ψῆφος is the counting-stone. In a positional number system built to balance around zero, the all-zeros word is three things at once — the additive identity, the geometric center, and the point every value is symmetric about. It is the stone the whole tower locks onto. Shown two ways: classical balanced ternary (radix 3, live) and the Generalized Balanced Ternary lattice (radix 7, from the 1991 thesis) — both centered on 0.
Digits are + (=+1), 0, − (=−1), place values 1, 3, 9, 27… Because the digits are signed and symmetric, negating a number = flipping every digit's sign, and 0,0,0 is the exact center of the range. Three trits reach exactly ±13 — and 13 = (3³−1)/2 = +++, −13 = −−−. Slide the value; watch it move symmetrically about the center stone.
the 27 three-trit words — a 3×3×3 lattice, 0,0,0 dead center; the lit node is your value
A word of 3 balanced trits spans [−13, +13] — that is 27 values, symmetric, with 0,0,0 at the exact middle. Add a plane and the reach is (3ⁿ−1)/2: 1, 4, 13, 40, 121…
i13 13 = (3³−1)/2 is the reach of a 3-plane balanced word from its 0,0,0 center (and 13 = 111 in ordinary base 3). A 4-plane / thirteen-symbol reading of this — which symbols, which net — is ROOT0's own (i13); here I only show the arithmetic fact the center-lock rests on.
checking…Kitto's 1991 thesis studies the GBT (Gibson & Lucas, 1982): a hierarchical, aggregate address system on the hexagonal lattice. A hexagon + its 6 neighbours is a level-1 aggregate (7 cells); that + its 6 neighbours is level-2 (49); and so on — 7ᵏ cells at level k, with address 0 at the geometric center of every level. Its focal theorem: EGBTₙ is ring-isomorphic to the q-adic integers, q = 2ⁿ⁺¹−1 (2-D → the 7-adic integers).
0 holds the center as the aggregate scales — the level-map fixes the origin. idealized the flower-of-flowers packs exactly 7ᵏ cells with 0 at center; the true GBT rep-tile has a rotated fractal boundary — this is the clean arrangement, not the exact tiling.
The 6 nonzero digits sit at the sixth roots of unity around the 0-center; they form the cyclic group ℤ/6 under multiplication (rotation, carry-free), and the six unit vectors sum to 0. The carry is circular: a carry out of the top position wraps back to position 0 (mod q), and the all-ones sequence is identified with all-zeros — the closure that pins the system to its center.
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Kitto (near line 447): "If the hexagon with address 0 is centered at the origin of the plane, then 1+2 = 3, …" (partial quote). Theorem 3.3.6: EGBTₙ ≅ the (2ⁿ⁺¹−1)-adic integers when n+1 and q are coprime.
The 0-center reading is drawn from two documents on the Internet Archive:
• An experimental study of the uses of ternary logic in digital computers — R. D. Friichtenicht, M.S. thesis, U.S. Naval Postgraduate School, 1964. The logic side: base three is put forward as near-optimal ("closest whole number to e"), and the three levels can be read as positive · zero · negative with zero at center — one of several namings the thesis lists; the device it actually built uses the unsigned levels {0,1,2} with mod-3 arithmetic (cyclic about 0). It demonstrated a working ternary half-adder from diodes and transistors — judged satisfactory only after the measured voltages were banded into three clean levels.
• Isomorphism theorem between the extended generalized balanced ternary numbers and the p-adic integers — W. Z. Kitto, Ph.D. thesis, University of Florida, 1991. The hierarchy side: GBT's 0-address as the fixed center of every nested aggregate, the circular carry, and EGBTₙ ≅ q-adic integers.
checking…