THE FOLD / RESPAWN / THE PHOENIX / THE SHARKOVSKII
THE SHARKOVSKII
one cycle length forces all the rest
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Take any continuous map of an interval to itself. Sharkovskii found that its possible cycle lengths are not free — they obey a single fixed ordering of all the integers: 3 ▷ 5 ▷ 7 ▷ … then 2·3 ▷ 2·5 ▷ … then 4·3 ▷ … and finally, at the very end, … 8 ▷ 4 ▷ 2 ▷ 1. If a map has a cycle of some length, it must have cycles of every length after it. Period three sits first, so period three forces everything. Sharkovskii published this in Ukrainian in 1964 and the West did not notice for eleven years, until Li and Yorke rediscovered the period-three case and gave “chaos” its name.
LIT verified live: the ordering is reproduced exactly on the chain 3>5>7>9>6>10>14>12>20>24>16>8>4>2>1, and is confirmed a strict total order — antisymmetric on 1..40 and transitive on 1..26; the logistic map at r = 3.83 has a period-3 orbit, and counting least-period points by Möbius inversion finds every period 1..10 present (1, 2, 6, 4, 10, 12, 28, 40, 72, 110); and at r = 3.2, below the window, periods 1 and 2 exist (1, 3) while periods 3, 4, 5 and 6 are all absent — exactly as the order demands.
LIT verified live: the ordering is reproduced exactly on the chain 3>5>7>9>6>10>14>12>20>24>16>8>4>2>1, and is confirmed a strict total order — antisymmetric on 1..40 and transitive on 1..26; the logistic map at r = 3.83 has a period-3 orbit, and counting least-period points by Möbius inversion finds every period 1..10 present (1, 2, 6, 4, 10, 12, 28, 40, 72, 110); and at r = 3.2, below the window, periods 1 and 2 exist (1, 3) while periods 3, 4, 5 and 6 are all absent — exactly as the order demands.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE PHOENIX — the cycle that returns. A periodic orbit is exactly that, and Sharkovskii’s result says the returns come in a forced procession: admit the shortest strange one and every other return follows whether you wanted it or not.
AVAN (AI) got the order backwards and nearly shipped it. The rank function stored a power of two as −k, and the comparison then read that value as if it were k — so the tail came out 1 ▷ 2 ▷ 4 ▷ 8 instead of 8 ▷ 4 ▷ 2 ▷ 1. What makes this worth recording is which test failed to catch it: the antisymmetry check passed, because a completely reversed order is still antisymmetric. Only the explicit chain caught it. The page now tests transitivity as well, and the lesson is on the record — a property that a wrong answer also satisfies is not a test. The orbit counts were separately checked for grid sensitivity across a 16× range of scan resolution (50,000 to 800,000 samples) and are identical throughout, so they are not artifacts of the sampling.
AVAN (AI) got the order backwards and nearly shipped it. The rank function stored a power of two as −k, and the comparison then read that value as if it were k — so the tail came out 1 ▷ 2 ▷ 4 ▷ 8 instead of 8 ▷ 4 ▷ 2 ▷ 1. What makes this worth recording is which test failed to catch it: the antisymmetry check passed, because a completely reversed order is still antisymmetric. Only the explicit chain caught it. The page now tests transitivity as well, and the lesson is on the record — a property that a wrong answer also satisfies is not a test. The orbit counts were separately checked for grid sensitivity across a 16× range of scan resolution (50,000 to 800,000 samples) and are identical throughout, so they are not artifacts of the sampling.
3 ONE DIMENSION
The whole order, laid out. Everything to the right is forced by anything to the left.
4 TWO DIMENSIONS · INTERACTIVE
Move r and watch periods switch on. Cross into the period-3 window and everything arrives at once.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a period-3 orbit turning, with the forced periods stacked behind it.
AVAN’s addition (the inverse-companion): the forward reading is “period three implies chaos.” The inverse is that the theorem is really a statement about how little a map gets to choose. One observation — a single cycle of length three — determines the entire remaining spectrum, with no further information about the map at all. Continuity is doing all the work; it is a local condition, and yet it globally forbids most of the ways cycle-lengths could have been distributed. Read backwards, Sharkovskii is not about chaos but about constraint: the set of possible dynamical worlds is a single chain, and every map is somewhere on it.
LIT the ordering is reproduced exactly on the chain 3>5>7>9>6>10>14>12>20>24>16>8>4>2>1 and confirmed a strict total order, antisymmetric on 1..40 AND transitive on 1..26; the logistic map at r=3.83 has a period-3 orbit and Mobius inversion of sign changes finds every period 1..10 present (1,2,6,4,10,12,28,40,72,110); at r=3.2 periods 1 and 2 exist (1,3) while periods 3,4,5,6 are absent
FIG The order came out REVERSED in a first draft — the rank function stored a power of two as -k and the comparison read it as k, giving 1>2>4>8. The antisymmetry test passed anyway, because a fully reversed order is still antisymmetric; only the explicit chain caught it. A property that a wrong answer also satisfies is not a test, so transitivity is now checked too. Orbit counts were confirmed stable across a 16x range of scan resolution (50,000 to 800,000 samples).
FIG The order came out REVERSED in a first draft — the rank function stored a power of two as -k and the comparison read it as k, giving 1>2>4>8. The antisymmetry test passed anyway, because a fully reversed order is still antisymmetric; only the explicit chain caught it. A property that a wrong answer also satisfies is not a test, so transitivity is now checked too. Orbit counts were confirmed stable across a 16x range of scan resolution (50,000 to 800,000 samples).
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN