Induced Semiclassical Gravity on a Timelike Braneworld Interface: Reconciling the Page–Geilker Anomaly with Gravitational Entanglement Witnesses

Abstract We recover four-dimensional General Relativity as the induced boundary dynamics of a timelike hypersurface Sigma in a five-dimensional Einstein bulk--without postulating an intrinsic four-dimensional Einstein-Hilbert action--and demonstrate that this geometry reconciles the Page-Geilker macroscopic sourcing anomaly with table-top gravitational entanglement witnesses. Varying the bulk action alongside a two-sided Gibbons-Hawking-York boundary functional cancels normal metric derivatives across Sigma, yielding the Lanczos-Israel junction condition and, under Z_2 reflection, the Shiromizu-Maeda-Sasaki field equations with emergent gravitational coupling 8 pi G_N / c^4 = kappa_5^4 lambda / 6. Because the normal bundle has structure group O(1) rather than U(1) and G_mu n vanishes on Sigma (Theorem 1), Maxwell electrodynamics is formulated natively on Sigma with its traceless Hilbert stress confined strictly to the tangential shear channel of tau_mu nu. Under cutoff holography (Z_bulk == Z_Sigma) and a pure joint state on H_Sigma tensor H_bulk, Schmidt spectral coincidence pairs boundary von Neumann entropy with bulk conditional negentropy (S_4D + S_5D = 0). Within the controlled regime of large central charge (c_eff >> 1), quadratic bulk fluctuations (||delta tau_mu nu|| << lambda), and small intra-bin pointer variance (epsilon_A / P_i^2 << 1), Theorem 2 derives from the real-time bulk-boundary path integral and the linearized Lanczos-Israel condition that the unconditional expectation sources the junction only while a transaction remains open (Phi < 0, P_SA = 0). Once apparatus work and visco-plastic dissipation book the Landauer threshold (Delta S_5D = -Delta S_4D < 0) at the constitutive interface temperature of Postulate 4, off-diagonal cross terms are suppressed by exp(Delta S_5D / 2 k_B) and each settled pointer bin selects an independent classical boundary-value problem sourced by _bin i in its future domain of dependence, resolving the Page-Geilker constraint without superposing bulk metrics. Conversely, inside the open window (Phi < 0, P_SA = 0), the static linearized boundary equation acts as an oscillator-free ADM lapse constraint (nabla^2 hat Phi_N = 4 pi G_N hat rho), predicting a Bose-Marletto-Vedral entanglement concurrence C in [0.081, 0.109] alongside preserved single-mass visibility V = 1 rather than Diosi washout across 14 ms-1.58 s. Keywords: braneworld gravity; Lanczos-Israel junction; Shiromizu-Maeda-Sasaki projection; Maxwell shear confinement; bipartite conditional entropy; Page-Geilker experiment; Bose-Marletto-Vedral protocol; constraint slaving; Diosi discriminator.