Thermodynamics of Scale: Entropic c-Functions, Data Processing, Spectral Thermodynamics, Fluctuation Relations, and the Arrow of Time

The irreversibility of the renormalization group (RG) is often described in thermodynamic language: information is "lost" under coarse-graining, c "decreases like an entropy", scale behaves like a temperature, and the RG arrow is compared with the arrow of time. This paper sorts such statements into four kinds (theorems, exact identities, standard physics applied to scale, and conjecture) and checks every exact statement numerically. (1) The c, F and a theorems state that c_UV > c_IR, a decrease from the ultraviolet to the infrared. In the entropic proofs this follows from strong subadditivity and Lorentz invariance, not from a thermodynamic law. For a massive lattice Dirac fermion, the Casini–Huerta function c(ℓ) = 3ℓ dS/dℓ decreases monotonically to zero. At the critical point it reproduces c = 1 times the finite-size chord factor (πℓ/N)cot(πℓ/N). (2) The monotonicity of relative entropy under quantum channels gives a rigorous H-theorem: if an RG step is a channel with an invariant state σ*, then D(Φⁿ(ρ)‖σ*) is non-increasing. This is verified on block restrictions of Gaussian fermion states. (3) Read the heat-kernel time τ as an inverse temperature on the Laplacian spectrum. Then S(τ) = ln P + τ⟨λ⟩ = −D(π_τ‖μ) is non-increasing, where μ is the density-of-states measure. The heat capacity C = τ²Var(λ) obeys C = d_s/2 − (1/2) dd_s/d ln τ, so on scaling plateaus the spectral dimension is twice a heat capacity. For hypercubic lattices C has a closed form, with a Schottky-type maximum of 0.6799 d and the equipartition limit d/2. (4) A physical protocol that changes the correlation length of a lattice field in time obeys the Jarzynski and Crooks relations exactly. For a sudden quench, the Jarzynski estimator has finite variance if and only if ω_f²/ω_i² > 1/2 for every mode. For finite ramp times, an exact Riccati equation for the tilted Gaussian density shows that the softening protocol keeps an infinite variance at every simulated ramp time. For the stiffening protocol the exact relative variances predict the observed standard errors. Jarzynski and Bennett acceptance-ratio estimates recover ΔF to within 0.015. Landauer's bound applies to physical erasure, not to the theorist's coarse-graining map. The identification of the RG arrow with the arrow of time, suggested by dS/CFT and by holographic c-theorems, is stated as a conjecture. Natural bulk candidates are the covariant entropy bound and the area law for past holographic screens. The conjecture does not explain the low-entropy initial state (the Past Hypothesis). This is the fifth paper of a series. The earlier papers are 10.5281/zenodo.23189575, 10.5281/zenodo.23193512, 10.5281/zenodo.23194419 and 10.5281/zenodo.23195881. Code, data and figures: https://github.com/Ruqing1963/scale-thermodynamics