No. Both computers did hard work, and earned credit toward proof of work in my example. I’m rather repeatedly saying energy can be used for two purposes (securing a ledger with a difficult computation that also means something), and I have no idea how else to explain this.
I’m not sure why your incentive would drop to zero. Say I swap in a universal simulation from the Big Bang instead of SHA2. A miner simulates a portion it chooses, which involves quintillions upon quintillions of gravitational fields. Eventually it simulates a rare event, like two black holes colliding or something. It somehow notates this as proof that it worked on the problem for a while, hashes that proof combined with the transactions, and boop, mined a block. Meanwhile, all along, we have learned science. I could think of other problems that work: SETI@home and the other BOINC apps come to mind, as does LHC analysis. Large search space, demanding algorithm, rare thing to find (SETI/Higgs candidates), and valuable science along the way. Some issues I could see are unpredictable nature of block time, verifiability being expensive, and problem spaces that move forward in time, like my first example. But I don’t see those as insurmountable.
Why would incentive drop to zero to search the space? I’m not following that claim, nor why you’re scare quoting important.