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 ℒₛ acting on C (ℤ₂), with weight 3ˢ on the odd inverse branch, topological pressure P (s) = log (1 + 3ˢ), and distinguished equilibrium parameter s* = log₃ (√3 + 1). The Cuntz-algebraic structure accompanies the 2-adic Ruelle transfer operator ℒₛ, 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 σₜ of automorphisms of O₂ derived from the transfer operator ℒₛ admits a unique (σₜ, 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) = (Σₒ ≥ ₒ䂷₈₍ (₊) s·M (k, s) ) / (k · Σₒ ≥ ₒ䂷₈₍ (₊) M (k, s) ) reproduces the empirical census within sampling error and satisfies lim₊→∞ 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 Sc (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 fc (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, 1ᵏ restricted to σ̂ₖ ≥ 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 log₃2 on every observed cycle. Computations for P ∈ 10⁶, 10⁷, 10⁸, 10⁹, 10¹0 establish P-invariance of f (k) = ⟨s⟩ₖ / k and the arithmetic law sₘin (k) = ⌊k·log₃2⌋ + 2.
Heewon Cha (Tue,) studied this question.