= Solution
Measure the logarithmic <surface brightness> slope in resolved radial annuli after accounting for instrumental blurring and foreground emission. Between a halo core and its outer boundary, <dark-matter decay radiation> predicts $d\ln I/d\ln R=-1$, whereas <dark-matter annihilation radiation> predicts $-3$; doubling radius therefore reduces the respective <surface brightness> by factors $2$ and $8$. These are quantitative templates to compare with independently mapped stellar and gaseous components.
An <exponential galactic disk> has $I_*(R)\propto e^{-R/R_d}$, so $d\ln I_*/d\ln R=-R/R_d$ and $I_*(2R)/I_*(R)=e^{-R/R_d}$. At $R=5R_d$ the latter is about $0.0067$, much faster decline than either halo template, and the projected emission should also be flattened along the <galactic disk>. A <galactic bulge> described by a <Sérsic profile> has $d\ln I_*/d\ln R=-(b_n/n)(R/R_e)^{1/n}$; for $n=4$ and $b_4\simeq7.67$, this is about $-3.4$ at $10R_e$ and continues to steepen, whereas a pure halo power law keeps a fixed slope.
For a <stellar halo> with outer luminosity density proportional to $r^{-p_*}$, the <projection of a spherical power-law emissivity> gives $I_*\propto R^{1-p_*}$. Illustratively, $p_*=3$--$4$ gives projected slopes $-2$--$-3$: then $I_d/I_*\propto R^{p_*-2}$ rises strongly outward, but annihilation can resemble the steeper stellar template. Gas-related gamma emission must be compared with observed gas columns and the <cosmic ray> distribution; it is not always proportional to optical light.
\b[A shallow, extended, approximately spherical $R^{-1}$ component is particularly distinctive; an $R^{-3}$ component needs additional evidence.] Fit the stellar, gas and halo templates together over a broad radial range, including morphology and emission spectra. A measured slope alone cannot uniquely identify <dark matter>, since stellar or gas profiles can coincide with a halo slope over a limited interval.
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