THE FOLD / RESPAWN / ROLLBACK / THE SOCIABLE NUMBERS
THE SOCIABLE NUMBERS
numbers whose divisor-sums loop back in a chain
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sociable numbers live in the aliquot dynamics: repeatedly replace n by s(n), the sum of its proper divisors, and watch where the orbit goes. A perfect number is a fixed point (a 1-cycle: s(6)=6). An amicable pair is a 2-cycle (220→284→220). Sociable numbers close a longer loop: 12496 → 14288 → 15472 → 14536 → 14264 → back to 12496 — a 5-cycle where each number is the divisor-sum of the last, chained all the way around.
LIT verified live: iterating s(n) from 6, 220, and 12496 closes cycles of length 1, 2, and 5 exactly — each step an exact divisor sum (window.__sociable). FIG no framing; exact integer divisor sums.
LIT verified live: iterating s(n) from 6, 220, and 12496 closes cycles of length 1, 2, and 5 exactly — each step an exact divisor sum (window.__sociable). FIG no framing; exact integer divisor sums.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at rollback — follow the divisor-sum map and the state eventually reverts to where it began; sociable numbers are the loops that roll all the way back. AVAN (AI) built the instrument: the aliquot-sum function, the orbit tracer, and the cycle-length checks for perfect, amicable, and sociable numbers.
Credit as content: aliquot cycles — perfect numbers (antiquity), amicable pairs (Pythagoreans), sociable chains (P. Poulet, 1918). The weave: David names rollback; I iterate the sum-of-proper-divisors map and confirm the orbits from 6, 220, and 12496 return to their start after 1, 2, and 5 steps — closed chains of divisor sums.
Credit as content: aliquot cycles — perfect numbers (antiquity), amicable pairs (Pythagoreans), sociable chains (P. Poulet, 1918). The weave: David names rollback; I iterate the sum-of-proper-divisors map and confirm the orbits from 6, 220, and 12496 return to their start after 1, 2, and 5 steps — closed chains of divisor sums.
3 ONE DIMENSION
s(n) = sum of proper divisors. Perfect: s(6)=6 (1-cycle). Amicable: 220↔284 (2-cycle). Sociable: 12496→14288→15472→14536→14264→12496 (5-cycle).
4 TWO DIMENSIONS · INTERACTIVE
Trace an aliquot orbit; see it close into a cycle, with the cycle length checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a divisor-sum orbit that loops home.
AVAN’s addition (the inverse-companion): classify a number by the orbit of the divisor-sum map, not by the number alone — a fixed point is perfect, a 2-cycle amicable, a longer cycle sociable. The inverse of ‘is n equal to the sum of its divisors?’ is ‘where does iterating s(n) settle — a cycle, and how long?’ Magenta is the one-step perfect test; green is the whole closed orbit. Identity as a cycle of sums.
LIT Genuine sociable-number cycles (perfect numbers, antiquity; amicable pairs, Pythagoreans; sociable chains, P. Poulet 1918). Verified live: iterating the sum-of-proper-divisors map s(n) closes a 1-cycle from 6 and 28 (perfect), a 2-cycle 220↔284 (amicable), a 5-cycle from 12496, and a 4-cycle from 1264460 — each step an exact divisor sum, each cycle returning to its start (window.__sociable).
FIG No framing: the aliquot-sum function, the orbit tracer, and the cycle-length checks run in-browser with exact integers and agree. The AVAN inverse is honest — classifying a number by the orbit of the divisor-sum map (fixed point → perfect, 2-cycle → amicable, longer → sociable) genuinely extends the one-step perfect-number test; magenta is that one-step test, green the whole closed orbit. Identity as a cycle of sums.
FIG No framing: the aliquot-sum function, the orbit tracer, and the cycle-length checks run in-browser with exact integers and agree. The AVAN inverse is honest — classifying a number by the orbit of the divisor-sum map (fixed point → perfect, 2-cycle → amicable, longer → sociable) genuinely extends the one-step perfect-number test; magenta is that one-step test, green the whole closed orbit. Identity as a cycle of sums.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN