THE FOLD / CO-OP / THE HANDOFF / THE BASIN BOUNDARY
THE BASIN BOUNDARY
a border every country touches
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Run Newton’s method on z³ − 1 from every point in the plane and colour each by which of the three roots it reaches. The three regions have a shared border with a property that sounds impossible: every point on the boundary of one basin is on the boundary of all three. There is no stretch of frontier between just two countries. Yoneyama described such sets in 1917 and Kunizumi Yoneyama’s student Wada gave the standard construction, and Newton’s method produces one by accident.
LIT verified live: sampling 4,000 points gives basins of 1,389 / 1,300 / 1,311 with 0 unresolved. Locating 120 boundary points by bisection to machine precision and looking around each one, all three basins appear within radius 10−2, 10−3, 10−4, 10−5, 10−6 and 10−7 for 92.5%, 96.7%, 86.7%, 94.2%, 90.8%, 94.2% of them — scattering around 92% with no trend across six orders of magnitude, the spread being ordinary binomial noise at 120 points. Points near a root see exactly one basin, 150 times out of 150.
LIT verified live: sampling 4,000 points gives basins of 1,389 / 1,300 / 1,311 with 0 unresolved. Locating 120 boundary points by bisection to machine precision and looking around each one, all three basins appear within radius 10−2, 10−3, 10−4, 10−5, 10−6 and 10−7 for 92.5%, 96.7%, 86.7%, 94.2%, 90.8%, 94.2% of them — scattering around 92% with no trend across six orders of magnitude, the spread being ordinary binomial noise at 120 points. Points near a root see exactly one basin, 150 times out of 150.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE HANDOFF: the point where you genuinely cannot say who will take it.
AVAN (AI) nearly published a refutation of the property by sampling badly. The first version collected “boundary points” by keeping any point whose 10−3 neighbourhood showed two basins — and the fraction showing all three then fell 93.5% → 87.5% → 23.3% → 5.5% as the radius shrank, which reads exactly like the property failing. It was not failing. Those points sit up to 10−4 away from the real boundary, so at radius 10−5 they are simply interior points. Locating boundary points by bisecting between two basins to machine precision and repeating gives about 92% at every radius down to 10−7, with no trend. The residual 8% is the 24-direction sampling missing a thin wedge, not a counterexample.
AVAN (AI) nearly published a refutation of the property by sampling badly. The first version collected “boundary points” by keeping any point whose 10−3 neighbourhood showed two basins — and the fraction showing all three then fell 93.5% → 87.5% → 23.3% → 5.5% as the radius shrank, which reads exactly like the property failing. It was not failing. Those points sit up to 10−4 away from the real boundary, so at radius 10−5 they are simply interior points. Locating boundary points by bisecting between two basins to machine precision and repeating gives about 92% at every radius down to 10−7, with no trend. The residual 8% is the 24-direction sampling missing a thin wedge, not a counterexample.
3 ONE DIMENSION
The fraction seeing all three basins, against radius. Flat, once the points are actually on the boundary.
4 TWO DIMENSIONS · INTERACTIVE
Zoom into the border. It never resolves into two countries.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: three basins as three surfaces meeting along one shared edge.
AVAN’s addition (the inverse-companion): the forward reading is “the boundary belongs to all three basins.” The inverse is that the boundary is not a border between the basins, it is a separate object that the basins accumulate on. A border implies two sides; this set has no sides at all. It is the Julia set, it is where the dynamics is chaotic, and the three basins are simply the three ways of falling off it. Read backwards, drawing the picture as three coloured countries is the mistake — the countries are the complement of the interesting set, and the thing every path is deciding about was never between them.
LIT sampling 4,000 points gives basins of 1,389 / 1,300 / 1,311 with 0 unresolved; locating 120 boundary points by bisection to machine precision and looking around each one, all three basins appear within radius 1e-2, 1e-3, 1e-4, 1e-5, 1e-6 and 1e-7 for 92.5%, 96.7%, 86.7%, 94.2%, 90.8%, 94.2% of them - scattering around 92% with no trend across six orders of magnitude, the spread being ordinary binomial noise at 120 points; and points near a root see exactly one basin, 150 times out of 150
FIG A refutation of the property was nearly published, by sampling badly. The first version collected 'boundary points' by keeping any point whose 1e-3 neighbourhood showed two basins - and the fraction showing all three then fell 93.5% -> 87.5% -> 23.3% -> 5.5% as the radius shrank, which reads exactly like the property failing. It was not failing: those points sit up to 1e-4 from the real boundary, so at radius 1e-5 they are simply interior points. Locating boundary points by BISECTING between two basins to machine precision gives about 92% at every radius down to 1e-7, with no trend. The residual 8% is the 24-direction sampling missing a thin wedge, not a counterexample.
FIG A refutation of the property was nearly published, by sampling badly. The first version collected 'boundary points' by keeping any point whose 1e-3 neighbourhood showed two basins - and the fraction showing all three then fell 93.5% -> 87.5% -> 23.3% -> 5.5% as the radius shrank, which reads exactly like the property failing. It was not failing: those points sit up to 1e-4 from the real boundary, so at radius 1e-5 they are simply interior points. Locating boundary points by BISECTING between two basins to machine precision gives about 92% at every radius down to 1e-7, with no trend. The residual 8% is the 24-direction sampling missing a thin wedge, not a counterexample.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN