= Differential sky map monopole degeneracy
{title2=$d_{ij}=T_i-T_j,\qquad A\mathbf1=0$}
An ideal differential measurement constrains two sky <temperatures> through $d_{ij}=T_i-T_j$. Represent every observed pair by a row of a <matrix> $A$, with $+1$ in column $i$, $-1$ in column $j$ and zeros elsewhere. Then $d=AT$ and $A\mathbf1=0$: adding any constant to the entire map changes no measurement. This <null space> is the angular <angular monopole> ambiguity, explaining why a directional comparison alone cannot measure the absolute <blackbody> <temperature> established by <FIRAS>.
For an original three-pixel example choose $T=(30,-10,-20)\,\mathrm{\mu K}$ relative to an unspecified mean. Measurements $d_{12}=40\,\mathrm{\mu K}$ and $d_{23}=10\,\mathrm{\mu K}$ determine the third difference $d_{31}=-50\,\mathrm{\mu K}$. The zero-mean condition recovers the chosen map, but adding $100\,\mathrm{\mu K}$ to every pixel gives identical differences. The closure sum $d_{12}+d_{23}+d_{31}=0$ checks consistency without establishing the mean.
More generally, the <graph> joining measured pixel pairs must be a <connected graph>. If $AT=0$, each edge forces equal endpoint <temperatures>; a path propagates that equality throughout each connected component. Thus a connected observation <graph> leaves exactly one undetermined constant, whereas disconnected coverage leaves one per component. Fixing the mean makes an ideal connected reconstruction unique; real <noise> calls for <least-squares solution> fitting and uncertainty analysis.
The figure chooses a <cosmic microwave background quadrupole> sky $T(\theta)=Q P_2(\cos\theta)$, with $Q=30\,\mathrm{\mu K}$ and <Legendre polynomial> $P_2(u)=(3u^2-1)/2$. On a great-circle scan, with $\phi$ measured from the chosen symmetry axis and ideal beams separated by $\alpha=60^\circ$,
$$
d(\phi)=T(\phi)-T(\phi+\alpha)
=\frac{3Q}{2}\sin\alpha\,\sin(2\phi+\alpha).
$$
Expanding $\cos^2\phi=(1+\cos2\phi)/2$ proves the formula; its peak magnitude is $45\sqrt3/2\simeq38.9711\,\mathrm{\mu K}$. A constant sky gives zero. This deliberately restricted scan also misses some angular patterns, so rejecting the <angular monopole> is not a proof of complete sky reconstruction. Broad, overlapping coverage supplies the missing constraints; finite <antenna beam patterns> additionally smooth small angular features.
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