origins

The digit function is the floor quotient from Euclidean division. It maps remainders to digit bins. Everything in the collision program starts here.

Change the base and the field changes with it. But the complement symmetry does not. Something underneath is base-independent.

Field glyphs sort the integers into three visible categories. The geometry is base-independent. The structure was there before the formalism.

The simplest self-referential equation produces the slowest-converging continued fraction and the threshold that separates the prime 3 from all others.

The integers expand outward through each range and contract back along the mirror path. The breathing pattern repeats at every magnification. The complement map is already here.

Multiplication tables on the circle of nine reveal mirror symmetries that persist at every scale. The palindromic structure is built into the place-value system.

In 2009, I drew a circle with nine positions and watched where the primes landed. The classes paired. The complements avoided each other. I called it polarity.