# Sources and claim boundaries

Snapshot: 16 September 2026. Authors: Khamit Kadyrbekov and Daniyal Kadirbekov.
The video is an explanation of research in progress, not an announcement of a
proof. Original narration and diagrams; no third-party portraits, audio or paper
pages are reproduced.

1. Definition, historical attribution and mathematical connections:
   Jeffrey C. Lagarias, *The 3x+1 Problem and its Generalizations* (1985;
   author-approved web edition 1996),
   https://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node1.html .
   The origin is historically nuanced; we say "traditionally associated with"
   Lothar Collatz, not that a precise invention date is established here.
2. Riho Terras, *A stopping time problem on the positive integers*, Acta
   Arithmetica 30 (1976), 241–252,
   https://doi.org/10.4064/aa-30-3-241-252 .
   Publisher record: https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/30/3/101028/a-stopping-time-problem-on-the-positive-integers .
3. C. J. Everett, *Iteration of the number-theoretic function f(2n)=n,
   f(2n+1)=3n+2*, Advances in Mathematics 25 (1977), 42–45,
   https://doi.org/10.1016/0001-8708(77)90087-1 .
   The abstract establishes almost-everywhere descent below the start.
   Independence and natural-density formulation also appear in Lagarias's
   exposition: https://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node4.html .
4. Daniel J. Bernstein and Jeffrey C. Lagarias, *The 3x+1 Conjugacy Map*,
   Canadian Journal of Mathematics 48 (1996), 1154–1169,
   https://websites.umich.edu/~lagarias/doc/bernstein.pdf .
   Exact 2-adic coding is a structural description, not universal convergence.
5. Terence Tao, *Almost all orbits of the Collatz map attain almost bounded
   values*, first submitted 2019, Forum of Mathematics, Pi 10 (2022), e12,
   https://arxiv.org/abs/1909.03562v7 .
   For every f(N) tending to infinity, the orbit minimum is at most f(N) for
   almost all starts **in logarithmic density**. This does not prove that
   almost all starts reach 1, or that all starts reach 1. Versions 6 and 7
   in July 2026 revise the same paper; they are not a new Collatz solution.
6. Project record, reports 423–425: internally reviewed work on a restricted
   family. Report 423 gives a conditional localization of a cubic contribution;
   424 gives a conditional depth bound; 425 bounds the degree of a first
   coefficient certificate under its hypotheses. Actual content/height bounds
   and universal orbit coverage remain open. No claim to improve Tao's theorem
   or to established publication priority is made.

The argument that universal strict descent implies convergence is elementary
strong induction. A finite collection of checked starts cannot, by itself,
quantify over all positive integers. These are explanations, not new results.

## Media production

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AI assistance: drafting, diagram implementation and internal argument review.
Green denotes internal checking, including AI-assisted checks; it does not denote
external peer review. Roles assigned to AI do not imply participation by named
historical mathematicians. Individual author duties have not been invented.