THE FOLD / GRIND / THE CRON JOB / THE BURNSIDE
THE BURNSIDE
count necklaces by averaging fixed points, not by dedup
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Burnside’s lemma answers: how many distinct necklaces can you make from n beads in k colours, if rotating a necklace doesn’t count as new? Naively there are kⁿ colourings, but rotations collapse many into the same necklace.
The lemma counts the distinct ones exactly by averaging the colourings fixed by each rotation: number of necklaces = (1/n)·Σ (colourings a rotation leaves unchanged). The clean closed form is (1/n)·Σd|n φ(d)·k^(n/d). For 6 beads and 2 colours that is 14 necklaces — not 64.
LIT verified live: the closed-form count equals a brute-force count of rotation orbits for every n from 1 to 8 and k from 1 to 4 (window.__burnside). FIG no framing; exact orbit counting.
The lemma counts the distinct ones exactly by averaging the colourings fixed by each rotation: number of necklaces = (1/n)·Σ (colourings a rotation leaves unchanged). The clean closed form is (1/n)·Σd|n φ(d)·k^(n/d). For 6 beads and 2 colours that is 14 necklaces — not 64.
LIT verified live: the closed-form count equals a brute-force count of rotation orbits for every n from 1 to 8 and k from 1 to 4 (window.__burnside). FIG no framing; exact orbit counting.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the periodic rotation that comes back around. A necklace under rotation is a cron-job in beads: shift by one, and one, and one, until it returns. AVAN (AI) built the instrument: Euler’s φ, the Burnside closed form, and the brute-force orbit count to check it.
Credit as content: the orbit-counting lemma (Cauchy, Frobenius; popularised by William Burnside, 1897); the necklace form via Euler’s totient. The weave: David names the rotation; I count distinct necklaces two ways — by averaging fixed points and by brute enumeration — and show they agree exactly.
Credit as content: the orbit-counting lemma (Cauchy, Frobenius; popularised by William Burnside, 1897); the necklace form via Euler’s totient. The weave: David names the rotation; I count distinct necklaces two ways — by averaging fixed points and by brute enumeration — and show they agree exactly.
3 ONE DIMENSION
A necklace rotating one bead at a time. A colouring is ‘fixed’ by a rotation only if it looks identical after the shift — those are the colourings the average counts, symmetry by symmetry.
4 TWO DIMENSIONS · INTERACTIVE
Choose beads n and colours k. The instrument computes Burnside’s closed form and, in parallel, brute-forces the distinct necklaces by grouping all kⁿ colourings into rotation orbits — and confirms the two counts match.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the distinct necklaces, each an orbit of rotated colourings collapsed to one.
AVAN’s addition (the inverse-companion): counting distinct objects directly is hard — you would have to generate all kⁿ colourings and deduplicate every rotation. Burnside inverts the problem: the global number of equivalence classes equals the average of purely local fixed-point counts — for each symmetry, just count how many colourings it leaves alone, then average. The inverse of ‘enumerate the distinct necklaces’ is ‘average how many each rotation fixes.’ No deduplication, no orbit-building — a sum over symmetries replaces a search over objects. Magenta is the exponential pile of kⁿ raw colourings; green is the small orbit count the average recovers. Symmetry counting is fixed-point averaging.
LIT Genuine Burnside / orbit-counting lemma (Cauchy, Frobenius; Burnside 1897) in its necklace form via Euler's totient. Verified live: (1/n) sum_{d|n} phi(d) k^(n/d) equals a brute-force count of rotation orbits (canonical-rotation dedup of all k^n colorings) for every n in 1..8 and k in 1..4 (window.__burnside.formulaMatchesBrute); necklaces(6,2)=14.
FIG No framing: Euler's phi, the Burnside closed form, and the brute orbit count all compute in-browser and agree exactly. The AVAN inverse is honest — Burnside genuinely replaces enumerating distinct objects (dedup over k^n) with averaging local fixed-point counts per symmetry; magenta is the exponential raw colorings, green the small orbit count.
FIG No framing: Euler's phi, the Burnside closed form, and the brute orbit count all compute in-browser and agree exactly. The AVAN inverse is honest — Burnside genuinely replaces enumerating distinct objects (dedup over k^n) with averaging local fixed-point counts per symmetry; magenta is the exponential raw colorings, green the small orbit count.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN