Exact Family Connections and Descent Certificates for the Collatz Map
Year: 2026
Type: preprint
Abstract: This independent research preprint presents a unified framework of exact family connections and descent certificates for the Collatz map. A combined theorem explains how a valid connection transfers termination through doubling and geometric families. The paper develops shared-continuation and inverse links, explicit residue domains, affine composition with retained offsets, and local repetition bounds including an infinite compensated family. The accompanying package contains 37,349 ordinary-step certificate records, 61 mixed-connection domains, a precisely scoped coverage register, verification code and selected longer-route certificates. A separate computation examines 500,000 odd starting integers and tests a proposed binary-length bound for a specified connection map. The remaining global task is to prove finite smaller-reference certification for every canonical base. That statement and the proposed quantitative bound remain open. The paper provides proved family relationships and a reproducible framework for further investigation; it does not claim a proof of the Collatz conjecture or established mathematical priority. This is personal research by Lee Harley, undertaken independently of employment. ChatGPT assisted with formulation, drafting, code, computation and preparation, as disclosed in the manuscript. The work has not been independently peer reviewed. This is the first public submission. Version 3.1 denotes the internal manuscript edition; earlier drafts were not submitted. Licensing: the paper, original figures and data are licensed under Creative Commons Attribution 4.0 International. Software is licensed under the MIT License, as specified in the accompanying package.
Author: Harley, Lee
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225433952