Scope and regime of validity

Scope and regime of validity

The results derived above apply to the following physical domain and assumptions.

1. Bulk-state domain. The same-readout witness requires that admissible initial data include normalizable bulk radiation. If the model definition fixes a unique stationary vacuum and excludes all homogeneous incoming data, the witness lies outside the restricted domain. The forced KK response and the effective-action equivalence remain, but the claimed instantaneous fiber must be reassessed for that domain.

2. Minimal action. A brane Einstein term, a Gauss–Bonnet term, additional bulk fields, finite brane thickness, a second brane, or broken $Z_2$ symmetry changes the junction condition, the propagator, and the coefficient relation. In such theories $D_1=D_2=0$ need not hold.

3. Positive-tension branch. The sign and stability statements use $\lambda>0$. The negative-tension branch can reverse kinetic signs or the effective Newton coupling; it is not a harmless sign convention.

4. Homogeneous sector. Equation (48) assumes an FRW brane, conserved brane matter, and no bulk matter flux. With bulk exchange, both $\dot\rho+3H(\rho+p)=0$ and $\mathcal U\propto a^{-4}$ are modified.

5. Weak-field sector. The coefficients $A_\Phi,A_\Psi$ assume a static weak source, $r\gg\ell$, and the regular retarded/no-incoming bulk condition. They are unreliable in the strong-field region and at $r\lesssim\ell$.

6. The state parameter $\mu$. Neither the sign nor the magnitude of $\mu$ is fixed by the projected brane equation. Positivity follows only after imposing a positive-mass regular AdS–Schwarzschild bulk. Treating dark radiation as a universal fixed prediction would overstate the model.

7. Observable versus decomposition. The induced metric and matter records are observable; the split $\mathcal C^{\mathrm{RS}}=\kappa_5^4\Pi-E$ is source-model dependent. A four-dimensional theory may reorganize the same total response into different fields or a nonlocal kernel.

8. Unrestricted alternatives. Nonlocality, colored noise, a branch cut, or preparation dependence does not by itself refute a Tier-1 theory. Each is permitted in the exact pushed-forward history law (95). A contrary conclusion requires an independently imposed restriction , finite-pole local quadratic dynamics, fixed perturbative content, or Markovianity. Locality, or finitely many interacting fields, is not enough.

9. Flat-brane tuning. The cross-regime relation was derived at $\Lambda_4=0$. A detuned or de Sitter brane changes the background junction relation and can modify the weak-field coefficients. Applying the flat-brane relation to a weakly curved universe is an approximation requiring $H\ell\ll1$ and negligible curvature/detuning effects; it is not a complete late-time cosmological fit.

Two further parameters are simply undetermined rather than failure modes. The numerical value of $\ell$ , equivalently $\lambda$ or $\kappa_5$ once $G_4$ is fixed , is not predicted; only bounds on it follow from data. And the theory is a five-dimensional effective field theory, so nothing here constrains its ultraviolet completion above the corresponding cutoff.

Decoupling limits.

These are consistent. As $\ell\to0$ at fixed $G_4$: $A_\Phi,A_\Psi\to0$, $\lambda\to\infty$, and $B\to0$. As $\rho/\lambda\to0$, $\mu/(a^4H^2)\to0$, and $r/\ell\to\infty$, ordinary four-dimensional GR is recovered in every sector.