Local Entropic Horizon Corrections in Inhomogeneous Cosmology: Density-Dependent Barrow Entropy and a Structural No-Go Result for Minimal Two-Scale Averaging

This work develops a self-consistent thermodynamic-elastic framework for inhomogeneous cosmology, formulated locally on apparent horizons, and uses it to delimit the dynamical role of local entropic corrections. Spacetime is treated as an entropic medium whose vacuum response is governed by a density-dependent Barrow entropy index Delta (delta) ; the coupling Delta (delta) is adopted as a phenomenological ansatz rather than derived from an underlying theory. From this construction a localised modified Friedmann equation is obtained, and the differential expansion of cosmic voids is described as a localised entropic-elastic relaxation of the vacuum. Three results follow, each supported by numerical integration of a two-scale (void/wall) Buchert partition. First, the framework supplies a microscopic thermodynamic origin for the equation-of-state form of the cosmological constant: a static, emptiness-driven relaxation saturates in late-time voids and reproduces the dynamics of a localised Lambda; this addresses the w ~ -1 behaviour, not the observed magnitude. Second, a dissipative, rate-driven relaxation is genuinely distinct from Lambda and generates non-Lambda dynamical dark energy in the quadrant w0 > -1, wa < 0, the region currently explored by CPL fits to DESI DR2 combinations, although it remains sub-dominant and no statistical fit is claimed. Third, within minimal two-scale averaging the entropic-elastic horizon term, in every formulation tested, remains either Lambda-degenerate or dynamically sub-dominant to geometric backreaction, halting near w0 ~ -0. 37; a robustness scan over six qualitatively different functional forms of Delta (delta) returns an identical bound. An analytic argument traces this to the structure of two-scale averaging: the saturating contribution is asymptotically Lambda-degenerate (the void empties to a coupling-independent attractor), and any rate-driven contribution is exactly proportional to the Buchert backreaction QD via the exact identity (fᵥ dot) ² = (3/2) fᵥ fw QD. The no-go is therefore structural within this class rather than an artefact of the chosen ansatz. The analysis is conditional by construction: it assumes a coupling of the form Delta (delta) and asks whether, granting it, acceleration can follow. The result identifies continuous inhomogeneous metrics (LTB/Szekeres) as the necessary arena for any quantitative test of the entropic-elastic driver.