Observations on the Thermodynamic Structure of the Collatz Map

We study the Ruelle transfer operator of the Collatz map on the 2-adic integers and prove a closed-form identity for the per-period odd-step statistic f(k) of ghost cycles via the effective Chebotarev density estimates of Thorner–Zaman (2019) and Sgobba (2023). The central object is the Ruelle transfer operator ℒ_s acting on C(ℤ₂), with weight 3^s on the odd inverse branch, topological pressure P(s) = log(1 + 3^s), and distinguished equilibrium parameter s* = log₃(√3 + 1). The Cuntz-algebraic structure accompanies the 2-adic Ruelle transfer operator ℒ_s, which encodes the 2-adic geometry of ℤ₂ and the 3-adic arithmetic of T₃ in a single operator-algebraic object. The operator-algebraic structure of the construction combines Siegel's χ₃ (Siegel 2024, 2022), which encodes the ℤ₂-to-ℤ₃ substrate, with Mori's identification of the Cuntz algebra O₂ ≅ C*(S₁, S₂) for the Collatz map (Mori 2025, Proposition 4.3.4). A pressure-shifted one-parameter group σ_t of automorphisms of O₂ derived from the transfer operator ℒ_s admits a unique (σ_t, β = 1)-KMS state equal to the Bernoulli measure with weights (2 − √3, √3 − 1). At the unique pressure parameter s* = log₃(√3 + 1) at which σ* coincides with the uniformiser of the ramified prime of ℚ(√3) above 2, the alignment is supplied by the conjugate-pair identity (√3 − 1)·3^(s*) = 2. The Möbius-primitive-count closed form f(k) = ( Σ_{s ≥ s_min(k)} s·M(k,s) ) / ( k · Σ_{s ≥ s_min(k)} M(k,s) ) reproduces the empirical census within sampling error and satisfies lim_{k→∞} f(k) = log₃2. On a census of 5430 ghost cycles, a multinomial χ² test rejects the offset c = 1 at χ²/dof ≈ 81 (Wilson–Hilferty z = 73.86, at the edge of the multinomial χ² approximation's validity), with c = 2 matching at χ²/dof = 1.17; the offsets c = 3 and c = 4 are excluded by structural impossibility of the predicted support S_c(k) on small-k bins of the census rather than by Pearson rejection; the asymptotic density-one form is also established in the paper. The second-order constant r₀ = (1 − log₃2)/(2·log₃2 − 1) ≈ 1.4094 in f_c(k) − log₃2 = (c − {k·log₃2} + r₀)/k + O(k^-2) admits two equivalent interpretations: as the Boltzmann conditional mean of the uniform measure on {0,1}^k restricted to σ̂_k ≥ log₃2, and as the second-order coefficient of the geometric tail of the binomial-coefficient envelope at s ∼ k·log₃2. The Cramér rate I(log₃2; √3 − 1) = 0.0243814… > 0 of the Bernoulli(√3 − 1) KMS state measures the relative-entropy distance from that KMS measure to the boundary parameter log₃2. At the finite truncation parameters N ∈ {8, 10, 12} of the integer-lift operator the measured spectral gap is 0.1425 ± 0.0012. A census of 75,536 ghost cycles on ℤ/2pℤ for primes p < 10^7 exhibits the dichotomy s/k > log₃2 on every observed cycle. Computations for P ∈ {10^6, 10^7, 10^8, 10^9, 10^10} establish P-invariance of f(k) = ⟨s⟩_k / k and the arithmetic law s_min(k) = ⌊k·log₃2⌋ + 2.