The Cosmic Web as a Generalized Murray Network: Transport-Optimal Branching Under Gravitational–Dark Energy Coupled Mass Flow

We propose that the cosmic web — the large-scale hierarchical structure of the Universe comprising voids, walls, filaments, and galaxy cluster nodes — is not a random fractal but a transport-optimal network obeying a generalized Murray's Law. Matter flows from underdense voids through walls into filaments and ultimately into cluster nodes, a parent-to-child transport hierarchy analogous to biological vascular networks, river deltas, and electrochemical electrode systems. We derive an effective gravitational transport exponent α_gravity(z) = 2 + δα × [Ω_Λ(z) / (Ω_Λ(z) + f_c × Ω_m(z))], interpolating between the pure gravitational infall limit (α = 2, analogous to ionic migration) and a dark-energy-modified regime (α ≈ 2.5–2.6) that includes the volume-priced upkeep cost of maintaining filament architecture against Hubble expansion. For the observed filament branching number κ = 2.5–4.0, the Murray-optimal parent-to-child filament diameter ratio ranges from 1.43 to 1.74 at z = 0. The framework predicts that existing simulation data (IllustrisTNG, EAGLE) are consistent with this result when the transport-active core diameter — defined by the velocity-weighted infall region — is distinguished from the total overdensity width. The critical observational test (Prediction P2) requires extraction of velocity-weighted filament diameters from existing simulation velocity fields, executable without new observational campaigns. A phenomenological connection between α_gravity and the dark energy equation of state w_eff = −1 + (2/3) × ((α − 2)/α) yields predictions in the range −0.87 to −0.78, encompassing DESI 2025 observational constraints. Five falsifiable predictions are defined using existing Euclid, SDSS, DESI, and simulation data. All numerical values have been independently verified by computation prior to submission. This is the twenty-third preprint published through the Presignal Research Initiative and the third in a series applying generalized Murray's Law across physical scales — from Nernst-Planck electrochemical transport in battery electrodes (DOI: 10.5281/zenodo.20270891) to gravitational-dark energy coupled mass flow in the cosmic web.