Attractlet attractlet model · § 7.4

Newton’s Method

A root-finder whose answers pull in every nearby guess, and pull a nudged guess back, for exactly as long as someone keeps running it.

Newton’s method finds where a function reaches zero. Start from a guess, follow the tangent line to where it crosses zero, and repeat: z → z − f(z)/f′(z). Near a root the guesses rush in. Nudge a settled guess a little and the next steps bring it back. Every root has a basin. By surface measurement this passes for an attractor: bounded, convergent, self-correcting. But the guesses produce none of it. The pull is the formula, applied from outside, one step at a time. Stop the solver and every guess stays exactly where it is.

Three Roots, Three Basins

f(z) = z³ − 1 on the complex plane

settled on a root—

largest move, last step—

solverrunning

converging

The pull is the formula’s, not the guesses’. Each step is computed and applied from outside. The basins are real, and a nudged guess really does return, but only while the solver runs. Stop it and the guesses freeze wherever they are, at rest or mid-flight. It is the numerical twin of the clock-driven pattern. Background colour: which root each point of the plane reaches. White rings: the three roots. The motion between steps is drawn smoothly for the eye; the guesses themselves change only once per step.

Why This Is an Attractlet

the kernel’s own classification

The kernel sorts attractlets into kinds. One is the Attractlet Model (Formal or Symbolic): a formal, mathematical, or symbolic representation that behaves like an attractor within its formalism, but whose persistence is “sustained by the computational, mathematical, or symbolic substrate,” which is itself sustained by sovereign attractors: the mathematicians, computers, or formal systems running the model. The kernel counts numerical solvers in this kind by name (ATTRACTLET, § 7.4).

Newton’s method is a numerical solver. Its roots do not hold the guesses; the next application of the formula moves them. Nothing in the picture produces that application. Take it away and there is no pull left to find.

The Shape of the Basins

elegance is not evidence

The border between two basins is not a clean line. Wherever two colours meet, the third is there too, at every magnification, which is why the picture is a fractal. It is a famous and beautiful picture.

None of that moves the classification. § 7.4 states it directly: “No attractlet may be promoted to attractor status by complexity, longevity, apparent stability, mathematical elegance, or external coupling strength.” A basin shows that a structure returns. It cannot show where the return comes from. Both an attractor and an attractlet have one.

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