From Einstein’s Cosmological Constant Λ to Entropic Geometry: A Dynamic Framework via the Haddad–Λ Tensor Ξ_{μν}(ϕ,R_{μν})

We present a novel theoretical framework in which the cosmological constant \(\Lambda\) is promoted from a fixed scalar to a _dynamical thermodynamic variable_ emerging from the quantum microstructure of spacetime. By extending the Einstein–Hilbert action through a non-minimal coupling between spacetime curvature and vacuum entropy density \(s_{\text{vac}}(x)\), we derive modified field equations of the form: \(R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} + \Lambda(x) g_{\mu\nu} + \Xi_{\mu\nu} = 8\pi G T_{\mu\nu},\) where the _local cosmological function_ is given by: \(\Lambda(x) = \Lambda_0 + \gamma \, \nabla_\alpha S^{\alpha}(x),\) with \(S^\alpha(x)\) denoting the entropy current associated with vacuum fluctuations and \(\gamma \in \mathbb{R}^+\) a coupling parameter. The correction tensor \(\Xi_{\mu\nu}\) incorporates the scalar field \(\phi(x)\) responsible for modulating vacuum entropy via: \(\Xi_{\mu\nu} = \alpha \left( \nabla_\mu \phi \nabla_\nu \phi - \frac{1}{2} g_{\mu\nu} \nabla_\alpha \phi \nabla^\alpha \phi \right) + \beta \, \phi^2 R_{\mu\nu}.\) The resulting field equations yield a _self-regulating cosmological dynamics_, in which the evolution of \(\Lambda(x)\) is governed by: \(\Box \phi + V'(\phi) = \delta \left( \frac{\partial s_{\text{vac}}}{\partial \phi} \right),\) establishing a direct link between quantum entropy production and large-scale acceleration. This framework unifies the early inflationary phase and the current dark energy epoch under a common dynamical mechanism, and addresses the longstanding discrepancy between quantum vacuum energy predictions and observed \(\Lambda_{\text{obs}} \sim 10^{-122} M_{\text{Pl}}^4\). We conclude by deriving testable predictions for next-generation cosmological probes such as _CMB-S4_, _Euclid_, and _JWST_, and show how deviations from \(\Lambda\)CDM at \(z > 1.5\) may serve as critical empirical signatures. This approach also paves the way for embedding vacuum thermodynamics within a topologically-constrained quantum gravity framework.