Feed energy into a network at some dots and drain it out at others. While gain and loss balance, every zero that counts stays on the ½ line. Push past the breaking point and they leave it in mirror pairs. Watch the network, its zeros, and its waves together.
Write each connection as two dot numbers joined by a dash, such as 1-2, 2-3, 3-1. A line like 4: 5 6 7 connects dot 4 to 5, 6 and 7. Add gain: 1 2 3 and loss: 4 5 6 lines to mark dots; without them every dot starts as neither. Bulk forms such as 1(primes) join one dot, or a whole set of dots, to a whole set at once (see below). Dot numbers start at 1, the lab lays the dots out for you, and a world written out from here carries a layout: line so it comes back as you left it. The same list works in the Swing Lab.
1(primes) joins dot 1 to every prime. primes(primes) joins all the primes to each other, even(odd) every even dot to every odd one, all(x+1) every dot to the next, all(2x) every dot to its double. Inside the brackets you can write a word (primes, composites, even, odd, all, squares, cubes, fib, tri, pow2, twins), a rule such as 3n or 3n+1, a range such as 5-12, a list such as 2,3,5, or x+1, 2x, x^2 for “relative to the dot on the left”; & filters, as in primes(x+2 & primes) for the twin primes. The left side takes the same words, so 3n+1(primes) works too. The words reach as far as the largest dot number in the list; add a line dots: 60 to set it.
Open the two Petersen graphs and slide the strength up slowly. The zeros creep along the ½ line but stay on it. At exactly 1, the strength of the connections joining the two worlds, every pair leaves the line at once.
Then pick the Petersen graph with one gain dot and one loss dot. Press “Find the breaking point”. The balance breaks the moment any gain is added. The Petersen graph has swings that repeat, and a lone pair of gain and loss splits them apart straight away. Joining two whole worlds spreads the energy evenly, and that is what holds the balance.
In the wave field, pick “A pulse from the picked dot” and press Play. Below the breaking point the energy sloshes back and forth between the worlds and stays bounded. Above it, the energy grows without limit.
Then open the E8 root network, the largest world here. Its zeros are already a strange sight with no gain at all: sixteen sit off the line, and the rest pile up at just three heights, because E8 has only five different swings. Press “Find the breaking point” (it takes a few seconds for 240 dots): the balance holds up to strength 1 and breaks there, just as the two joined worlds do.
This is a standard model from physics: a network with balanced gain and loss, the kind built with lasers and coupled circuits. Its zeros are worked out the same way as in Zeta Worlds, with the gain and loss added at each dot.
It shows how balance can hold zeros on a line, and how that balance can snap. It is not zeta. Bender, Brody and Müller proposed a balanced system whose notes would be zeta’s zeros in 2017. That proposal has gaps, and it is not accepted as a proof. Showing the balance can never break is the Riemann Hypothesis in another form.
Everything is computed in your browser. Each dot’s swing comes from the table of connections with gain added as +i·strength and loss as −i·strength on the dot itself. For a world where every dot has the same number of connections, each swing gives two zeros from zero² − swing × zero + (connections − 1) = 0, and the ½ line is fixed by the world without gain or loss. Other worlds use the full no-turning-back calculation. The main term is marked at the swing with the largest real part. The trivial zeros at +1 and −1 are left out of the picture. The wave field follows how a wave changes in time when the swings are its frequencies: bars show the real part of each dot’s amount, or its size squared.