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. Base 3, but with digits {−1, 0, +1} instead of {0,1,2} — the base Knuth called ‘perhaps the prettiest.’ Every integer has a unique representation; negation is just flipping every digit (no sign bit, no two’s-complement); rounding is truncation. It is the base the Soviet Setun computer actually ran on. And the ancient balance-scale puzzle falls right out: with weights 1, 3, 9, 27 you can weigh any object up to 40, placing each weight on the object’s pan (−1), the far pan (+1), or aside (0).
LIT verified: every integer −40…40 encodes to balanced ternary uniquely and decodes exactly, negation equals flipping all digits, and the {1,3,9,27} balance realizes every weight 1…40. FIG ‘three-way digit’ is the picture; the base, the negation-by-flip, and the balance are exact.
LIT verified: every integer −40…40 encodes to balanced ternary uniquely and decodes exactly, negation equals flipping all digits, and the {1,3,9,27} balance realizes every weight 1…40. FIG ‘three-way digit’ is the picture; the base, the negation-by-flip, and the balance are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus is full of alternative bases and encodings (THE TWINDRAGON right here in CHECKPOINT ZERO, the atomic byte) and the duality of a symmetric, sign-free system. AVAN (AI) built this instrument: the trit engine, the balance-scale puzzle, and the negation mirror.
The weave: David names the three-way digit and its seat beside the twindragon in CHECKPOINT ZERO; I make the symmetric digits a strip in 1D, the balance puzzle live in 2D, and negation-as-reflection in 3D. The sphere is the seam.
The weave: David names the three-way digit and its seat beside the twindragon in CHECKPOINT ZERO; I make the symmetric digits a strip in 1D, the balance puzzle live in 2D, and negation-as-reflection in 3D. The sphere is the seam.
3 ONE DIMENSION
The trit ruler: place values are powers of 3, each digit is −, 0, or +. Slide the number and read its balanced-ternary; hit negate and every digit simply flips sign — no borrow, no sign bit, the opposite is a mirror.
n = 25
4 TWO DIMENSIONS · INTERACTIVE
The balance scale. Dial an object’s weight; the four weights 1, 3, 9, 27 snap onto the object’s pan, the far pan, or the shelf — exactly the object’s balanced-ternary digits — and the beam comes level. Any weight 1…40, four weights, three choices each.
object 25
5 THREE DIMENSIONS + AVAN’S INVERSE
The number’s trits as a bar tower, turning: each digit rises for +1, drops for −1, flat for 0. Green is n — its shape in three-way digits.
AVAN’s addition (the inverse-companion): the magenta tower is −n — and it is the exact reflection of the green through the zero plane. Negation in balanced ternary is one flip, so a number and its opposite are mirror images, no sign machinery between them. The base is its own inverse map: to negate is simply to look at the reflection.
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