THE FOLD / SPAWN / NULL ISLAND / THE NEGABINARY
THE NEGABINARY
count in base minus-two, no sign needed
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Negabinary is base −2: the same digits {0,1} as binary, but place values are powers of −2 — 1, −2, 4, −8, 16, −32, … The alternating signs mean a single unsigned digit string represents every integer, positive and negative, with no sign bit and no two’s-complement. −6, for instance, is 1110 (= −8 + 4 − 2). Encoding just repeatedly takes the bit n mod 2 and divides by −2 — and the representation of each integer is unique.
LIT verified live: over every integer from −2000 to 2000, decode(encode(n)) returns n, and all representations are distinct (window.__negabinary). FIG no framing; exact integer arithmetic.
LIT verified live: over every integer from −2000 to 2000, decode(encode(n)) returns n, and all representations are distinct (window.__negabinary). FIG no framing; exact integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the origin where positive and negative meet; negabinary spans both sides of zero from one unsigned string, no sign attached. AVAN (AI) built the instrument: the mod-2 / divide-by-−2 encoder, the alternating-place-value decoder, and the round-trip + uniqueness checks.
Credit as content: negative-base numeration (Vittorio Grünwald 1885; studied by Knuth). The weave: David names null-island; I encode integers in base −2 by taking bits and dividing by −2, decode by alternating place values, and confirm every integer round-trips to a unique string — signs carried by the base itself.
Credit as content: negative-base numeration (Vittorio Grünwald 1885; studied by Knuth). The weave: David names null-island; I encode integers in base −2 by taking bits and dividing by −2, decode by alternating place values, and confirm every integer round-trips to a unique string — signs carried by the base itself.
3 ONE DIMENSION
Place values 1, −2, 4, −8, 16, … So 1110 = 1·(−8) + 1·4 + 1·(−2) + 0 = −6. The alternating signs let one digit string reach any integer, no sign bit.
4 TWO DIMENSIONS · INTERACTIVE
A number, its base-−2 digits with their signed place values, and the round-trip checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: negatives represented with no sign.
AVAN’s addition (the inverse-companion): represent signed integers with no sign bit and no two’s-complement — let the base carry the signs by making it negative, so alternating place values reach below zero. The inverse of ‘store magnitude plus a sign’ is ‘use base −2 — one unsigned string is already signed.’ Magenta is the separate sign bit; green is the signless negabinary string. The sign folded into the base.
LIT Genuine negative-base (base −2) numeration (Vittorio Grünwald 1885; treated by Knuth). Verified live: for every integer n in −2000..2000, the mod-2/divide-by-−2 encoder and the alternating-place-value decoder round-trip (window.__negabinary.roundTrip), and all digit strings are distinct — a unique representation per integer (window.__negabinary.unique) — exact integer arithmetic.
FIG No framing: the encoder, the decoder, and the round-trip + uniqueness checks run in-browser with exact integers and agree. The AVAN inverse is honest — representing signed integers with no sign bit by letting a negative base carry the signs (alternating place values reach below zero) genuinely replaces magnitude-plus-sign; magenta is the separate sign bit, green the signless negabinary string. The sign folded into the base.
FIG No framing: the encoder, the decoder, and the round-trip + uniqueness checks run in-browser with exact integers and agree. The AVAN inverse is honest — representing signed integers with no sign bit by letting a negative base carry the signs (alternating place values reach below zero) genuinely replaces magnitude-plus-sign; magenta is the separate sign bit, green the signless negabinary string. The sign folded into the base.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN