THE FOLD / SPAWN / CHECKPOINT ZERO / THE ZECKENDORF
THE ZECKENDORF
every integer, one sum of non-consecutive Fibonaccis
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Zeckendorf’s theorem. We write numbers in base ten, or base two — place values 1, 10, 100 or 1, 2, 4, 8. But you can also use the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21… as place values, and something remarkable happens: every positive integer has exactly one representation as a sum of Fibonacci numbers no two of which are consecutive.
Finding it is greedy: subtract the largest Fibonacci number that fits, repeat. 100 = 89 + 8 + 3. And you will never need two neighbours — because any two consecutive Fibonacci numbers add up to the next one, so using both is always replaceable by one. The ‘no two adjacent’ rule is exactly what makes the representation unique.
LIT verified live: for every integer 1…1000 this page confirms the greedy Zeckendorf digits sum back to n, use no two consecutive Fibonacci numbers, and are the only such representation (a brute-force count of valid representations returns exactly 1) — window.__zeck.allSumBack && noConsecutive && allUnique. FIG no framing needed; this is Zeckendorf’s theorem, checked number by number.
Finding it is greedy: subtract the largest Fibonacci number that fits, repeat. 100 = 89 + 8 + 3. And you will never need two neighbours — because any two consecutive Fibonacci numbers add up to the next one, so using both is always replaceable by one. The ‘no two adjacent’ rule is exactly what makes the representation unique.
LIT verified live: for every integer 1…1000 this page confirms the greedy Zeckendorf digits sum back to n, use no two consecutive Fibonacci numbers, and are the only such representation (a brute-force count of valid representations returns exactly 1) — window.__zeck.allSumBack && noConsecutive && allUnique. FIG no framing needed; this is Zeckendorf’s theorem, checked number by number.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in CHECKPOINT ZERO, beside THE TWINDRAGON and THE THREE-WAY DIGIT — the spawn domain of exotic ways to write a number. Base −1+i, balanced ternary, and now the Fibonacci base: each a different alphabet for the same integers. AVAN (AI) built the instrument: the greedy peel, the no-adjacent digits, the uniqueness count.
The weave: David gathers the strange numeral systems; I make this one legible and checkable — the Fibonacci digit-string in 1D, the greedy tiling in 2D, the golden place-values in 3D. The sphere is the seam. Credit: Édouard Zeckendorf (theorem published 1972; C. G. Lekkerkerker, 1952).
The weave: David gathers the strange numeral systems; I make this one legible and checkable — the Fibonacci digit-string in 1D, the greedy tiling in 2D, the golden place-values in 3D. The sphere is the seam. Credit: Édouard Zeckendorf (theorem published 1972; C. G. Lekkerkerker, 1952).
3 ONE DIMENSION
The Fibonacci digit-string of a number: a 1 over each Fibonacci place value that is used, a 0 elsewhere — and never two 1s in a row. That single forbidden pattern (‘no 11’) is the whole reason the representation is one-of-a-kind.
4 TWO DIMENSIONS · INTERACTIVE
Dial a number and watch greedy Zeckendorf peel off the largest Fibonacci that fits, then the next, tiling the number with non-adjacent Fibonacci blocks. The digits light up with never two together; the sum and the uniqueness check confirm it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The Fibonacci place-values as a turning ladder of golden blocks, each the sum of the two below it — green, the whole scale.
AVAN’s addition (the inverse-companion): the magenta blocks are the ones chosen for your number, and I draw the forbidden link between any two neighbours. Most number systems are defined by what digits you may use; Zeckendorf is defined by what you may not place — two adjacent Fibonacci blocks. And that prohibition is not arbitrary: two neighbours always fuse into the block above them, so forbidding the pair is forbidding redundancy. Uniqueness is carved out by a rule of absence. The green is what exists; the magenta bond is the pairing the system refuses, and in that refusal every number gets exactly one name.
LIT Genuine Zeckendorf representation (Zeckendorf's theorem, published 1972; Lekkerkerker 1952). Verified live: for every integer 1..1000 the greedy digits sum back to n, use no two consecutive Fibonacci numbers, and a brute-force count of valid non-consecutive representations returns exactly 1 (window.__zeck.allSumBack && noConsecutive && allUnique, all true). Because consecutive Fibonaccis sum to the next, forbidding adjacency forbids redundancy — that is the uniqueness, demonstrated not asserted.
FIG No metaphor is doing the work: the Fibonacci place values, the greedy peel, and the uniqueness count are the theorem itself, checked number by number. Calling the digit rule 'no 11' is the only framing.
FIG No metaphor is doing the work: the Fibonacci place values, the greedy peel, and the uniqueness count are the theorem itself, checked number by number. Calling the digit rule 'no 11' is the only framing.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN