Thank you, this is interesting stuff! Can you expand a little on how this principle would help? In the Mayer Vietoris sequence there is only the difference of two such restriction maps. What makes me a bit skeptical is that my intuitive argument works in any dimension, that is, not just for the $3$-torus, whereas using the "half-lives, half-dies" principle would work only in dimension $3$ (or, any odd dimension?).