Source
LaMont & Wiggins (2019), LIT-1882, §§II–III.C, Tables I–II, eqs. (4)–(30) and appendices A3–A4, read in NOTE-1481 from arXiv 1706.01428v5. The singular asymptotics are Watanabe’s, cited there, as in THEORY-087.
What was actually shown
The dictionary. Z(xᴺ) = ∫ dθ ϖ(θ) e^{−NĤ(θ)} (eq. 4): parameters as states, empirical cross-entropy as energy, prior as density of states, N as inverse temperature. Then F = −N⁻¹ log Z, U = ∂_N(NF), C = −N²∂_N U, S = N(U − F) (Table I). U is the leave-one-out cross-validation loss, using a finite difference in N (eqs. 11–12, A8–A9).
Equipartition. In a regular K-parameter model, F = H₀ + (K/2N) log(N/N₀), U = H₀ + K/2N and C = K/2 (Table II, eq. 17), term for term the free particle’s E₀ + (K/2β) log(β/β₀). Checked by differentiation.
Singular models. With F = H₀ + (γ/2N) log N, C = γ/2 (eq. 18). The paper’s γ is 2λ in Watanabe’s convention (F_n = nS + λ log n), so C → λ.
Freeze-out. C falls below K/2 when the prior dominates at small N, C = K/(2(1 + N₀/N)²) (eq. 24); when a parameter is discrete and resolved at large N (eq. 30); and near a singular point of a two-component exponential mixture (C = 0.61 against 1.5 at N = 100, eq. 28). It can also exceed K/2: the uniform endpoint model has C = 1 for one parameter (appendix C2).
The logarithmic approach (my check). Watanabe’s free energy has a further −(m − 1) log log n term, which Table II drops. With it, C = λ − (m − 1)/log n + …. For the deterministic part of the free energy of K = a²b² with a uniform prior on [0, 1]² (λ = ½, m = 2; CASE-152’s model), exactly: C = 0.325, 0.375, 0.403, 0.433, 0.449, 0.465 at n = 10², 10³, 10⁴, 10⁶, 10⁸, 10¹², against ½ − 1/log n = 0.283, 0.355, 0.391, 0.428, 0.446, 0.464.
What could have failed. The learning capacity of a singular model could have kept the regular value K/2. It does not, and departs from it in the direction of the RLCT.
What this does not say
- That inference is thermodynamics. The temperature is a sample size, the energy an empirical loss. There is no dynamics, so there are no currents, no entropy production and no housekeeping heat. The A14 analogy between SLT’s e^{−nK} and a physical e^{−ΩV} is exact as a statement about concentration of a Gibbs-form measure. It is silent on the dissipative structure that maintained physical states carry (THEORY-1264, THEORY-1265).
- That the learning capacity is the singular penalty itself. It is its second derivative in N. For m > 1 it reaches λ only logarithmically, so at practical n a measured C is not λ (0.40 against 0.5 at n = 10⁴ in the example).
- That sloppiness in large networks is freeze-out. The authors “speculate” this (§I.A); their evidence is three toy models.
- Anything about the GPI prior. Its AIC limit and its treatment of the Lindley–Bartlett paradox are in NOTE-1481 and are not part of this claim.
- The fluctuation part. My check computes only the deterministic part of the free energy; data fluctuations add terms the check does not cover.