In Proof-of-Work the cost of the work is what keeps the network honest. If the work has value then an attacker is free to invest as many resources as I want into subverting the network. Even a failed attack can still be profitable, just less so.

In another scenario, where the works value is less then the cost you're still hoping that at no point in the future will an attacker figure out a way to do the work at a net profit.

The only way the network can be trusted is if the work has definitely now and always, 0 value.

Not littering has value. However, if I don't litter, it doesn't benefit me, and I cannot profit off of it; no matter how eco-friendly I am, I get no value from it.

Am I wrong in saying that the work has negative value? And there are different degrees of that. Bitcoin's negative value is larger.