Is there a method or tool(s) to prove network outages?

[randy at psg.com (Randy Bush)]

if you do not control the rf end [0], then i assume the upstream
supplies it and is really selling you connectivity to behind the rf cpe.
so you should show you do not have that connectivity, the rf is a red
herring.  use smokeping or any other tool from immediately behind the rf
cpe with a target of the first layer three hop beyond your network.

but, if the upstream is in solid denial and is not actually accepting
that they have a contractual obligation, measurement and technology are
not going to help you.

randy

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[0] - literally NO control, e.g. not antenna placement, power supplied,
      ...