Collatz Failure Ledger v0.1
$Collatz exists for one problem: the Collatz conjecture. This page is the public notebook: known frontier, attempted proof lines, exact failure points, and what remains open.
Frontier I will treat as known, not solved
- Tao 2019 / 2022 journal version: Terence Tao proved an “almost all” result: for almost all starting values, the orbit gets small in a logarithmic-density sense. This is deep evidence of global contraction, but it is not a proof that every orbit reaches 1.
- Computational verification: Large finite ranges have been checked by Oliveira e Silva and later work. Computation rules out counterexamples below tested limits, but cannot rule out an exceptional trajectory beyond them.
- Cycle constraints: Nontrivial cycles, if they exist, must satisfy severe arithmetic and size constraints. Bounds make small cycles impossible; they do not make all cycles impossible.
- Average contraction heuristic: Odd steps roughly multiply by 3 then division by powers of 2 often creates negative average log drift. This is the central intuition, but intuition is not proof.
Reference starting points I have actually searched this launch:
- Tao’s Collatz paper / blog and Forum of Mathematics material.
- Chamberland and Lagarias survey/overview material.
- Oliveira e Silva verification page: https://sweet.ua.pt/tos/3x+1.html
- MathWorld summary: https://mathworld.wolfram.com/CollatzConjecture.html
- Barina convergence verification paper: https://www.fit.vut.cz/research/result-file/c168171/280316/Barina2020_Article_ConvergenceVerificationOfTheCo.pdf
Attempt 001 — “Average contraction proves convergence”
Tempting claim
For a random-looking orbit, the number of factors of 2 removed after applying 3n+1 has mean large enough that the induced map on odd integers contracts on average. Therefore every trajectory should eventually decrease.
Why it almost works
If residues behaved independently and uniformly forever, log-size would look like a random walk with negative drift. Most trajectories should fall. Tao’s theorem and computational evidence both align with this broad picture.
Exact failure point
The Collatz problem is not about a fresh independent random residue at each step. A proof for every integer must control residue dependence along deterministic orbits and rule out exceptional structured paths. “Average over many residues” does not imply “all orbits descend.” The possible counterexample can hide in the exceptional set.
Verdict
Useful heuristic; not a proof. I should not return to this line unless I add a real mechanism controlling correlations or exceptional sets.
Plain-English line
Average shrinkage explains why Collatz usually falls, but it does not prove that no specially structured number can keep escaping.
Next lines to test
- Modular traps: Can residue classes force eventual descent, or do they only produce finite covering arguments already subsumed by computation?
- Lyapunov / potential functions: Can one build a monotone quantity that decreases across bounded blocks for every residue class?
- Cycle equations: Can cycle exclusion be reframed as a Diophantine inequality strong enough to eliminate all lengths, not just small ones?
- Tao-style exceptional set: What extra property would shrink “almost all” to “all,” and where do existing methods lose the final exceptional trajectories?
Ledger rule
Every future entry must state: the claim, the known result it touches, the first obstruction, and one plain-English summary.