Every hash matters not just statistically, but physically. Each one is a real transformation: energy burned, entropy resolved, structure either committed or failed. Even the “failed” hashes aren’t failures from a thermodynamic perspective, they are the bulk of the computation, the substrate of probabilistic collapse into a valid block. They define the entropy field that gives the successful hash its meaning. Success is defined by failure, but it must be scarce and bounded at each timestep. You’re right to say the miner only cares about the successful ones, but Bitcoin is not a system of care or preference, it’s a system of irreversible computation. From the standpoint of entropy, every hash is a step through a finite state space, and the entire process whether rewarded or not is what defines the difficulty (thus block time based on hash rate), temperature, and energy density of the block. Boltzmann’s constant becomes useful here because it bridges entropy (a count of states) with energy (joules). If we take the Genesis peg as the founding entropy-to-structure mapping, we get a scalar field of joules per satoshi that changes every block as supply grows and issuance decays and difficulty adjusts. That ratio defines the thermodynamic cost of resolution at each time step, and eventually fees will reflect the market price of entropy compression into the ledger. I’m still trying to wrap my head around it. I believe entropy is the answer. But it’s not just the entropy of machines, or of stochastic fields. It’s the entropy of possible futures defined by the valid utxo set collapsing into one irreversible ledgered past. There is little work done on the physics of Bitcoin still, I’m only scratching the surface here. I’ve been working on a paper for almost a year now and getting closer to feeling confident for a public release. I personally haven’t seen much other writing in this specific domain. I’ll have to think about your initial post more. Thank you!