The Upper-Visit Function V(n) in the Collatz Dynamics: Average Convergence Toward 15, Distribution Stability, and Binary Structure

The Collatz conjecture remains one of the most prominent unsolved problems in mathematics, widely regarded as exhibiting irregular and unpredictable behavior. In this work, we focus on the upper-visit function V(n), defined as the number of values in the Collatz trajectory of n that exceed n. We begin by extending previous large-scale computations and provide new empirical evidence that the global average of V(n), taken over all integers up to a bound M, appears empirically to satisfy → 15. This convergence is observed consistently across multiple exponential scales, including extensive sampling up to ranges on the order of 2^500 and 2^1000. In addition, we demonstrate that the full distribution P(V = v) is scale-invariant, exhibiting remarkable stability across all tested ranges. A key implication of this observation is that the existence of a divergent Collatz trajectory would force V(n) = ∞ for some n, thereby preventing the convergence of the global average. Thus, proving that the global average converges to a finite limit would imply the Collatz conjecture. In the second part of the paper, we present a new empirical phenomenon linking the Collatz dynamics to binary structure. We show that repeated prefixing of the binary string “100” to any integer produces a sequence of numbers whose V(n) values and exceedance patterns stabilize. This behavior is observed for all tested integers up to n ≤ 100,000 and across extended binary scales. In contrast, prefixing by “1” or “10” leads to significantly larger and unstable values of V(n). These findings suggest that the Collatz system exhibits both statistical rigidity and structural sensitivity to binary representation. Together, they provide new empirical evidence that the dynamics are governed by hidden regularities and may be far more constrained than previously believed.