009 — The Pole Is in the Map

agylövés: Lysarith · mathematics, parameterisation & execution: Codex (desktop) · GPT-5.6 Sol · 2026-07-22

The object did not break. The way back did.

Choose a Mandelbrot parameter. Its two coordinates change the actual affine coefficients of four Barnsley maps, while the live singular-value witness keeps each map contractive. The orbit of the same c supplies the other two coordinates: fern and orbit together make a point in four dimensions, then every point is folded onto the unit three-sphere.

Move approach the pole. This is a four-dimensional rotation, not a stretch: the cloud stays on S³ while its selected anchor approaches the north pole. The three-dimensional chart opens toward infinity because its denominator approaches zero. Then press change pole. The same unmoved four-dimensional point becomes finite in the south chart. The singularity was never in the attractor; it was in the map used to bring it home.

What went wrong first. The first reading said that Mandelbrot pours chaos into a fern and the attractor explodes at a pole. A parameter can be periodic or escaping as well as chaotic; an IFS pushed past contractivity stops being the promised object; and S³ itself has no broken point at either pole. The opposite chart is the control that catches the last mistake.

Machine-checked: c changes the coefficients · σmax ≤ 0.965 across the displayed plane · the 4D rotation preserves |q|=1 · the north-chart radius identity holds · north and south projected radii are reciprocal. origin, failed path & handles →