𝗟𝗲𝗮𝗻 𝟰 𝗙𝗼𝗿𝗺𝗮𝗹𝗶𝘇𝗮𝘁𝗶𝗼𝗻 𝗼𝗳 𝗖𝗼𝗹𝗹𝗮𝘁𝘇 𝗘𝘅𝗶𝘁-𝗟𝗮𝘆𝗲𝗿 𝗖𝗼𝗻𝘃𝗲𝗿𝗴𝗲𝗻𝗰𝗲
I am sharing a new methodological record in Lean 4.
This work formalizes a specific part of the Collatz dynamics. We call this the exit-layer fragment. It covers odd numbers that reach a power-of-two in one step.
This is not a proof of the full Collatz conjecture. It is a formal record of a known mathematical observation.
Key technical details:
• We use coinductive Stream' coalgebras to represent orbits. • We define a halt-at-1 variant of the Collatz step. • This makes the orbit eventually constant at 1. • We prove that exit-layer numbers reach this constant stream. • The proof relies on the standard Mathlib axiom triple.
A notable result:
The theorem head_collatzOrbit is completely axiom-free. Running #print axioms on this theorem returns an empty set. This places it below the standard classical axiom base.
What this paper does not do:
- It does not solve the Collatz conjecture.
- It does not break the Cases 5–8 trailing-1-bits wall.
- It does not change the analytic results of Terence Tao.
- It does not address the unsolved parts of Janik (2026).
The goal is to provide a clean, axiom-free formalization within our observed range. We use the coalgebraic perspective from Kim (2008) and apply it to Lean 4.
This is a methodological contribution to the Rei-AIOS series. It focuses on axiom-base minimization and formalizing known structures.
Optional learning community: https://t.me/GyaanSetuAi