= Solution
<Kaluza-Klein theory> explains apparently four-dimensional physics as the low-energy limit of a higher-dimensional theory with compact spatial directions. In units $\hbar=c=1$, take a circle coordinate $y\sim y+2\pi R$. A periodic scalar has the <Fourier series>
$$
\Phi(x,y)=\frac1{\sqrt{2\pi R}}\sum_{n\in\mathbb Z}\phi_n(x)e^{iny/R}.
$$
Substitution into its higher-dimensional <kinetic term>, followed by integration over $y$, makes different Fourier modes orthogonal. The internal derivative contributes $n^2/R^2$ to the lower-dimensional squared mass, so
$$
\boxed{m_n^2=M^2+n^2/R^2.}
$$
The quantized internal momentum produces the <Kaluza-Klein tower>. A real higher-dimensional field satisfies $\phi_{-n}=\phi_n^*$; the tower does not introduce independent complex modes for both signs. More general compact spaces replace $n^2/R^2$ by <Laplacian eigenvalues>, with boundary conditions determining the spectrum.
At energies $E\ll R^{-1}$, nonzero <Kaluza-Klein modes> cannot be produced and may be integrated out, leaving a four-dimensional <effective field theory>. <Dimensional reduction> retains the zero modes at leading order; integrating out the massive modes also generates interactions suppressed by the <Kaluza-Klein compactification scale>. The higher-dimensional metric decomposes into a four-dimensional metric, vectors from mixed components, and scalar fields from internal components. For a circle, these include an Abelian <gauge field> and a <radion>. Internal coordinate shifts induce the vector's <gauge symmetry>, while the <radion> describes the radius. Pure circle reduction alone does not automatically give the chiral matter spectrum of the <Standard Model>.
The gravitational scale follows by integrating the <Einstein-Hilbert action> over an unwarped internal volume $V_n$:
$$
\frac{M_*^{n+2}}2\int d^{4+n}x\sqrt{|G|}\,\mathcal R_{4+n}
\longrightarrow\frac{M_*^{n+2}V_n}2\int d^4x\sqrt{|g|}\,\mathcal R_4,
\qquad\boxed{M_4^2=M_*^{n+2}V_n.}
$$
Here $M_4$ is the <reduced Planck mass>. For $n$ equal circle radii, $V_n=(2\pi R)^n$. The <Planck mass from compactification volume> relation allows weak four-dimensional gravity even when $M_*$ is much lower. A conventional metric-origin gauge coupling scales as $g_{\rm KK}\sim(M_4R)^{-1}$, so an order-one coupling in that simplest construction tends to place $R$ near the gravitational length scale.
In the <brane-world scenario>, ordinary matter and its <gauge fields> can instead be localized on a three-dimensional spatial <brane>, while gravity explores the bulk. The absence of ordinary-matter excitations along the extra dimensions then permits larger radii than a model in which all fields propagate there. In <large extra dimensions>, gravity changes approximately from a $1/r$ potential for $r\gg R$ to a $1/r^{n+1}$ potential for $M_*^{-1}\ll r\ll R$. For illustration, $M_*$ of order a TeV and $n=2$ give $R$ of order $0.1$ mm; one extra dimension would require an astronomically large radius.
A warped <Randall–Sundrum model> uses
$$
ds^2=e^{-2k|y|}\eta_{\mu\nu}dx^\mu dx^\nu-dy^2,
\qquad m_{\rm IR}=e^{-kL}m_0.
$$
For the usual doubled orbifold interval, $M_4^2=M_5^3(1-e^{-2kL})/k$. The relevant lengths are the bulk curvature radius $k^{-1}$, the proper separation $L$, and the inverse warped excitation scale, of order $(ke^{-kL})^{-1}$. Thus the first mass need not equal $1/L$. \b[Compact volume dilutes gravity; warping redshifts local masses.] Both mechanisms require consistent field localization and <radius stabilization>; choosing a large radius or separation is not itself a dynamical explanation of it.
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