ABSTRACT We propose a phenomenological cosmological framework, denoted PXT UP, in which the observable universe is modeled as a thermodynamically open system interacting with an external background environment. The interaction is encoded through a scalar response function g(T, x) depending on the trace of the energy--momentum tensor, leading to effective modifications of gravitational dynamics across different density regimes. At low densities, the framework reproduces accelerated cosmic expansion without invoking a strictly constant cosmological term. At intermediate (galactic) densities, the response function generates effective gravitational enhancements that can mimic dark matter phenomenology without introducing new particles. At extreme densities, the model allows for non-singular black hole interiors characterized by energy exchange with the background. We present the theoretical structure of the model, derive the modified field equations, and illustrate its phenomenological consequences through simplified numerical examples. The framework does not claim a complete replacement of The Standard Cosmological Model (Lambda-CDM), but provides a unified perspective for exploring dark sector phenomena, cosmic tensions, and black hole physics within an open-system interpretation of gravity. Observable implications and potential falsifiability are briefly discussed. 1. INTRODUCTION Modern physics is fragmented between Quantum Mechanics and General Relativity. Ad-hoc solutions like WIMPs or scalar fields remain undetected. PXT UP approaches from the Principle of Nested Reality, assuming the 4D universe is a brane within a high-entropy Background. Interactions between the "Interior" and "Exterior" manifest as density-dependent gravitational modifications. 2. RELATED WORK AND CONCEPTUAL CONTEXT The PXT UP framework is related to several existing approaches in modified gravity and cosmology, while differing in both interpretation and implementation. Models based on modified gravitational actions, such as $f(R)$ and $f(R,T)$ gravity, introduce explicit dependence on curvature scalars or the trace of the energy--momentum tensor to account for late-time acceleration and structure formation. While PXT UP also employs a $T$-dependent interaction, it interprets this dependence as an effective response arising from an open-system interaction rather than as a fundamental modification of the gravitational Lagrangian. Energy exchange cosmologies and interacting dark energy models allow for non-conservation of the energy--momentum tensor at the effective level. In contrast, PXT UP interprets local non-conservation as a signature of energy flux between the observable universe and an external background, rather than interactions between dark sector components. Emergent and entropic gravity proposals similarly seek to explain dark matter phenomenology without new particles. PXT UP differs by introducing a continuous density-dependent response function that interpolates between cosmological, galactic, and compact-object regimes within a single formal structure. Finally, several approaches to non-singular black holes replace classical singularities with effective cores or modified causal structures. In PXT UP, high-density objects are interpreted phenomenologically as regions where energy exchange with the background becomes dominant, providing an alternative perspective on singularity resolution. These connections place PXT UP within the broader landscape of phenomenological extensions of general relativity, while maintaining a distinct open-system interpretation. 3. THEORETICAL FRAMEWORK 3.1. Action and Field EquationsModified Einstein-Hilbert Action:S = ∫ d⁴x √(-g) [ R / (16πG) + L_m + g(T, x) ] Generalized Field Equations:G_μν = 8πG ( T_μν + T_μν_eff ) Where T_μν_eff is the effective stress-energy tensor arising from background interaction. 3.2. Local Non-ConservationA key consequence is the local violation of energy conservation (signifying an open system):∇^μ T_μν = Q_ν Q_ν represents the flux: Q_ν > 0 (Source - Expansion) and Q_ν T_crit). Hubble Constant Single static parameter (Leading to Tension). Evolving H(z). Coupled to mean cosmic density. The g(T) function evolves as the universe expands and dilutes. Table 2: Comparison of Free Parameters Parameter Lambda-CDM PXT UP Physical Interpretation in PXT UP Baryon Density Omega_b Omega_b Standard baryonic matter (Identical). Dark Sector Omega_c (Cold Dark Matter) rho_0, alpha rho_0: Galactic density threshold. alpha: Screening slope. Expansion Driver Omega_Lambda Lambda_eff Effective osmotic pressure from the Background. Extreme Physics (None / Singular) T_crit, beta T_crit: Puncture threshold. beta: Leakage rate. Total Parameters 6 6 PXT UP explains more phenomena with equivalent complexity. II. TECHNICAL APPENDICES APPENDIX A: MATHEMATICAL FOUNDATION OF THE g(T) ANSATZ The proposed interaction Lagrangian g(T) is not arbitrary but is derived from the Principle of Saturated Response. We model the observable universe as a brane interacting with a bulk reservoir. The response function must satisfy three phenomenological conditions: Vacuum Limit (T -> 0): The membrane is relaxed; external pressure maximizes.Limit: g(T) -> -2 * Lambda_eff (Accelerated Expansion). Galactic Limit (T ~ rho_0): The membrane stiffens; the derivative g'(T) is non-zero.Effect: Generation of Virtual Gravity (Dark Matter mimicry). Critical Limit (T > T_crit): The membrane ruptures.Effect: Formation of an energy sink (Black Hole leakage). The Unified Master Equation: g(T) = [ -2 * Lambda_eff / (1 + (T / rho_0)^alpha) ] - [ beta * Theta(T - T_crit) * (T / T_crit - 1)^2 ] Where: T is the trace of the energy-momentum tensor. Theta is the Heaviside step function. alpha controls the steepness of the galactic screening. beta controls the magnitude of the black hole energy flux. APPENDIX B: VARIATIONAL PRINCIPLE AND FIELD EQUATIONS We start with the modified Einstein-Hilbert Action S: S = ∫ d⁴x √(-g) [ R / (16πG) + L_m + g(T) ] To derive the field equations, we perform the variation with respect to the inverse metric g^μν. The principle of least action (δS = 0) yields: δS = ∫ d⁴x √(-g) [ G_μν - 8πG (T_μν + T_μν_eff) ] δg^μν = 0 Derivation of the Effective Tensor:The variation of the interaction term g(T) requires the chain rule, noting that T = g^μν T_μν.The resulting Modified Einstein Field Equation is: G_μν = 8πG ( T_μν + T_μν_eff ) Where the effective stress-energy tensor T_μν_eff is explicitly defined as: T_μν_eff = (2 * g'(T)) * T_μν + [ 2p * g'(T) - g(T) ] * g_μν Physical Interpretation: The term 2 * g'(T) modifies the effective coupling constant G_eff. When g'(T) > 0, gravity is enhanced (Dark Matter regime). When g(T) becomes highly negative (Black Hole regime), it acts as a sink term. APPENDIX C: PYTHON SOURCE CODE FOR NUMERICAL VERIFICATION The following Python code was used to generate the phenomenological plots (Figures 1-3) presented in the main text. It utilizes standard libraries (numpy, matplotlib) to simulate the behavior of the g(T) function across cosmological and local scales.