I don't know what "ER peaks" means, but I think I get the idea. (Did you mean "EM", i.e., electromagnetic?)
Suppose you see a spiral galaxy that's 100,000 light-years across, and you're seeing it nearly edge-on. Suppose the entire galaxy's brightness increases and decreases significantly with a 1-year cycle.
It's not possible (or rather, it's vanishingly unlikely) that this is the result of all the stars pulsating in unison. The near edge of the galaxy is 50,000 light-years closer to us than the center, which is 50,000 light-years closer than the far edge. That means that the light that we're seeing now from the far edge must have originated 100,000 years sooner than the light we're seeing now from the near edge. Since nothing can travel faster than light, there is no natural phenomenon that could keep all the stars over that 100,000 light-year expanse synchronized with each other. And even if there were, they would appear to be synchronized only as seen from one particular direction.
If a 100,000 light-year galaxy appears to pulse with a 1-year cycle, then the pulses are coming from a much smaller body, probably at the galaxy's core. And that body is probably substantially smaller than 1 light-year, because the waves within it that cause it to pulsate are probably moving substantially slower than the speed of light.
If you see a light source pulsating with a 1-year cycle, then it must be less than 1 light-year across. If it's any bigger, then (a) whatever waves cause it to pulsate will take more than a year to cross it, so it can't stay synchronized with itself, and (b) even if it could, the pulses would be "blurred out" because we simultaneously see parts of the light source at different distances.
In your example, you got it backwards. If a black hole visibly pulses on a 10-minute cycle, then the body that's emitting the light must be smaller than 10 light-minutes.