The Energy Cost of a Hash Trial: Proof-of-Work Priced from the Thermodynamic Floor Up

Proof-of-work priced from the thermodynamic floor up. One double-SHA-256 trial costs 1.5e-11 J on frontier hardware (Antminer S21 Pro, 15 J/TH); the strict Landauer floor at 300 K is 7.35e-19 J per trial (256-bit erasure only), and even a generous floor counting every gate evaluation as a full bit erasure is 8.22e-15 J. Network totals, per-block energy at the 10 November 2026 difficulty, and the two regimes of efficiency improvement (fixed difficulty versus post-retarget equilibrium) are computed from cited inputs by the accompanying script. Model calculations are labelled simulated, not measured.