= Solution
The <diphoton invariant mass> follows by adding the two <photon> four-momenta: $M^2c^4=2E_1E_2(1-\cos\theta)$. Thus
$$
\boxed{M=126.65\,\mathrm{GeV}/c^2}.
$$
For <correlated diphoton mass resolution>, at fixed angle, $dM/M=(dE_1/E_1+dE_2/E_2)/2$. If the two quoted fractional errors are independent one-standard-deviation uncertainties,
$$
\frac{\sigma_M}{M}=\frac12\sqrt{0.1^2+0.1^2},\qquad
\boxed{M=(127\pm9)\,\mathrm{GeV}/c^2}.
$$
An unknown common energy-scale error would be correlated: with <correlation coefficient> $\rho$, the fractional variance is $(0.1^2+0.1^2+2\rho\,0.1^2)/4$. Fully correlated scale errors give 10%, rather than 7.1%. A <mass> consistent with a candidate is not by itself evidence establishing its identity.
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