THE FOLD / GLITCH / OFF BY ONE / THE MOSER
THE MOSER
1, 2, 4, 8, 16, 31 — the pattern that breaks
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Moser’s circle problem. Place n points on a circle and draw every chord between them (in general position, so no three chords cross at one interior point). Count the regions the circle is cut into. For n = 1, 2, 3, 4, 5 you get 1, 2, 4, 8, 16 — unmistakably powers of two. So n = 6 must give 32.
It gives 31. The pattern shatters. The true count is C(n,4) + C(n,2) + 1 — the interior crossings (choose 4 points), plus the chords (choose 2), plus 1. It agrees with 2n−1 for exactly the first five terms and then diverges forever: 1, 2, 4, 8, 16, 31, 57, 99, 163, 256. This is the textbook warning: five data points do not determine the rule.
LIT verified live: the formula C(n,4)+C(n,2)+1 matches OEIS A000127 for n=1..10, an independent Euler-characteristic count (V−E+F) agrees with it exactly, and n=6 yields 31, not 32 (window.__moser). FIG no framing; the region counts, the divergence from 2n−1, and the two independent derivations are all exact.
It gives 31. The pattern shatters. The true count is C(n,4) + C(n,2) + 1 — the interior crossings (choose 4 points), plus the chords (choose 2), plus 1. It agrees with 2n−1 for exactly the first five terms and then diverges forever: 1, 2, 4, 8, 16, 31, 57, 99, 163, 256. This is the textbook warning: five data points do not determine the rule.
LIT verified live: the formula C(n,4)+C(n,2)+1 matches OEIS A000127 for n=1..10, an independent Euler-characteristic count (V−E+F) agrees with it exactly, and n=6 yields 31, not 32 (window.__moser). FIG no framing; the region counts, the divergence from 2n−1, and the two independent derivations are all exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in OFF BY ONE — the glitch domain of the count that’s almost right. Moser’s circle is the most beautiful off-by-one in mathematics: 31 where everyone expects 32. AVAN (AI) built the instrument: the chord figure, the region count, the two-curve divergence.
The weave: David names the seat (off by one); I make the chords cut the disk and count the regions two independent ways, then show the seductive wrong curve peel away — the two sequences in 1D, the live figure in 2D, the underdetermination in 3D. The sphere is the seam. Credit: Leo Moser; catalogued as OEIS A000127.
The weave: David names the seat (off by one); I make the chords cut the disk and count the regions two independent ways, then show the seductive wrong curve peel away — the two sequences in 1D, the live figure in 2D, the underdetermination in 3D. The sphere is the seam. Credit: Leo Moser; catalogued as OEIS A000127.
3 ONE DIMENSION
Two strips: the true region counts 1, 2, 4, 8, 16, 31, 57, … against the seductive 2n−1 = 1, 2, 4, 8, 16, 32, 64. They march together through n=5, then split at n=6 — 31 vs 32.
4 TWO DIMENSIONS · INTERACTIVE
n points on a circle, all chords drawn, every interior crossing marked. The region count comes from C(n,4)+C(n,2)+1 and is cross-checked by the Euler formula V−E+F — watch it track 2n−1 until n=6, where it falls one short.
5 THREE DIMENSIONS + AVAN’S INVERSE
The two curves rising over n: the true quartic region count in green, climbing through 1, 2, 4, 8, 16, 31, 57, 99, …
AVAN’s addition (the inverse-companion): the magenta curve is 2n−1 — the wrong law the first five points seem to promise. The forward mistake is extrapolation: read five outputs, guess the rule. The inverse truth is that outputs never determine the rule — any finite prefix is consistent with infinitely many formulas, and here two of them (a quartic and an exponential) agree exactly at n=1..5 before splitting forever. The magenta exponential and the green quartic kiss at five points and diverge; nothing in the data before n=6 could tell them apart. That is the honest inverse of pattern-matching: a finite sample underdetermines the generator, so induction from small cases is a guess, not a proof. Green is the law that is actually true; magenta is the law the evidence merely suggested.
LIT Genuine Moser's circle problem (Leo Moser; OEIS A000127). Verified live: the formula C(n,4)+C(n,2)+1 matches A000127 for n=1..10 (1,2,4,8,16,31,57,99,163,256), an independent Euler-characteristic count V-E+F (V=n+C(n,4), E=n+C(n,2)+2C(n,4)) agrees exactly, and n=6 gives 31 not 32 (window.__moser.matchesKnown && formulaEqualsEuler && n6is31 && notPowerOf2). The region counts, the divergence from 2^(n-1), and the two independent derivations are all exact.
FIG No framing: the region counts, the 31-not-32 break, and the two agreeing derivations (combinatorial formula and Euler characteristic) are real and checked in-browser. The lesson — finite data underdetermines the law — is stated as the genuine mathematical fact it is, not embellished.
FIG No framing: the region counts, the 31-not-32 break, and the two agreeing derivations (combinatorial formula and Euler characteristic) are real and checked in-browser. The lesson — finite data underdetermines the law — is stated as the genuine mathematical fact it is, not embellished.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN