Take any odd number x — negative numbers allowed — multiply by 3, add 1, then divide by 2 until the result is odd again. Call that one step (the map T). A loop is a number that returns to itself after k steps. Exactly four loops are known. This page is about the arithmetic that anchors each of them to a near-miss between the powers of 2 and the powers of 3 — and about why the easy supply of loops is a finite stock, already known to be spent in the fourteenth century. Everything marked live is computed in this page in exact integer arithmetic as you interact; the stated result numbers are transcribed from the project's verification record. Companion to Reduced coordinates for the Collatz map (DOI 10.5281/zenodo.21421120) and The 3-adic mirror of the reduced Collatz dynamics (DOI 10.5281/zenodo.21303918).
One colour per loop, everywhere: the same colour marks a loop's orbit, its anchor on the ladder, and its rows in the exhaustive map. Hovering a loop anywhere highlights it everywhere.
Why near-misses? A loop that returns to itself after k steps has multiplied by 3-and-a-bit k times and halved m times in all, and closing exactly forces an exact balance: x · q = R, where q = 2m − 3k measures how nearly the two towers collide and R is an integer the step pattern determines. The towers never meet exactly — q = 0 is impossible — so no “perfect” loop exists: every loop must ride a near-miss, and must win the divisibility q | R. At |q| = 1 that win is free, since 1 divides everything; otherwise it is a lottery. That is why the ladder below is this page's first picture.
Powers of 2 and powers of 3 never collide — every power of 3 is odd, and the only odd power of 2 is 1 — but they have near-misses. For each k, the value 3k lands strictly between two consecutive powers of 2. The bar drawn between them is that gap, and 3k splits it into a lower part q₋ = 3k − 2⌊k·log₂3⌋ and an upper part q₊ = 2⌈k·log₂3⌉ − 3k. The two parts fill the gap exactly, and their sum is exactly the lower power of 2 — the envelope identity q₊ + q₋ = 2⌊k·log₂3⌋, checked live below at every k shown. This violet bar is q's first costume; the same object returns twice more, in panels 3 and 4.
Three times in all of history the gap part is exactly 1: 2–3 (at k = 1, from below), 4–3 (at k = 1, from above), and 9–8 (at k = 2, from below). Those three are drawn in the colours of the loops they anchor — panel 6 explains why a gap part of 1 hands out a loop for free. At k = 7 the lower part is 139: not 1, but the anchor of the −17 loop, the one lottery win. And k = 1 is the unique exact tie: q₊ = q₋ = 1 — the only k at which neither side is nearer.
Here are the four known loops, drawn by actually running the map in this page. Each is labelled with its anchor data (k, m, q): k steps, m halvings in total across the loop, and q = 2m − 3k — the same gap number as on the ladder above, now attached to a loop. The arrows say what each step does: ×3+1, then the number of halvings written on the arrow.
The moral is worth stating flatly: loops are real. Any argument of the form “the map tends to shrink, so loops cannot exist” — any parity or average-speed argument — proves too much, because it would forbid these four. What actually decides whether a candidate loop exists is a divisibility fact about q, which the next panel shows.
A candidate loop is described by its step pattern: the word (s₁, …, sₖ) listing how many halvings follow each ×3+1. The word fixes m = s₁+…+sₖ and so q = 2m − 3k, and for each rotation r — the same pattern read from a different starting point — an integer Rr. A genuine loop must satisfy xr · q = Rr at every rotation, so q must divide every Rr: apparently k separate conditions. They are one condition. The exact identity 2sᵣ·Rr+1 = 3·Rr + q (the transport recurrence — Remark 12.6.1.1 in the project record, ledger entry L-A1) carries divisibility by q from each rotation to the next. Watch the verdict column: it agrees at every rotation — all ✓ or all ✗ — whatever word you try.
What about taking a loop's word and repeating it — going round twice, three times, five times? The repeated word passes the divisibility test whenever its base does, but it never buys anything new. That is a proved exact law (ledger entry L-A2): for P = Bj with j ≥ 2, gcd(qP, R₀) = |qP| / qred(B), and this is forced to be greater than 1 — so a repeated word is divisible exactly when its base is, and then it merely retraces the base's loop j times. Below, the repeated word wears its base's colour: it belongs to that house, not to a new one. In the exhaustive k ≤ 10 map of panel 6, 384 repeated words occur, and every one obeys this law.
The default demo is the k = 5 positive sector's one false alarm: the +1 word (2) repeated five times gives qP = 781 and gcd = 781 — it passes divisibility, and what it reconstructs is the number 1 walking its one-step loop five times, not a new loop.
Back to the ladder. At each k the near-miss has two sides: is 3k closer, in plain distance, to the power of 2 below it (q₋ smaller) or above it (q₊ smaller)? The negative world's loops anchor from below, the positive world's from above — so this is a question about which sign of the map gets the better near-misses. Plotting the fractional part {k·log₂3} for each k gives the walk below: the points never repeat and fill the interval evenly, and the lower side wins exactly when the point falls below the threshold log₂(3/2) = 0.5849625007… — about 58.5% of the time. That is a Benford-type law: “the negative side wins” is the same event as “the second binary digit of 3k is 0”. Every dot's side below is decided by exact integer comparison, not by the plotted position.
In the verification record the count is exact to k = 105 (58,496/99,999 = 0.5849658) and measured at k = 107 as 0.5849626, against the constant 0.5849625007. The toggle asks the other natural question — which side is closer in ratio — and there the split is 50/50. Both answers are right — to different questions. Divisibility lives on q as an integer, so the additive one is the cycle-relevant one. k = 1 is the unique exact tie (q₊ = q₋ = 1), so the law reads “for k ≥ 2”.
When a gap part is exactly 1 — |q| = 1 — the divisibility condition of panel 3 is free: 1 divides everything, so every step pattern with that (k, m) closes into a genuine loop. That free stock is exactly three tickets, all dealt by k = 2: one north, two south. And there is never another: |2a − 3b| = 1 has no further solutions — by a remainder-mod-8 argument plus a two-line factoring, arithmetic available in the fourteenth century.
The k ≤ 10 strip is a verification exhibit — the story confirmed completely for every k ≤ 10 — not a proof method: k has no ceiling. What remains is exactly the open question, and it needs the two ingredients coupled — the two-towers geometry of panels 1 and 5, and the divisibility arithmetic of panels 3 and 4. No finite piece of arithmetic alone settles it, and this page claims nothing beyond what is shown.