Collatz research workbench: literature, first passage, orbit clocks and exact constructions

This is the archival home of the PolyClank Collatz research workbench: an ongoing, open human–LLM collaboration studying the mathematics of the Collatz map. The ordinary map sends an even positive integer n to n/2 and an odd one to 3n+1. The workbench reconstructs the literature, develops exact comparison maps and new deductions, and makes the arguments available with source files, reproducible calculations and explicitly scoped formal proofs.

GitHub workbench · Reading guide · PolyClank discussion and contributions

Mathematical programme

The wider workbench contains source-faithful exposition of Maxwell C. Siegel’s Hydra and numen constructions; a separate study of clocks and first-passage measures in Terence Tao’s theorem; and independent work on weighted odd-step paths, arithmetic histories, affine transport, finite spectral encodings, integral comparisons and bounded-support constructions. These strands have separate manuscripts and attribution. Exact maps and proofs, rather than a shared vocabulary, establish their connections.

First passage, natural density, clocks and height

The principal reader in this edition is Conditioning and first-passage fibres in the natural-density argument. It reconstructs the joint Syracuse offset–valuation calculation and the arithmetic first-passage fibres, develops the uniform kernel and timed-target estimates, and compares the arguments of Tao, Allikvere, Mazur and Shaik. It also credits Inselmann’s earlier drift-envelope results.

One resulting synthesis controls three clocks and the height of the same orbit prefix. Let d = log(4/3), with natural logarithms. For every β,e > 0 there is A = A(β,e), independent of B,c,X, such that for every X ≥ 2, integer B ≥ 2 and 0 < c < 1/17.232, all but at most C_c X(log B)^−c + C_(β,e) X(log(X+2))^−e starting integers n ≤ X have a prefix ending at an odd value ≤ B. Its odd-step, shortcut-step and ordinary-step counts are bounded, respectively, by j log(n)/d + A(log(n+2))^4/5, for j = 1,2,3, and its largest ordinary value is at most n^1+β. A shortcut odd step combines 3n+1 with one halving; a Syracuse return removes all its powers of two.

The proof joins a height-controlled passage to a polylogarithmic level with a timed first hit of B on the original deterministic orbit. A lower bound for time already spent leaves only O((log n)^4/5) odd returns in the tail. Exact valuation telescoping then controls the two other clocks and the tail height, without assuming independence or restarting a probability law at the landing point. The fixed-target estimate implies passage strictly below every f(n) → ∞ for a natural-density-one set, with clocks independent of f. It does not imply that every orbit reaches 1.

Further results include an elementary endpoint-separation estimate, a kernel exponent c < 1/17.232 using the cited Rhin input, refinements of terminal scalar estimates, and exact finite timeout and completed-block height bounds. The reader gives the hypotheses, proofs, dependencies and limitations of each result. These are workbench reconstructions and deductions, not a claim to have corrected Tao’s theorem or independently certified every source manuscript.

Sources and verification

The principal sources are Tao’s v7 paper, Allikvere’s v2 manuscript, Mazur’s natural-density argument, Shaik’s v3.2.4 manuscript and its pinned source, and Inselmann’s v3 paper. Precise source versions and locators appear in the manuscript and claim records.

The archive is a complete snapshot of the public GitHub workbench at commit e56508d9102bf16442ec9a1bc21fb0a7e1f29898, including its earlier readers and research programmes. Start with the separately available 168-page PDF, which is also the record’s preview; its complete LaTeX source is supplied alongside it. The current first-passage package contains six cumulative proof chapters, 33 finite checking scripts and their labelled local reports, figures, and five narrowly scoped Lean source modules with recorded axiom outputs. All 33 finite suites passed in the local workbench after typesetting repairs; two self-contained arithmetic checks were also rerun against the distributed package. Finite tests do not prove the analytic density theorem; the Lean modules certify discrete clock, witness, reflection, dyadic-prefix and completed-height statements, not the whole analytic argument. Historical readers and their separate verification records retain their dates and scope; inclusion in this archive is not a fresh verification of every historical result. The GitHub reading guide provides routes through the collection.

Collaboration and reuse

The public collaboration name is Kokuno Yumeto. Work was developed with ChatGPT 5.6 Sol and GPT-6 Astra, in Ultra mode in Codex. Literature authors retain their own attribution. To contribute, fork the GitHub repository, let your model read the relevant proofs and primary sources, commit its arguments and reproducible checks, and submit a pull request or link your fork in the PolyClank discussion. Contributions should explain the mathematics clearly enough for another mathematician to follow.

This edition continues the Zenodo series formerly titled Affine Packets, Cyclic Orbit Packets, Dyadic Dilation, Lambert–Mahler Series, Hydra/Numen Structure, and Exact Rational Periodic Orbit Enumeration in the Odd-Step Collatz Algebra. Earlier editions remain available in the version history. This is ongoing research, not a completed proof of the Collatz conjecture.