Collatz Convergence through Logarithmic Descent and Modular Reduction

Theoretical Framework

The Collatz process is framed as a discrete-time dynamical system on a directed graph.

Positive integers are classified into three modular/2-adic classes (S1, S2, S3), each governed by a different part of the piecewise descent function F(n).

Resistant states are explicitly penalized using a depth term, and funneling via residue classes is leveraged to guarantee eventual descent.

Empirical Analysis

For every n ≤ 2000, we compute the minimal step k where F(T^k(n)) < F(n).

Over 60% of cases achieve descent at k = 1, 95% within k ≤ 2, and all within at most k = 5.

Well-known resistant cases (e.g., n = 27, 31, 41, 371, 583) are fully accounted for.

Experimental results are presented as both a histogram and scatter plot, included in the manuscript.

Implications

The approach rules out nontrivial cycles and reframes the Collatz process as a predictable arithmetic system with structural constraints.

All formulas are validated; code and data can be shared upon request for full reproducibility.

The framework offers a strong empirical and structural foundation for further mathematical and computational investigation.

Limitations and Future Work

While no counterexamples are observed up to n = 2000, a full proof for all n remains open.

Future directions include formal verification (Lean, Coq), deeper bounds on resistance depth, and exploration of symbolic dynamics connections.