From the Casimir Effect to Space-Entropy: A New Experimental Frontier for Emergent Gravity

We develop an experiment-first, QFT/entanglement–informed programme to test whether the Casimir interaction can be operationally reconstructed in an entropic form. The key technical step is to identify a finite, geometry-dependent information functional I(d)I(d)I(d) (related to mutual information) and combine it with the modular/KMS scale of the plate separation, yielding an entropic-force relation F(d)=kBTeff(d) ∂dI(d)F(d)=k_B T_{\rm eff}(d)\,\partial_d I(d)F(d)=kBTeff(d)∂dI(d). In the ideal EM parallel-plate limit, a heat-kernel/zeta matching fixes the universal coefficient and reproduces the standard Casimir pressure without introducing an ad-hoc distance ansatz. On the experimental side, we propose a differential/null protocol comparing smooth and engineered-rough (or fractal) coatings on the same chip, designed to suppress common-mode systematics (calibration drift, patch potentials, residual electrostatics) while making geometry-controlled deviations measurable. Two operational observables are emphasized: (i) the local exponent flow αeff(d)\alpha_{\rm eff}(d)αeff(d) extracted from differential force curves, and (ii) a characteristic memory time τS\tau_SτS probed via multi-cycle (history-dependent) measurement sequences. The goal is not to replace Lifshitz/Casimir theory, but to isolate falsifiable signatures that would distinguish purely electromagnetic/material explanations from an entanglement/entropy-based effective description. This record contains the full manuscript and appendices, including uncertainty budgeting and reproducible numerical estimates for realistic separations around d∼10−7d\sim10^{-7}d∼10−7–10−6 m10^{-6}\,\mathrm m10−6m.