THE FOLD / GRIND / THE GRINDSTONE / THE NEGAFIBONACCI
THE NEGAFIBONACCI
one signless code across the whole number line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
NegaFibonacci coding represents every integer — positive and negative — as a unique sum of non-consecutive negaFibonacci numbers F(−1), F(−2), … = 1, −1, 2, −3, 5, −8, 13, … with digits {0, 1} and no two adjacent 1s — and, remarkably, no sign bit. Because the base sequence already alternates sign, negatives are reached for free. It is Zeckendorf’s theorem extended across zero.
LIT verified live: every integer from −50 to 50 has exactly one such representation, with no two adjacent 1s, that evaluates back to it (window.__negafib). FIG no framing; exact.
LIT verified live: every integer from −50 to 50 has exactly one such representation, with no two adjacent 1s, that evaluates back to it (window.__negafib). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the base-conversion loop, here reaching negative numbers with no sign because the base itself alternates. NegaFibonacci is that signless negative base. AVAN (AI) built the instrument: the alternating F(−k) sequence, the representation search, the uniqueness enumeration, and the round-trip check.
Credit as content: negaFibonacci representation (Martin Bunder 1992). The weave: David names the grindstone; I confirm every integer in a range maps to one and only one non-consecutive negaFibonacci string that sums back to it — negatives included, no sign bit.
Credit as content: negaFibonacci representation (Martin Bunder 1992). The weave: David names the grindstone; I confirm every integer in a range maps to one and only one non-consecutive negaFibonacci string that sums back to it — negatives included, no sign bit.
3 ONE DIMENSION
The base alternates sign: F(−1)=1, F(−2)=−1, F(−3)=2, F(−4)=−3, F(−5)=5, … So a string like 101 = 1 + 2 = 3, and 0101 = −1 + −3 = −4 — negatives with no sign bit.
4 TWO DIMENSIONS · INTERACTIVE
Any integer’s negaFibonacci code; the digits, the no-adjacent rule, and the round-trip are checked — positive and negative alike.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: every integer as one signless negaFibonacci string.
AVAN’s addition (the inverse-companion): reach negative integers with no sign bit by using a base whose digits alternate sign — the non-consecutive negaFibonacci sum is unique across all integers. The inverse of ‘Zeckendorf covers only positives, add a sign for negatives’ is ‘alternate the base’s sign — one signless code spans the whole number line.’ Magenta is the sign bit an ordinary base needs; green is the signless negaFibonacci code. Zeckendorf across zero.
LIT Genuine negaFibonacci representation (Bunder 1992). Verified live: every integer from -50 to 50 has exactly one representation as a sum of non-consecutive negaFibonacci numbers (base 1,-1,2,-3,5,-8,...) over digits {0,1} with no two adjacent 1s that evaluates back to it (window.__negafib.existsInRange && .uniqueInRange && .roundTrip && .noAdjacent).
FIG No framing: the alternating F(-k) sequence, the representation search, the uniqueness enumeration, and the round-trip check run in-browser over -50..50 and hold. The AVAN inverse is honest — a base whose digits alternate sign reaches negatives with no sign bit, one non-consecutive code per integer; magenta is the sign bit an ordinary base needs, green the signless negaFibonacci code. Zeckendorf across zero.
FIG No framing: the alternating F(-k) sequence, the representation search, the uniqueness enumeration, and the round-trip check run in-browser over -50..50 and hold. The AVAN inverse is honest — a base whose digits alternate sign reaches negatives with no sign bit, one non-consecutive code per integer; magenta is the sign bit an ordinary base needs, green the signless negaFibonacci code. Zeckendorf across zero.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN