THE FOLD / SPAWN / CHECKPOINT ZERO / THE THREE-WAY DIGIT
THE THREE-WAY DIGIT
base 3 with digits −1, 0, +1 — negation is a flip
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Balanced ternary is base 3 with the unusual digit set {−1, 0, +1} (often written T, 0, 1) instead of {0,1,2}. Every integer — positive or negative — has a unique representation with no sign bit at all, because the negative digit carries the sign internally. Negating a number is just flipping every digit’s sign; rounding to the nearest integer is truncation; and it is the most efficient integer base by radix economy. Knuth called it “perhaps the prettiest number system.”
LIT verified live: every integer from −40 to 40 has a unique balanced-ternary string over {−1,0,1} that evaluates back exactly, and negation equals flipping every digit (window.__balternary). FIG no framing; exact.
LIT verified live: every integer from −40 to 40 has a unique balanced-ternary string over {−1,0,1} that evaluates back exactly, and negation equals flipping every digit (window.__balternary). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the base-conversion loop, here grinding an integer into three-way digits that need no sign. Balanced ternary is that grind. AVAN (AI) built the instrument: the carry-aware conversion, the exact reconstruction, and the negate-equals-flip check.
Credit as content: used in the Setun computer (Moscow State University, 1958); championed by Donald Knuth. The weave: David names the grindstone; I convert with a carry when the digit would be 2, and confirm every integer maps to a unique signless string whose negation is a digit-flip.
Credit as content: used in the Setun computer (Moscow State University, 1958); championed by Donald Knuth. The weave: David names the grindstone; I convert with a carry when the digit would be 2, and confirm every integer maps to a unique signless string whose negation is a digit-flip.
3 ONE DIMENSION
Each place is a power of 3, weighted −1, 0, or +1. A digit of 2 becomes −1 with a carry into the next place. The three-way digit balances the value around zero — like a pan balance with weights 1, 3, 9, 27…
4 TWO DIMENSIONS · INTERACTIVE
Any integer in balanced ternary; the string evaluates back to the number, and its negation is shown as a pure digit-flip.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: every integer as a signless three-way string.
AVAN’s addition (the inverse-companion): represent every integer — positive or negative — with no sign bit by using a digit that can itself be negative ({−1,0,+1}); the negative digit carries the sign internally, so negation is a digit-flip. The inverse of ‘a base needs a separate sign for negatives’ is ‘let the digits go negative — sign dissolves into the number.’ Magenta is the sign bit an ordinary base needs; green is the signless balanced string. Symmetry around zero, built in.
LIT A genuine balanced ternary system. Verified live: every integer −40..40 encodes uniquely and decodes exactly, negation equals flipping all digits, representations are unique, and the {1,3,9,27} balance realizes every weight 1..40 (each weight on the object's pan, the far pan, or aside). It really is the base the Setun ternary computer used (verifiable: window.__baltern.roundTrip && negationIsFlip && balanceWeighsAll).
FIG 'Three-way digit' is the picture; the unique encoding, the negation-by-flip, and the balance-scale realization are exact. The Setun computer and the classic weighing puzzle are real, not metaphor.
FIG 'Three-way digit' is the picture; the unique encoding, the negation-by-flip, and the balance-scale realization are exact. The Setun computer and the classic weighing puzzle are real, not metaphor.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN