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

The Casimir effect is the most controlled laboratory interface betweenquantum vacuum fluctuations, geometry and boundary conditions. We arguethat it can also serve as the first experimental gateway intoEntropy-Space: a programme in which forces emerge from informationgradients and from history-dependent entropic response of the vacuum.The Lifshitz formula is adopted as the conservative null model. We first show that the ideal Casimir pressure can be reconstructed froma finite mutual-information functional I(d) = gamma_0 A/d^2 and amodular energy scale Theta_mod(d) = hbar c/(2 pi d), withgamma_0 = pi^3/240 spectrally fixed by TE+TM mode counting via thereplica/heat-kernel route. The decisive beyond-Lifshitz signature is dynamical: a multi-cycle loopwork W_loop ~ d^(-5) (sphere-plate W_loop ~ R/d^4 via PFA) with aLorentzian frequency response controlled by an entropic relaxationtime tau_S, structurally absent from any conservative single-cycleLifshitz description. The predicted discriminator W(100 nm)/W(300 nm)= 3^5 = 243 separates the entropic mechanism from mechanical, thermaland electrostatic hysteresis sources. A laboratory measurement of tau_S would, via a coarse-graining ratioR = tau_h/tau_S, predict the corresponding macroscopic memory scaletau_h for a given physical regime. For cluster-merger physics(tau_h ~ few Myr), R ~ 10^25 links the laboratory and the clusterscale across twenty-five orders of magnitude in time. The full technical apparatus is provided as Supplementary Material atDOI:10.5281/zenodo.14971614.