Galaxy Rotation Curves and Dark Energy: A Gravitational Entropy Perspective ------The GEE Framework: Temperature Conversion Drives Hubble Parameter

Abstract This paper proposes a phenomenological framework (the GEE framework) based on gravitational dissipative structures and the holographic principle, in which the observational effects conventionally attributed to "dark matter" and "dark energy" in the standard cosmological model are reinterpreted as effective projections of gravitational degrees of freedom during coarse-graining. The central physical picture of this paper is the isomorphism of Prigogine dissipative structures at two scales: at the thermodynamic scale, the most ubiquitous Prigogine dissipative structure is the cyclone; at the gravitational scale, the most ubiquitous manifestation is the galactic rotation, structurally isomorphic to the cyclone. Terrestrial typhoons, Jupiter's Great Red Spot, Martian global dust storms, and Venusian polar vortices are all quasi-steady ordered structures formed by open systems maintained by a continuous energy flow, far from equilibrium, and self-organized through nonlinear positive feedback. Galaxies, galaxy clusters, and accretion disks are quasi-steady rotational structures formed by self-gravitating systems during virialization. In the Prigogine framework, both share the same abstract structure: open system, far from equilibrium, nonlinear positive feedback, continuous entropy production, and self-organized ordered structure. The physical identity of the GEE framework is precisely a Prigogine dissipative-structure theory in a long-range-force system. Yet structural isomorphism is not physical identity. From cyclone to galactic rotation there are three hard fractures: first, opposite signs of heat capacity -- the cyclone is a positive-heat-capacity system, the galactic rotation is a negative-heat-capacity system; second, different energy currencies -- the cyclone corresponds to the thermal energy of the canonical ensemble, the galactic rotation to the gravitational potential energy of the microcanonical ensemble; third, a time-scale gap of about 20 orders of magnitude -- cyclone lifetimes are days to centuries, galactic-rotation lifetimes are billions of years. Of the three fractures, only the second can be softened by a temperature-equivalent reduction. This paper defines Here is the local gravitational temperature in the microcanonical-ensemble sense; when the self-gravitating system reaches local thermal equilibrium it is approximately equal to the microcanonical statistical temperature ; in virial approximation it equals the dynamical temperature is the ratio of two temperature scales, used to compare local gravitational potential depth with thermal kinetic energy density; it does not represent a phase transition between two thermodynamic temperatures. has negative heat capacity and does not possess the stable thermodynamic limit of the canonical ensemble; is the canonical-ensemble thermodynamic temperature. The first and third remain hard fractures: negative heat capacity solves the mechanical stability problem of the galactic rotation, but not the energy-source problem; the galactic rotation still requires external energy flows , to maintain the far-from-equilibrium state, so that entropy production persists. The cyclone and the galactic rotation are structurally isomorphic in mathematical form but opposite in the sign of heat capacity: after the cyclone's energy flow is cut, drops, rises, and the structure collapses; after the galactic rotation's energy flow is cut, rises, rises, but the structure is maintained by negative-heat-capacity self-regulation. They are two ends of the dissipative-structure spectrum, not the same mechanism; this paper calls this "structural isomorphism, opposite mechanisms." As a Prigogine dissipative structure, the galactic rotation also requires a continuous external energy flow, and the input energy required to maintain the dissipative structure must be at least several times the dissipation rate; in the late epoch the main energy source is the cosmic expansion energy . Negative heat capacity solves only the mechanical stability problem, not the energy-source problem. Convention: Throughout this paper we adopt natural units unless otherwise stated. GEE assumes zero particle mass for the dissipative-structure gravitational gradient: The framework introduces a dimensionless effective gravitational entropy order parameter and a dissipative-structure order parameter , and fuses gravitational relaxation with the holographic principle: gravitational relaxation provides the dynamical background for the formation of dissipative structures, while the holographic principle provides the energy-density formulas for the dissipative-structure gravitational gradient and for the effective energy density driving cosmic accelerated expansion. The two meet through the gravitational entropy . serves as a specified proportionality relation, stipulating that the energy released by gravitational relaxation is proportional to the system mass, but it does not directly enter the fusion equation. Hubble parameter evolution: Planck's 67.4 is a model-dependent inference of today's , not the actual expansion rate at recombination. Therefore this paper does not claim that GEE has solved the Hubble tension. In GEE, is a shared fit parameter; is obtained by solving the full background equations together with the GEE components; the temperature-conversion order parameter is only a phenomenological function controlling delayed interaction; it is not directly identified with . Core statement: GEE is currently a phenomenological dissolution framework. It provides a parameter table, a fitting pipeline, and schemes for MCMC and Bayes-factor calculations. Before the full background solution and MCMC are completed, it cannot claim to have explained flat rotation curves, to have solved the Hubble tension, or to have obtained support from three independent observational datasets. Prigogine virial equation: The standard virial theorem assumes a closed, dissipationless system, giving In the GEE framework, the comparison between the local gravitational temperature and the local thermodynamic temperature introduces a Prigogine dissipative term , and the virial equation is generalized to where The self-consistent equation gives , . This is an algebraic domain lower bound, not an independent phase-transition criterion. Formation of a dissipative structure requires separately and . is the ratio of two temperature scales, used to judge the relative size of local gravitational potential depth and thermal kinetic energy density. is the local gravitational temperature in the microcanonical sense and does not have canonical-ensemble heat capacity; is the canonical-ensemble thermodynamic temperature. When , ; when , , and the dissipative term provides additional effective gravity. The Prigogine virial equation predicts that the standard virial theorem may overestimate mass; whether three observational datasets support it must be rechecked. Holographic dissolution of the dissipative-structure gravitational gradient: GEE does not use particle components. The observational effects conventionally attributed to "dark matter" are reinterpreted in GEE as the effective projection of the galactic dissipative-structure gravitational gradient under a holographic cutoff. We adopt a holographic model with the Granda-Oliveros (Ricci) cutoff. This cutoff is introduced as a phenomenological assumption; its first-principles derivation from the first law of thermodynamics on the horizon is left to Sec. 15.1. In a universe containing only baryons and radiation, the background interaction is In the background continuity equations, flows from the expansion energy density to the dissipative-structure gravitational gradient: The effective power-law index is defined as a derived quantity: During the dissipative-structure-dominated era, if is approximately constant, its average is denoted . From the continuity equation one obtains hence Since , , algebraically If early is growing, i.e. , then Thus there is no epoch with . is a derived quantity, and The present density ratio only constrains normalization and cannot by itself determine . The dimension of the interaction term is Phenomenological construction of the k-essence Lagrangian: The k-essence Lagrangian is constructed phenomenologically from the holographic energy density and the k-essence definition ( is a phenomenological assumption): During the dissipative-structure gravitational-gradient-dominated era, taking , the Lagrangian is approximately where has dimensions of energy density. The exponent is determined by the effective sound speed: A typical value is , corresponding to an effective sound speed corresponds to extreme fine-tuning; its naturalness is left to Sec. 15.1. The relation between and has been computed from the k-essence equation of motion given the form of : During the dissipative-structure gravitational-gradient-dominated era, , . Without interaction For large , this exponent is approximately , i.e. the non-interacting case gives , which does not match the required slope ; if is introduced, the background evolution must be re-solved. is a consequence of the phenomenological assumption; its applicability requires numerical verification. Dissolution of the effective energy density driving cosmic accelerated expansion: GEE does not use material components. The observational effects conventionally attributed to "dark energy" are reinterpreted in GEE as the effective energy density driving cosmic accelerated expansion. It is generalized to a two-layer structure consisting of a holographic component and an early peak (EDE) component, coupled to the dissipative-structure gravitational gradient through an interaction. The early peak fraction is dominated by Jeans-mass halos at . Cross-validation using both the Press-Schechter and Sheth-Tormen mass functions yields