Robert Polak

Nontrivial Cycles of the Collatz Conjecture: The Smooth Model and the Diophantine Bridge

Part 1 of a two-paper series on the Collatz conjecture. We study nontrivial cycles in a smooth model and build an exact Diophantine bridge Real→Integer. We prove the smooth model has no nontrivial cycles; the only discrete fixed point compatible with the original map is 1. For an...

Analysis of the Collatz Conjecture: Excluding Infinite Growth.

Part 2 of a two‑paper program toward an analytical proof of the Collatz conjecture. Using the modulo‑3 “One‑Way Gate”, the 3‑adic contraction of T(n)=(3n+1)/2^{v2(3n+1)}, and an exact Diophantine reduction with a telescoping identity, we exclude infinite growth of Collatz traject...