Bitcoin as a Thermodynamic System: Phase Transitions at Halving Events

We model Bitcoin as a thermodynamic system where the hash rate plays the role of temperature, the price plays the role of pressure, and the halving events are phase transitions. The block reward \(R(t) = 50 \cdot 2^{-\lfloor t/T_H \rfloor}\) defines a deterministic energy input schedule; the hash rate \(H(t)\) measures the physical energy expenditure of the network; and the price \(P(t)\) reflects the market's valuation of the resulting output. We derive an equation of state relating these quantities and show that halving events satisfy the Ehrenfest classification of a second-order phase transition: the first derivative of the miner free energy with respect to the block reward is continuous, while the second derivative — capturing the price-volatility response — is discontinuous across the halving boundary. The spectral representation of the return density (Nagy, 2026a) provides a candidate order parameter: the zeroth Fourier coefficient \(A_0\) of the return density undergoes a structural shift at the halving, analogous to the susceptibility divergence in a magnetic transition. The equation of state, halving dynamics, and the 21M supply cap are formally verified in Lean 4; the Ehrenfest classification and spectral order parameter remain analytical results supported by empirical observation from four historical halvings (2012, 2016, 2020, 2024).