This is not a proof and not a cipher. It is a way of seeing. Multiplying two numbers is easy; pulling them back apart — factoring — is hard, and that small asymmetry is the hinge the whole modern lock-and-key world hangs on. The Rig is a moving picture of that asymmetry. Two primes become the two legs of a triangle. The triangle is launched up the axes and grows. It blooms into circles. And somewhere in the growing and the blooming you can watch, with your eyes, why a number can carry a secret that even its owner can only open with a key they kept.
Everything here is honest about what it is: a visualization. Where it touches real cryptography, it says so. Where it is only a beautiful way to think, it says that too.
I — THE LAUNCH
Two rockets up the axes
The Rig is built by two clocks. One climbs the horizontal axis, the other the vertical, and each notches off one prime at a time — 2, 3, 5, 7, 11, 13 — like two rockets rising, leaving a vapor trail of primes behind them. The triangle is the span stretched taut between the two rising tips: two prime legs, and the long diagonal closing them. Its area is half their product — half a semiprime. That product is the public number; the two legs are the secret split that made it.
When the two rockets climb at the same thrust, the triangle stays balanced — tall as it is wide. When one outruns the other, it leans into a thin sliver. Watch the bloom: the same two legs spin out circles, one per side, and where the legs are equal the circles are equal and the flower is round.
II — THE SCAN
The shape is the difficulty
Now tick the Rig not by free primes but by the real factor-pairs of each semiprime in turn. Each beat the legs become the true factors of the next number — and the triangle's shape becomes a difficulty meter. Most semiprimes have a small factor, so most triangles are thin slivers: lopsided, easy to pull apart. But at the rare balanced ones — where the two primes are close — the triangle snaps fat and the flower blooms round.
Those round blooms are the strong numbers, the hard-to-factor ones. They fall where the primes run close together. The whole sequence becomes a landscape: a long run of thin sheared slivers, snapping occasionally to a fat round bloom exactly at the balanced semiprimes.
III — THE HINGE
Where the Rig is really a lock
Here is the one place the Rig stops being only a picture and touches a real, deployed cryptosystem. Take the triangle's two legs — two primes p and q — and form the area, the public number N = p·q. Square any number and wrap it around N — the map x² mod N — and you have the Rabin function, a genuine one-way lock whose hardness is provably equal to factoring.
And the Rig's parts map onto it exactly. The four-fold symmetry of the bloom is the four square roots the lock always has. And the hinge — the move that pins one vertex and stands the flattened triangle back up — is the trapdoor: given two of those roots, a single greatest-common-divisor splits N back into p and q. The hinge that un-flattens the triangle is the secret that un-multiplies the number. Same move, two languages.
The Rig builds the key, the squaring is the lock, and the hinge is the trapdoor — the move that stands the flat thing up, available only to whoever kept the two legs.
So the whole machine is one honest picture of a real system: choosing two balanced primes is key generation (that is the scan); flattening them into one public number is the modulus; squaring mod that number is the lock; and the hinge — swing one leg into a circle and back — is the private trapdoor. The secret was never the number. It was always the two legs, and the hinge that remembers them.
IV — THE SHADOW
The leg you can see but never stand on
Stand the Rig a different way: let N be the long side — the hypotenuse — and swing the angle until one leg falls exactly on a true factor p. You can do it; the leg lands clean on a whole prime. But the moment it does, the other leg is forced to become p·q̆, where q̆ = √(q²−1) — an irrational that falls just short of the second prime q and never reaches it.
It earns its own mark. Write it q̆ — q with a breve, the symbol for short — because that is exactly what it is: the prime q, shortened. It can never be a whole number, because q̆² = q² − 1, and no two perfect squares differ by one except zero and one. So the Rig can see the second prime — it sits right there in the figure — but can never stand a leg on it. One factor caught; the other only shadowed, held at the breadth of a breve.
The triangle still closes on N exactly, by a clean cancellation: (p·q̆)² + p² = N². The minus-one buried inside q̆ and the small leg p² undo each other perfectly — that cancellation is the triangle closing. The price of standing on one prime is a shortfall of exactly p² in the other leg: the integer you gain on one side is the irrationality you owe on the other, and they balance to N² every time.
The whole secret, in N alone
Now take away the factors and keep only what anyone is given — the number N itself. The triangle stops being a single shape. The one rule you can still write is a² + b² = N², and every angle satisfies it: each θ gives a legal right triangle with legs a = N·cosθ and b = N·sinθ. In terms of N alone, the triangle is not a triangle at all. It is a circle — a quarter-circle of radius N, every point on it a different possible split.
The factorization is one angle on that circle — the rare θ where a leg falls exactly on a divisor of N (the red dots). But nothing about the circle points to them. To find them you must swing, and test, angle by angle. N gives you the circle. It withholds the angle. That missing angle, on a circle anyone can draw, is the whole of the wall — and q̆ is the shadow every other angle casts, the second prime shortened, present in the figure and absent from the integers.
N hands you the circle and keeps the angle. The factors are there — one caught, one shadowed — but the turn that finds them is the secret.
V — THE LEAP
A prime is a leap, not a place
One last way of seeing, and it is the strangest. To travel from one prime to the next you would have to cross the continuum between them — and between any two points there are infinitely many more. Taken literally, the rocket never arrives; this is Zeno's old trap. So the Rig does not travel. It leaps. Each prime is not a place the rocket crawls to but the amplitude of an instant jump — a kick of that size, a measure of resistance cleared in no time at all.
You measure the toll, not the road. The road between primes is infinite; the toll — the gap, the resistance — is finite, and that is what you pay. The rocket tunnels: it appears at the next prime without ever crossing the distance, the way a struck bell does not travel to its note but simply rings. The motion between the ticks is not lost. It never existed. There were only the kicks.
And because each Rig, flattened, takes up no depth, you can stand infinitely many of them side by side — a deck of flat states, every semiprime, every setting, every moment of the climb, laid out at once and read across in a single glance. The same flatness that makes the wall strong leaves the third dimension free to compare. You do not move through the states. You lay them down, and look.
You measure the toll, not the road. The road is infinite; the toll is finite — and the leap is free.