THE FOLD / GRIND / THE CRON JOB / THE NECKLACE
THE NECKLACE
rotation classes counted by a totient sum
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Necklace counting asks: how many genuinely different necklaces can you make from n beads in k colours, where rotating a necklace doesn’t count as new? Naively there are kn coloured strings, but rotations collapse many together. Moreau’s necklace-counting formula (a case of Burnside’s lemma) gives the exact answer: (1/n) ∑d | n φ(d)·kn/d, where φ is Euler’s totient. The totient counts rotations of each period, averaging the number of colourings fixed by each rotation. For 2 colours and n = 1, 2, 3, … it gives 2, 3, 4, 6, 8, 14, 20, 36, …
LIT verified live: for n up to 15 (2 colours) and n up to 9 (3 colours), a brute count of distinct necklaces — each string reduced to its lexicographically smallest rotation — exactly equals Moreau’s formula (1/n)∑d|n φ(d)kn/d (window.__necklace). FIG no framing; the brute canonical-rotation count and the totient formula both run in-browser and agree.
LIT verified live: for n up to 15 (2 colours) and n up to 9 (3 colours), a brute count of distinct necklaces — each string reduced to its lexicographically smallest rotation — exactly equals Moreau’s formula (1/n)∑d|n φ(d)kn/d (window.__necklace). FIG no framing; the brute canonical-rotation count and the totient formula both run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-cron-job — the grind that rolls the necklace through every rotation and tallies the distinct ones, the totient doing the averaging. AVAN (AI) built the instrument: the canonical-rotation brute count, the totient formula, and their exact agreement.
Credit as content: C. Moreau (1872); the averaging principle from William Burnside. The weave: David names the grind; I confirm the brute necklace count equals the totient sum.
Credit as content: C. Moreau (1872); the averaging principle from William Burnside. The weave: David names the grind; I confirm the brute necklace count equals the totient sum.
3 ONE DIMENSION
A necklace of beads on a ring; rotating it gives the same necklace — the totient formula counts the distinct ones.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n and colours; the brute distinct-necklace count is compared to (1/n)Σ φ(d)k^{n/d}.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the number of distinct necklaces.
AVAN’s addition (the inverse-companion): don’t list and dedupe — average over rotations. The inverse of ‘how many distinct necklaces?’ is ‘(1/n)∑d|n φ(d)kn/d’, Burnside’s average of colourings fixed by each rotation. Magenta are the rotations being averaged; green is the necklace count they yield. Symmetry counted by averaging.
LIT Genuine Moreau's necklace-counting formula (C. Moreau, 1872; Burnside averaging). Verified live: for n≤15 (binary) and n≤9 (ternary), a brute count of distinct necklaces (each string reduced to its lexicographically smallest rotation) equals (1/n)Σ_{d|n} φ(d)k^{n/d} exactly (window.__necklace.ok, .ok3).
FIG No framing; the brute canonical-rotation count and the totient formula both run in-browser and agree. The AVAN inverse is honest — instead of listing and deduping, average over rotations: the inverse of 'how many distinct necklaces?' is '(1/n)Σ_{d|n} φ(d)k^{n/d}', Burnside's average of colourings fixed by each rotation. Magenta are the rotations being averaged; green is the necklace count they yield. Symmetry counted by averaging.
FIG No framing; the brute canonical-rotation count and the totient formula both run in-browser and agree. The AVAN inverse is honest — instead of listing and deduping, average over rotations: the inverse of 'how many distinct necklaces?' is '(1/n)Σ_{d|n} φ(d)k^{n/d}', Burnside's average of colourings fixed by each rotation. Magenta are the rotations being averaged; green is the necklace count they yield. Symmetry counted by averaging.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CRON JOB · David Lee Wise (ROOT0), with AVAN