= Solution
In conventional <Kaluza-Klein theory>, bulk fields have a <Kaluza-Klein tower> with masses of order $1/R$. If ordinary matter propagates in the extra dimension, the absence of accessible excited modes requires a small $R$: a TeV compactification scale corresponds to roughly $10^{-19}$ m; Planck-scale compactification would be much smaller, of order the <Planck length>. This is a scale estimate, not a universal radius limit independent of the fields and spectrum.
An unwarped <brane-world scenario> localizes ordinary fields on a <brane> while gravity explores a large compact bulk. The volume relation
$$
M_4^2=M_*^{n+2}V_n
$$
makes gravity weak without a fundamental Planck-scale cutoff. In the original https://arxiv.org/abs/hep-ph/9803315[large-extra-dimension model], $M_*\sim\mathrm{TeV}$ permits approximately millimetre/submillimetre dimensions for $n=2$; more dimensions are smaller. Ordinary-field KK bounds no longer limit these gravity-only radii, although gravitational and other constraints do. This replaces the large fundamental scale hierarchy by a large compact volume needing stabilization.
A warped <brane-world scenario> instead uses a position-dependent metric factor. In the two-brane <Randall–Sundrum model>, $m_{\rm IR}=e^{-kL}m_0$, so $kL\sim35$ generates a weak scale from a Planck-sized input without a millimetre proper separation; for $k$ of Planck order, $L$ is tens of inverse-Planck lengths. The https://arxiv.org/abs/hep-th/9906064[single-brane version] even allows infinite proper size through <gravity localization in a noncompact warped dimension>; it is not by itself the two-brane redshift hierarchy construction. Thus \b[warped proper size can be unbounded], while finite hierarchy models need specified warping and <radius stabilization>.
<Supersymmetry> addresses the <hierarchy problem> differently: boson/fermion cancellations protect scalar masses against quadratic sensitivity to high scales, with residual corrections controlled by <soft supersymmetry breaking>. It stabilizes a chosen hierarchy rather than explaining its initial size. Volume dilution and the https://arxiv.org/abs/hep-ph/9905221[warped redshift mechanism] explain scale separation geometrically; neither automatically determines its moduli or removes all tuning.
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