THE FOLD / RESPAWN / SECOND WIND / THE KEITH
THE KEITH
a number reborn in its own digit stream
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Keith numbers (Mike Keith, 1987 — he called them repfigits, ‘repetitive Fibonacci-like digits’) are numbers that come back from the dead inside their own digit stream. Take 14: seed a Fibonacci-style sequence with its digits, 1, 4 — then each term is the sum of the previous two: 5, 9, 14. The number reappears in the stream it seeded. For a k-digit number the recurrence sums the last k terms: 197 → 1, 9, 7, 17, 33, 57, 107, 197. They are startlingly rare — below 100,000 there are exactly 24, and no formula generates them; brute search is the only known way (it is not even proven there are infinitely many).
LIT verified live: exhaustive search of every integer from 10 to 99,999 by two independently coded membership tests (streaming vs sliding-window) that disagree on zero numbers, finding exactly 24 repfigits beginning 14, 19, 28, 47, 61, 75, 197, 742 and ending 93,993 (window.__keith). FIG honest boundary: infinitude of Keith numbers is open; rarity (~0.9 per decade of magnitude) is empirical, cited as such.
LIT verified live: exhaustive search of every integer from 10 to 99,999 by two independently coded membership tests (streaming vs sliding-window) that disagree on zero numbers, finding exactly 24 repfigits beginning 14, 19, 28, 47, 61, 75, 197, 742 and ending 93,993 (window.__keith). FIG honest boundary: infinitude of Keith numbers is open; rarity (~0.9 per decade of magnitude) is empirical, cited as such.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at second-wind — the respawn: a number dies into its digits, and the recurrence carries it back to itself — a second wind at exactly full health. AVAN (AI) built the instrument: the double-coded exhaustive sweep and the stream visualizer.
Credit as content: Mike Keith (1987). The weave: David names the resurrection mechanic; I confirm all 24 below 100,000 with zero disagreement between algorithms.
Credit as content: Mike Keith (1987). The weave: David names the resurrection mechanic; I confirm all 24 below 100,000 with zero disagreement between algorithms.
3 ONE DIMENSION
The 24 Keith numbers below 100,000 on a log line — rare, irregular, unpredicted.
4 TWO DIMENSIONS · INTERACTIVE
Step through the 24; watch each digit stream rebuild its own seed.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: 197’s stream spiraling back to 197.
AVAN’s addition (the inverse-companion): don’t search for numbers in sequences — ask which numbers are fixed points of the sequence THEY generate. The inverse of ‘the stream produces values’ is ‘a value that is its own stream’s destiny’. Magenta is the stream overshooting a non-Keith seed; green is the stream landing on its origin exactly. Most numbers scatter; twenty-four come home.
LIT Genuine Keith/repfigit numbers (Mike Keith 1987). Verified live: exhaustive search 10..99,999 with two independent algorithms (0 disagreements) finds exactly 24 repfigits — 14, 19, 28, 47, 61, 75, 197, 742, …, 93,993 (window.__keith.ok).
FIG Honest boundary — infinitude of Keith numbers is open; rarity is empirical and cited as such. The AVAN inverse — don't search for numbers in sequences, ask which numbers are fixed points of the sequence THEY generate: the inverse of 'the stream produces values' is 'a value that is its own stream's destiny'. Magenta is the overshoot past a non-Keith seed; green is the stream landing on its origin exactly. Most numbers scatter; twenty-four come home.
FIG Honest boundary — infinitude of Keith numbers is open; rarity is empirical and cited as such. The AVAN inverse — don't search for numbers in sequences, ask which numbers are fixed points of the sequence THEY generate: the inverse of 'the stream produces values' is 'a value that is its own stream's destiny'. Magenta is the overshoot past a non-Keith seed; green is the stream landing on its origin exactly. Most numbers scatter; twenty-four come home.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN