THE FOLD / GRIND / THE GRINDSTONE / THE PERSISTENCE
THE PERSISTENCE
multiply the digits, repeat — 277777788888899 resists 11 times
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Multiplicative persistence. Take a number, multiply its digits together to get a new number, and repeat until you reach a single digit. The number of steps is the number’s persistence. 39 → 27 → 14 → 4 — three steps, persistence 3.
Almost every number collapses in a handful of steps. The astonishing fact is how hard it is to be stubborn. The smallest number with persistence 11 is 277777788888899 — and despite searching every number up past 10233, no one has ever found a number with persistence greater than 11 in base 10. It is conjectured that 11 is the ceiling, but it has never been proven. The record-holders climb 10, 25, 39, 77, 679, 6788, 68889, … and then just stop rising.
LIT verified live: each record-holder has exactly its stated persistence, 277777788888899 collapses in exactly 11 steps, and no number below 100000 exceeds persistence 7 (window.__persistence). FIG the persistence values and the record are exact; that 11 is the maximum is conjecture, not proven — carried honestly.
Almost every number collapses in a handful of steps. The astonishing fact is how hard it is to be stubborn. The smallest number with persistence 11 is 277777788888899 — and despite searching every number up past 10233, no one has ever found a number with persistence greater than 11 in base 10. It is conjectured that 11 is the ceiling, but it has never been proven. The record-holders climb 10, 25, 39, 77, 679, 6788, 68889, … and then just stop rising.
LIT verified live: each record-holder has exactly its stated persistence, 277777788888899 collapses in exactly 11 steps, and no number below 100000 exceeds persistence 7 (window.__persistence). FIG the persistence values and the record are exact; that 11 is the maximum is conjecture, not proven — carried honestly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE GRINDSTONE — the grind domain of wearing a thing down. Persistence is exactly a grindstone: multiply the digits and grind any number, however huge, down to a single digit. AVAN (AI) built the instrument: the collapse chain, the step-by-step grinder, the resist-collapse record.
The weave: David names the seat (the grindstone); I make numbers collapse under repeated digit-multiplication and show which ones resist longest — the chain in 1D, the grinder in 2D, the collapse-vs-resistance asymmetry in 3D. The sphere is the seam. Credit: Neil Sloane (1973), who introduced multiplicative persistence; record-holders in OEIS A003001.
The weave: David names the seat (the grindstone); I make numbers collapse under repeated digit-multiplication and show which ones resist longest — the chain in 1D, the grinder in 2D, the collapse-vs-resistance asymmetry in 3D. The sphere is the seam. Credit: Neil Sloane (1973), who introduced multiplicative persistence; record-holders in OEIS A003001.
3 ONE DIMENSION
The collapse chain of the record 277777788888899 — eleven links, each the digit-product of the last, ending at a single digit. The longest such chain any base-10 number is known to make.
4 TWO DIMENSIONS · INTERACTIVE
Pick a number and grind it: each step multiplies the current digits into the next number, until one digit remains. Step through the record-holders 10, 25, 39, 77, 679, …, 277777788888899 and watch the persistence climb to 11.
5 THREE DIMENSIONS + AVAN’S INVERSE
The record-holders as a rising staircase of persistence — green, the smallest number that resists collapse for 0, 1, 2, …, 11 steps, climbing then flattening.
AVAN’s addition (the inverse-companion): the magenta is an ordinary number’s collapse — steep and instant. The forward map is trivially easy: any number, however astronomical, grinds to a single digit in a few steps. The inverse is the frontier: ‘what is the smallest seed that resists collapse for k steps?’ Collapsing is free; resisting collapse is a search with no known ceiling proof. The record seeds are built of only 2s, 3s, 7s, 8s, 9s — never a 0, never digits whose product breeds a 0 — a delicate recipe the inverse must discover. The green resistance-staircase rises to 11 and then, as far as anyone has ever computed, stops; the magenta collapse plunges every time. The asymmetry is the point: forward is a triviality, backward is an open problem. Green is how hard it is to stay uncrushed; magenta is how easy it is to be crushed.
LIT Genuine multiplicative persistence (Neil Sloane 1973; record-holders OEIS A003001). Verified live: each record-holder (10,25,39,77,679,6788,68889,2677889,26888999,3778888999,277777788888899) has exactly its stated persistence, 277777788888899 collapses in exactly 11 steps, and no number below 100000 exceeds persistence 7 (window.__persistence.recordsCorrect && record11). The persistence values, the collapse chains, and the record are all exact (digit products stay within safe integer range). HONEST CAVEAT: that 11 is the MAXIMUM is a conjecture (searched past 10^233, never proven), flagged as FIG-open.
FIG No false framing: the persistence values, the record-holders, and the 11-step collapse of 277777788888899 are real and reproduced in-browser. The striking claim — that no number exceeds persistence 11 — is carried explicitly as an unproven conjecture backed by search, not asserted as theorem.
FIG No false framing: the persistence values, the record-holders, and the 11-step collapse of 277777788888899 are real and reproduced in-browser. The striking claim — that no number exceeds persistence 11 — is carried explicitly as an unproven conjecture backed by search, not asserted as theorem.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN