Bulk-to-Brane Projection, Dynamical Nonclosure, and Observable Residues in Randall–Sundrum Gravity

Paper VII provides the physical realization layer of the seven-paper Shadow Theory sequence through the positive-tension, two-sided Randall–Sundrum II model.

It constructs an explicit gauge-inequivalent bulk gravitational-wave pair with the same instantaneous brane readout but different later brane behaviour, establishing that the full RS2 source evolution does not descend to a deterministic law on the instantaneous brane state. The paper derives the exact projected brane Einstein equation, evaluates its homogeneous cosmological correction and linear weak-field Kaluza–Klein response, and obtains a parameter-free relation linking the high-density cosmological correction to the leading long-distance gravitational-potential residues.

It also constructs the corresponding minimal dynamical completion and shows the exact interpretive limit: RS2 can be distinguished from suitably restricted four-dimensional alternatives, but not from an unrestricted four-dimensional history-level theory reproducing the same complete brane records. The result is therefore a concrete, model-relative physical witness, not an unrestricted claim about the ontology of our universe.