , and outer stellar halos and tidally processed satellites are identified as the primary observational targets. The framework provides a mathematically rigorous and observationally testable alternative to the Markovian assumption underlying ΛCDM relaxation theory, and establishes the structural parameter α as a unifying quantity linking non-equipartition, the S8 suppression, and rotation-curve stabilisation within the broader CIOU paradigm.