Jungle of Base Nodes in the Collatz (3x+1) Problem: Precise Localization and Density Distribution within Ultra-Large Intervals

【Abstract】This paper introduces a novel framework for locating and analyzing "jungles" in the Collatz iteration landscape. We redefine the base node as \frac{4^N - 1}{3}, termed the tail number in forward iteration and the base node in the reverse tree. Our Core Theorem establishes that for any odd number X_i with a sufficiently large pure odd iteration count (typically X_i \geq 50), there exists a jungle interval [X_i - \Delta X, X_i + \Delta X] wherein numbers generated from the same base node are densely distributed and share identical pure odd iteration counts. Furthermore, the interval width \Delta X increases monotonically with the pure odd iteration count. By initiating continuous base number calculations from high-iteration base numbers using our second Python algorithm, we can precisely compute jungle intervals and visualize their distribution densities—capabilities that extend with remarkable precision to ultra-large numerical ranges (from 10^{20} up to 10^{1000} and beyond). Pre-computed compressed data files are appended for empirical verification.This paper establishes a fundamental dichotomy in base node propagation density based on modular arithmetic. For base nodes of the form \frac{4^N - 1}{3}: if the base node is congruent to 2 \pmod{3}, it undergoes compressive propagation, yielding a higher density of generated base numbers within the natural odd number sequence, as exemplified by the base node 341 (341 \equiv 2 \pmod{3}), which propagates via the rule \frac{2X_i - 1}{3}; conversely, if the base node is congruent to 1 \pmod{3}, it undergoes expansive propagation, resulting in a lower density of generated base numbers, as exemplified by the base node 85 (85 \equiv 1 \pmod{3}), which propagates via the rule \frac{4X_i - 1}{3}. This modular classification provides a precise predictive framework for jungle density distribution in Collatz iteration landscapes.