Closest I can think of that’s related to the this question is the Landauer principle
https://physicsworld.com/a/landauer-principle-passes-quantum-muster
Erasure of information has associated with it a minimum amount of waste heat.
If you calculated the relativistic mass for the amount of energy dissipated, you might get your answer. But I might be just blowing a bunch of hot air.
I know its been a while since I read your response and I have been thinking about it from time to time. One of the things that sticks out in my mind is the term “Relativistic”. I was just watching a video today about micro black holes and their possible explanation if they even exist. Could this be an explanation for dark matter? The thing that I found fascinating was why if black holes lose energy “Hawking radiation” at all and eventually evaporate down, would this explain dark matter if they eventually reached a stable point? The explanation seemed to be that they simply get too small to quantumely vibrate internally anymore. Apparently they were all created after the first few micro nanoseconds which didn't actually exist after the "Big Bang". That whole relativity thing again. The guy that was giving the presentation asked the obvious question. What then happens to all that left over information? Interesting thought huh?