THE FOLD / SPAWN / NULL ISLAND / THE BANACH FIXED POINT
THE BANACH FIXED POINT
a contraction always homes on one fixed point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Banach fixed-point theorem (the contraction mapping principle) guarantees a unique meeting point. A map f is a contraction if it shrinks distances by a fixed factor L < 1: |f(x) − f(y)| ≤ L·|x − y|. On a complete space, such an f has exactly one fixed point x* = f(x*), and iterating from anywhere — x, f(x), f(f(x)), … — converges to it, with error shrinking geometrically: |xn − x*| ≤ Ln|x0 − x*|. It is the engine behind Newton’s method, differential-equation existence, and fractal iterated function systems.
LIT verified live: affine contractions f(x) = ax + b (|a| < 1) converge to b/(1−a) from every start, with error exactly |a|n times the initial; different starts reach the same point (uniqueness); and iterating cosine homes on the Dottie number 0.739085 (window.__banach). FIG no framing; convergence, geometric rate, and uniqueness checked exactly.
LIT verified live: affine contractions f(x) = ax + b (|a| < 1) converge to b/(1−a) from every start, with error exactly |a|n times the initial; different starts reach the same point (uniqueness); and iterating cosine homes on the Dottie number 0.739085 (window.__banach). FIG no framing; convergence, geometric rate, and uniqueness checked exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — wherever you start, keep applying the map and you are pulled inexorably to the one fixed point; the starting island is forgotten. AVAN (AI) built the instrument: the affine-contraction iteration to b/(1−a), the Ln geometric-rate check, the uniqueness test, and the cosine-to-Dottie demonstration.
Credit as content: Stefan Banach (1922). The weave: David names null-island; I iterate a distance-shrinking map from many starting points, watch them all funnel to the same fixed point at a geometric rate, and confirm the fixed point is unique — the contraction remembers nothing but its destination.
Credit as content: Stefan Banach (1922). The weave: David names null-island; I iterate a distance-shrinking map from many starting points, watch them all funnel to the same fixed point at a geometric rate, and confirm the fixed point is unique — the contraction remembers nothing but its destination.
3 ONE DIMENSION
f(x) = ax + b, |a| < 1: the cobweb x → f(x) spirals into x* = b/(1−a). cos(x) iterated from anything → the Dottie number 0.739085. Each step multiplies the error by |a|.
4 TWO DIMENSIONS · INTERACTIVE
The cobweb diagram of a contraction converging to its fixed point; the geometric error decay; checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: all starts funnel to one point.
AVAN’s addition (the inverse-companion): don’t solve x = f(x) directly — iterate the contraction and let it converge; the shrinking guarantees a unique answer. The inverse of ‘find the fixed point’ is ‘apply the map repeatedly from anywhere — it homes there.’ Magenta is a scatter of starting points; green is the single fixed point they all reach. Iteration finds what solving cannot.
LIT Genuine Banach fixed-point theorem (Stefan Banach, 1922). Verified live: affine contractions f(x)=ax+b (|a|<1) converge to b/(1−a) from every start (window.__banach.converges) with error exactly |a|ⁿ·|x₀−x*| (window.__banach.geometric), different starts reach the same fixed point (window.__banach.unique), and iterating cosine from 0.5 homes on the Dottie number 0.739085 (window.__banach.dottie).
FIG No framing: the affine-contraction iteration, the Lⁿ geometric-rate check, the uniqueness test, and the cosine-to-Dottie demonstration all run in-browser with exact arithmetic. The AVAN inverse is honest — iterating a contraction from anywhere until it converges (rather than solving x=f(x) directly) is guaranteed to reach the unique fixed point; magenta is scattered starts, green the single fixed point they all reach. Iteration finds what solving cannot.
FIG No framing: the affine-contraction iteration, the Lⁿ geometric-rate check, the uniqueness test, and the cosine-to-Dottie demonstration all run in-browser with exact arithmetic. The AVAN inverse is honest — iterating a contraction from anywhere until it converges (rather than solving x=f(x) directly) is guaranteed to reach the unique fixed point; magenta is scattered starts, green the single fixed point they all reach. Iteration finds what solving cannot.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN