Representational Symmetry and Halo Structure: A Structural Constraint on the Density–Anisotropy Relation

Numerical simulations of collisionless dark matter halos reveal a persistent correlation between the local density slope and velocity anisotropy of particle orbits. Although widely reproduced, the physical origin of this density–anisotropy relation remains uncertain. Representational Symmetry and Halo Structure: A Structural Constraint on the Density–Anisotropy Relation This paper proposes a structural interpretation of the dark matter halo density–anisotropy relation within the Emergent Reality Architecture (ERA) framework. Numerical simulations consistently show that halos populate a restricted band in the parameter space defined by the logarithmic density slope (γ) and the velocity anisotropy parameter (β). Despite its robustness across cosmological simulations, the underlying mechanism producing this relation remains unclear. Within ERA, physical systems operate under finite representational capacity. Trajectories that require excessive configurational updating cannot be sustained indefinitely by entities with limited differentiation capacity. When applied to collisionless dark matter halos, this principle suggests that orbital families producing large gravitational gradient exposure may become structurally inaccessible. The paper introduces a measure of orbital complexity based on the time-averaged gravitational jerk exposure experienced along particle trajectories. Evaluating this quantity for particle orbits in spherical halos reveals that the complexity of a trajectory depends primarily on how deeply the orbit penetrates into the halo’s gravitational gradient field. Radial orbits that plunge toward the halo center experience rapidly increasing gradient exposure and therefore impose larger configurational demands. Extending this calculation to an ensemble of halo orbits shows that the aggregate orbital complexity depends on the distribution of low-angular-momentum trajectories within the system. Imposing a finite complexity threshold produces a structural accessibility boundary in the (γ, β) plane that links the halo density slope to the allowable orbital anisotropy. The resulting prediction defines a family of curves describing the maximum anisotropy accessible to halos with a given density slope. For parameter values consistent with cosmological simulations, the predicted slopes reproduce the sign and approximate magnitude of the empirically observed density–anisotropy relation. This interpretation suggests that the density–anisotropy relation reflects a structural constraint on accessible orbital configurations rather than solely the outcome of dynamical relaxation processes. In this view, halo structure emerges from the requirement that collisionless systems remain below a threshold of sustainable orbital complexity. The paper concludes by outlining observational and simulation-based tests of this mechanism, including the use of dwarf spheroidal galaxies as anchor systems for calibrating the complexity threshold and generating falsifiable predictions for halo anisotropy across different environments. Related work: This paper builds on the Emergent Reality Architecture (ERA) framework developed in Emergent Reality Architecture: Probability, Gravitation, and Physical Regimes (Zenodo). ERA proposes a layered structural interpretation of physical regimes in which quantum behavior, null propagation, and gravitation arise from representational constraints rather than independent dynamical mechanisms. The present work applies this framework to the phase-space structure of collisionless dark matter halos. A complementary interpretation of dark matter phenomenology is developed in Representational Limits and Dark Matter: Spatial Suppression and Halo Structure, which explores how structural restrictions on localization pathways may explain the extended halo configurations observed in cosmological structure formation. Additional conceptual clarifications within general relativity appear in Orbits as Inertial States: Rethinking Rest in General Relativity and Geodesics as Regimes of Persistence. Together these works examine how persistence, ordering, and metric structure emerge across different physical regimes. Revision notes (Version 7.1): Expanded Particle-Verified Validation (Section 8.6): Transitioned from a single-halo analysis of Aq-A-2 to a comprehensive evaluation of all five Aquarius level-2 halos. Processing 170–192 million particles per halo allows for direct dark matter measurements without relying on stellar tracer assumptions. Established Geometric Invariance Across Mass Scales: Included a summary table (Table 1) demonstrating that the geometric component ln λ remains constrained to the range (0.103 to $0.127) across systems spanning four orders of magnitude in halo mass. Explicit Falsifiability Conditions (Section 2): Added three specific conditions that would rule out the framework: if χMD varies systematically with mass/history, if the geometric and anisotropy decomposition fails to separate cleanly, or if the shell-bracketing condition rmin < rs < rmax is violated. Identification of Systematic Bias in Literature: Noted that published profile figures often underestimate velocity anisotropy βs at the scale radius compared to raw particle data; correcting this bias is essential for accurate pipeline calibration and explains why published-figure ln λ estimates are systematically higher than particle-verified values. Robustness against Morphology and History: Confirmed that ln λ remains within the predicted band even for halos with recent major mergers (Aq-C-2) or elevated asphericity (Aq-D-2), proving the geometric component is insensitive to formation history. Integrated Curvature Tensor Derivation (Appendix B): Added a formal derivation showing that shell width is encoded in the local quadratic geometry of the NFW potential at the γ = 2 transition. Achieved below 0.45% agreement between this geometric prediction and exact pipeline results. Refined Accessibility Boundary Framing: Clarified that the ERA boundary defines a structural restriction within the dynamically admissible region allowed by the Jeans equations, rather than acting as a replacement for Jeans equilibrium. Confirmed Shell-Bracketing: Explicitly verified the condition rmin < rs < rmax across all five Aquarius halos, supporting the prediction that the scale radius sits within the accessible orbit family. Independence Test (Aq-E-2): Demonstrated that the geometric component ln λ is independent of orbital anisotropy state. Aq-E-2, with β_s = 0.072—the lowest in the suite—produces ln λ = 0.107, within 0.004 of the Draco value despite an anisotropy term four times smaller. This confirms that ln λ reflects NFW potential geometry rather than orbital state. Correspondence: Peter Nowicki — peternowicki@proton.me