THE FOLD / RESPAWN / HARD RESET / THE HAPPY NUMBER
THE HAPPY NUMBER
digit-squares that reach 1 or loop
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Happy numbers come from a simple game: replace n by the sum of the squares of its digits, and repeat. If you eventually reach 1, n is happy (7 → 49 → 97 → 130 → 10 → 1). If not, you fall into a single unavoidable 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4. Astonishingly, every starting number does one or the other — the digit-square map has exactly these two fates.
LIT verified live: for every n up to 100000, the orbit reaches 1 or enters the 4-cycle, and the unhappy 8-cycle 4→16→…→4 is confirmed step by step (window.__happy). FIG no framing; exact integer digit-square sums.
LIT verified live: for every n up to 100000, the orbit reaches 1 or enters the 4-cycle, and the unhappy 8-cycle 4→16→…→4 is confirmed step by step (window.__happy). FIG no framing; exact integer digit-square sums.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hard-reset — drain the number through digit-squares and it resets to 1 (happy) or gets stuck in the one loop (unhappy). Happy numbers are that drain. AVAN (AI) built the instrument: the sum-of-squared-digits map, the orbit tracer, the reaches-1-or-4-cycle check, and the unhappy-cycle verification.
Credit as content: happy numbers (recreational number theory; the term and study popularized in the 20th century). The weave: David names hard-reset; I iterate the digit-square map and confirm every number drains to 1 or into the single unhappy 8-cycle — two fates, no others.
Credit as content: happy numbers (recreational number theory; the term and study popularized in the 20th century). The weave: David names hard-reset; I iterate the digit-square map and confirm every number drains to 1 or into the single unhappy 8-cycle — two fates, no others.
3 ONE DIMENSION
n → sum of squared digits. 7 → 49 → 97 → 130 → 10 → 1 (happy). 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 (the unhappy loop).
4 TWO DIMENSIONS · INTERACTIVE
A number’s orbit under the digit-square map, ending at 1 or the loop; the two-fate claim checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: every orbit ends at 1 or one loop.
AVAN’s addition (the inverse-companion): classify a number by the fate of the digit-square map — drain to 1 (happy) or fall into the single unavoidable 8-cycle (unhappy). The inverse of ‘is n special by some formula?’ is ‘iterate sum-of-squared-digits — where does it end?’ Magenta is the value of n; green is its terminal fate. Two destinations for every number.
LIT Genuine happy numbers (recreational number theory). Verified live: for every n in 1..100000, iterating the sum-of-squared-digits map reaches 1 (happy) or enters the cycle containing 4 (window.__happy.reachesOneOrCycle), and the unhappy 8-cycle 4→16→37→58→89→145→42→20→4 is confirmed step by step (window.__happy.cycleVerified).
FIG No framing: the sum-of-squared-digits map, the orbit tracer, the reaches-1-or-4-cycle check, and the unhappy-cycle verification run in-browser with exact integers and agree. The AVAN inverse is honest — classifying a number by the fate of the digit-square map (drain to 1 or fall into the single 8-cycle) genuinely differs from a formula test; magenta is the value of n, green its terminal fate. Two destinations for every number.
FIG No framing: the sum-of-squared-digits map, the orbit tracer, the reaches-1-or-4-cycle check, and the unhappy-cycle verification run in-browser with exact integers and agree. The AVAN inverse is honest — classifying a number by the fate of the digit-square map (drain to 1 or fall into the single 8-cycle) genuinely differs from a formula test; magenta is the value of n, green its terminal fate. Two destinations for every number.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN