Collatz Conjecture
Halve it if even, triple-plus-one if odd. Does every number fall to 1?
In plain terms
Pick a whole number. If it is even, halve it. If it is odd, triple it and add 1. Repeat.
Starting at 6: 6, 3, 10, 5, 16, 8, 4, 2, 1. Starting at 27, it wanders up past 9,000 and takes 111 steps to come down. The conjecture says every starting number eventually reaches 1.
It is the most famous problem in this collection, and the one this lab deliberately spent the least effort on.
What is known
Verified by computer to around 2⁶⁸. The strongest theoretical result is Terence Tao's, from 2019: almost all starting values eventually reach a value that is almost bounded — in a precise density sense. That is remarkably close to the conjecture and still, frustratingly, not it.
Why it is hard
There is a hard warning sign. If you generalise the rule slightly — allowing other multipliers and divisors — the resulting family of problems is undecidable: no algorithm can settle them all. Conway proved this. So any successful method must be specific to 3n+1 and must break for its near neighbours, which rules out every general technique in advance.
The heuristic picture makes it worse. On average an odd step multiplies by 3/2 and the following even steps divide by more, so orbits drift downward like a random walk with negative bias — which strongly suggests the conjecture is true and gives no purchase on proving it, because a single exceptional orbit is all it takes.
Erdős's assessment is still quoted because it is still accurate: mathematics is not yet ready for such problems.
What a breakthrough would mean
Nothing practical. No application depends on this.
What it would mean is a method for controlling the interaction between multiplication and addition on the integers — the thing that makes the arithmetic of 3n+1 so opaque. That is a genuine gap in mathematics, and it is why the problem retains serious attention despite being a magnet for cranks.
In this lab it is a deliberate long shot with a minority budget, kept for a different reason: the space of failed Collatz approaches is large and unusually well documented, which makes it the best available test of whether a library of recorded dead ends is useful at all. It was queued and never worked, so the honest status is that nothing has been tried here.
What this lab tried
Standing: Map First sweep done (001, swarm-executed): the lens taxonomy has weak purchase -- 17/22 reports collapse onto two classical families, both closed (residue-Lyapunov certificates killed at every finite memory by the -1 tower; cycle-equation sieving no purchase beyond known bounds). Parity-entropy barrier certified. 001's one nonstandard lead -- the graph-structure census -- is now CLOSED by 002: the map being a function forces every unlabelled graph invariant to be constant or closed-form, so the whole graph-decomposition family is priced out by a one-paragraph argument. Live leads are now the labelled/dynamical ones only: the escape-fraction anomaly (002 lead 2), survivor-decay and tilted-moment maps (001 leads 2-3).
Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.
Attempt 001informedMap
Swarm sweep (11 lenses x 2 stances, 22 gpt-5.6-luna workers, $0.084): 17/22 routes collapse to two classical families — residue-Lyapunov certificates KILLED at every finite memory (-1 tower, re-derived exact to L=200), cycle-equation sieve no purchase; parity-uniformity entropy barrier certified k<=16 after correcting a false worker claim; 3 leads, best: dominator/treewidth census of truncated Collatz digraphs.
- Open gaps
- Family-A kill covers bounded-lookahead finite-memory certificates over residue states only; magnitude-carrying or unbounded-adaptive certificates are untouched
- all literature attributions are [T]-grade worker/director memory; the lead-1 novelty claim (no dominator-theoretic Collatz work) needs the literature check before an attempt leans on it
Attempt 002informedRefuted
Queue item 21 CLOSED: all three unlabelled invariants on the truncated Collatz digraph are degenerate because the map is a function -- dominator tree IS the orbit tree (idom(v)=T(v), checked to B=2^18), treewidth is exactly 2 for every B, and the min cut is |S_k| = (2^{k-1}-(-1)^{k-1})/3 = J_{k-1} ~ 2^k/6, decided by the one-step inequality 3r+1>2^{k+1} with no dynamics in it; B drops out of the cut entirely. Closed form proved, machine-checked k<=22, max-flow k<=12, and reproduced by 3 independent gemini-3.7-flash implementations (one of which supplied a cleaner proof).
- Verifies
- attempt 001
- Open gaps
- The collapse is proved for UNLABELLED invariants on the forward functional digraph only; labelled/weighted objects on the same graph (transfer operators, 001 lead 3's tilted moments) are untouched and this is not evidence about them
- The escape-fraction by-product (non-monotone, drop at k=12->13, plateau ~0.404 to k=22) is measured and brute-force confirmed but unexplained; lead 2
- All literature attributions remain [T]-grade: the novelty pre-step was web search only, no PDFs read, so 'no dominator-theoretic Collatz work exists' is a negative search result, not a survey
Go deeper
Problem statement and published background · Full prior-art notes · Machine-readable index
Found something, or think we got it wrong?
Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.
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scripts/blind.sh collatz ../workMore on blind and informed mode.