A stick balanced on a cart that follows your hand. It shrinks, three centimetres a second, and the shorter it gets the faster it falls, until it falls faster than you can answer. The length where you drop it is your score, lower is better, and it is not really about the stick: it is about how late you see. A line from the theory reads your reaction off the length.
The cart chases your hand with a stiff spring and a damper, so what you control is where the base of the stick goes, as when you balance a broom on your palm. The stick is a uniform rod; its mass stays, its length goes. Today's world adds small gusts at the tip. The score is the stick's length, in centimetres, when it fell, so lower is better, and the floor is three. The rungs on the ladder are linear hands: four numbers on the state, read late by a fixed reaction and run on the same shrinking stick with today's gusts. The line in the corner is the best such hand at every reaction, found by search on this very model with no gusts at all; the rungs sit a little above it at the short end, because a hand near its limit forgives nothing. A fall between two rungs says what reaction a linear hand would need to drop it there, which is a fair reading of yours, screen and all. Runs by hand are not replayed by a link: your hands were the controller.
An inverted stick of length L falls with a time constant near √(2L/3g): a metre stick takes a quarter of a second to double its lean, a ten-centimetre one eighty milliseconds. Feedback has to act well inside that time, and yours cannot act at all inside your reaction, which is two hundred milliseconds or so for a thing you see. So there is a length below which every lean you notice has already grown past what your hand can undo, and no gains, no strategy, no practice moves it much, because it is set by the product of the unstable pole and the delay. The line makes that concrete: the shortest stick a linear hand holds against its own delay, and it climbs with the square of the delay. Halve your reaction and the stick shortens by four. That is the shape of every fundamental limit in this series: not a wall in the plant, but a wall in the pair of plant and information.
For a plant with one unstable pole p and a pure delay τ in the loop, the achievable margins shrink with the product pτ: the loop must cross over above p to catch the instability and below about 1/τ to keep its phase, and when those two frequencies meet there is nothing left to design with. Delayed feedback on a pendulum (the human balancing literature, Stépán and Milton among others) gives a critical length of order gτ²; the constant depends on what the hand can do, and here it is about two, found by trying every linear hand on this plant.
The lesson is not the constant but the exponent. Reaction enters squared, so the difference between a quick and a slow player is a stick four times as long, and the difference between a person and a 240 Hz loop with a few milliseconds of latency is a stick a hundred times shorter. Time Machine buys back a known delay with a model; nothing buys back an unknown one.
The sensitivity S = 1/(1 + L) says how much a disturbance gets through. Its logarithm, integrated over all frequencies, is a constant that no controller changes: zero for a stable plant, positive by π times the sum of the unstable poles otherwise. So any band where |S| < 1 (disturbances rejected) must be paid for by a band where |S| > 1 (disturbances amplified), and an unstable plant starts in debt.
The delay is what makes the debt unpayable. Without one the excess area can be spread thinly over high frequencies, where it costs nothing. A delay caps the bandwidth at about 1/τ, so the same area must fit under a lower ceiling, and it piles up until the peak of |S| is the whole margin. A short stick is a large p, a slow hand is a small ceiling, and the shortest stick is where the pile meets it.