Abstract

Abstract

We study, within the positive-tension, two-sided, $Z_2$-symmetric Randall–Sundrum model (RS2), the precise sense in which the instantaneous state of a brane observer fails to determine its own future, and the precise sense in which that failure can and cannot be detected. Selecting RS2 is an additional physical postulate; nothing below derives it. For a brane of tension $\lambda>0$ with five-dimensional coupling $\kappa_5^2$, bulk cosmological constant $\Lambda_5$, induced metric $q_{\mu\nu}$, and brane stress tensor $T_{\mu\nu}$, the Israel condition and the Gauss–Codazzi identity give the exact projected equation $G_{\mu\nu}[q]+\Lambda_4q_{\mu\nu}=\kappa_4^2T_{\mu\nu} +\mathcal C^{\mathrm{RS}}_{\mu\nu}$ with $\mathcal C^{\mathrm{RS}}_{\mu\nu}=\kappa_5^4\Pi_{\mu\nu}-E_{\mu\nu}$, where $\Pi_{\mu\nu}$ is quadratic in $T_{\mu\nu}$ and $E_{\mu\nu}$ is the electric projection of the bulk Weyl tensor. We construct an explicit gauge-inequivalent pair of bulk data with identical brane readout and different brane futures, proving that the source evolution admits no deterministic quotient flow on the instantaneous brane Cauchy state over the full RS2 source domain. We then evaluate the projection exactly in the homogeneous Friedmann sector, obtaining a positive $\rho^2$ term and a bulk-state-dependent dark-radiation term $\mu/a^4$. In the linear weak-field sector the Gaussian/Schur-complement elimination of the bulk is exact at linear tree level; evaluating the resulting response asymptotically for $r\gg\ell$ , a controlled expansion, not an exact closed form at arbitrary distance , gives the long-distance potentials $\Phi=-\tfrac{G_4M}{r}(1+\tfrac{2\ell^2}{3r^2}+\cdots)$ and $\Psi=-\tfrac{G_4M}{r}(1+\tfrac{\ell^2}{3r^2}+\cdots)$. Eliminating the common AdS radius $\ell$ between the two regimes yields the parameter-free cross-regime relation $A_\Psi=\tfrac12A_\Phi$, $B=\tfrac{\kappa_4^4}{24}A_\Phi =\tfrac{\kappa_4^4}{12}A_\Psi$, where $H^2\supset B\rho^2$ and $A_\Phi,A_\Psi$ are the leading $1/r^3$ potential coefficients. Violation falsifies minimal RS2; agreement at a nonzero value is a parameter-saving cross-regime success but does not falsify a four-dimensional theory that treats the three coefficients as independent. Finally, an elementary pushforward lemma fixes the exact interpretive boundary: for every brane-only protocol, including adaptive ones, the induced Tier-1 history law reproduces the RS2 record distribution exactly, and in RS2 this obstruction is realized concretely by the four-dimensional gravity-plus-CFT description, which reproduces the same $1/r^3$ residue. The result is therefore positive at the level of model-relative projection and nonclosure, and negative at the level of unrestricted ontological discrimination.