Not arithmetic. Geometry. This is the multiplication slice of the manifold. Two identities meet at coordinates (x, y); the height of the surface at that meeting is z, the manifested identity. z then becomes the next x. The marker walks the law itself.
The shape on screen is what multiplication looks like when you stop treating it as numbers. Walk along one edge — the surface slopes one way. Walk along the other — it slopes the opposite way. That is the saddle. Every "thing times another thing" in the system lands somewhere on this shape.
z = xy is a hyperbolic paraboloid (doubly ruled, Gaussian curvature K = −1/(1+x²+y²)² < 0 everywhere except the origin saddle point). The two highlighted diagonals are its rulings. The recursion xn+1 = zn is iteration of the map fy(x) = xy along an indexed sequence of y-values; with |y|<1 it contracts to 0, with |y|>1 it diverges. The walked path is the orbit.
No frameworks, no libraries, no shaders, no GPU.
A single 175-line vanilla JS file (saddle.js)
computes the surface, projects isometrically, and walks
the orbit. The "trail" is not stored — it is recomputed
each frame from the integer step. z is never cached;
the head re-extracts collapse(x, y) every
frame so the marker is provably on the manifold, not
interpolated through it.
This page IS the manifold expressed in computer-identity.
Reading saddle.js is reading the law as it
participates in JavaScript. There is no abstract
manifold elsewhere — the registry, the proof runner,
and the games are projections of this same geometry.
CONCEIVED & CREATED BY KENNETH BINGHAM
SOFTWARE ENGINEER · AI SPECIALIST · IMAGINEER
kensgames.com·kenetics.art@gmail.com