Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism

Abstract

The Collatz map is a simple deterministic transformation whose orbit structure remains highly nontrivial. A recent direction-phase decomposition partitions each orbit into upward and downward steps, and numerical observations indicate that the number of upward phases, $N_{\uparrow}$, follows an approximate Gamma distribution. In this work, we provide a mechanistic explanation for this statistical regularity by modeling the occurrence of upward phases in the odd-compressed, or Syracuse, version of the Collatz map as a homogeneous Poisson process. From the mean-field logarithmic balance and the geometric distribution of $2$-adic valuations, we derive closed-form expressions for the Gamma parameters: the scale parameter $\theta = 2/(2-\log_2 3)^2 \approx 11.61$ is constant, whereas the shape parameter $K$ grows logarithmically with the maximal initial value $X_0=2L+1$. We also analyze the closure conditions for periodic orbits, showing that nontrivial cycles are severely constrained, which supports the plausibility of the statistical framework. Numerical validation for $L$ ranging from $10^5$ to $10^{15}$ confirms the theory with relative errors below $3\%$, and a bias-corrected mean estimate reduces the error to $10^{-3}$--$10^{-2}\%$. These results establish a quantitative link between the arithmetic properties of the Collatz map and Gamma-type statistics, and suggest possible extensions to generalized Collatz-type problems.