Empirical discrimination

Empirical discrimination

A pre-registered model test

Fit the three phenomenological coefficients of (82)–(83) without using RS2 in the fits, and form the null combinations $$\boxed{\; D_1=A_\Phi-2A_\Psi, \qquad D_2=24B-\kappa_4^4A_\Phi . \;}\tag{115}$$ Minimal RS2 predicts $D_1=D_2=0$ with $A_\Phi,A_\Psi,B\ge0$.

In the nonrelativistic sector, $$\frac{\Delta|\mathbf a|}{|\mathbf a|} =\frac{3A_\Phi}{r^2}+O(r^{-4}),\tag{116}$$ whereas relativistic propagation probes $\Phi+\Psi$ and therefore the different combination $(A_\Phi+A_\Psi)/2=3A_\Phi/4$. The two probes are not degenerate, which is what prevents a single rescaling of $G_4M$ from absorbing the signal.

Short-range torsion-balance experiments already test the inverse-square law at submillimetre separations [16]. Such data should be fitted to the full RS2 KK spectral potential built from (56), convolved with the extended attractor and detector geometry, rather than by translating a Yukawa exclusion curve. The truncated $1/r^3$ potential may be used only after the fitted parameter range verifies $r\gg\ell$; near $r\sim\ell$ it is uncontrolled. Primordial-abundance and expansion data constrain the nonstandard high-density expansion and the separate radiation-like term [17]. A combined analysis must fit $B$ and $\mu$ separately: during radiation domination $\rho^2\propto a^{-8}$ whereas dark radiation scales as $a^{-4}$, so a single-epoch expansion residual can mix their amplitudes, but multi-epoch data and equation-of-state transitions separate them.

This is a falsification test, not a discovery forecast. Existing short-distance bounds already make the corresponding high-energy cosmological term small at late epochs, and a simultaneous precision measurement of both static potentials at microscopic distances is technically difficult. At present the cross-regime relation is more useful as a consistency condition on any claimed anomaly than as a near-term detection channel.

Spectral test

A time-dependent brane source couples to a massless pole plus a continuum. A response measurement with sufficient bandwidth could compare a finite-pole local model against the positive KK spectral measure $$\rho_{\mathrm{KK}}(m)=u_m(0)^2\ge0, \qquad m\in[0,\infty).\tag{117}$$ A resolved threshold continuum, or the corresponding power-law late-time tail, would exclude a pre-specified finite-pole free or tree-level linear alternative. It would not exclude an interacting four-dimensional conformal sector or continuum model, which has the same analytic structure (Remark 6.1).

What no experiment of this kind can decide

No statistic constructed solely from induced brane fields can decide between RS2 and the exact Tier-1 process functional built from RS2. This is not a limitation of experimental sensitivity; it follows from equality of probability measures, (96). An experiment that couples directly to a bulk coordinate could distinguish the source fiber, but it would thereby enlarge the observable algebra and would no longer be confined to the readout used in the theorem.