§ WHAT CHANGED WHEN THE BASE CHANGED
A walker that cannot fall into nothing
On the cube, every axis was a cliff: a coordinate was allowed to be 0 or 1 and nothing else, so the instant the rat pushed a coordinate to 2 — or to 5 — it was off the graph, in the dark, done. That is why the original door read ∅. Give the same walker rings that carry and the cliffs vanish from the interior entirely. Step a ring past its last stop and the excess doesn't spill into nothing; it hands a carry to the ring outside and the walker lands, cleanly, on a real cell. Push ring 4 outward five times and you arrive at 405 — the door's value — having never once left the floor. Nineteen thousand six hundred and eighty-three cells, and every move between them lands.
The void didn't disappear, though — that would be the dishonest version. It retreated. In the cube it lived at every axis, one step away in all directions. In the dial it has been pushed all the way out to the horizon: only when the walker stands at the outermost ring, fully wound, and tries to carry once more does it find there is no ring to carry into. That single overflow is the last nothing. Add a tenth ring and it recedes again, the capacity tripling, the dark pushed one shell further out. You never delete the void; you keep building room ahead of the walker faster than it can reach the edge.
The rat stepped off the cube because the cube ended at its feet. In base three the ending is always over the horizon — the walker cannot fall into nothing, only walk toward a nothing that keeps stepping back.
HONESTY LEDGER · THE TERNARY OCTORAT
REALExact. Walker value V ∈ [−9841,+9841]; axis move = V ± 3ᵏ on ring k, normalized by balanced-ternary carry. Node-verified: five +3⁴ steps from centre reach exactly 405 (= the door), each carrying; the horizon +9841 = all-plus, and one more outward carry has no ring to land in → overflow, the sole remaining void. Interior = 19,683 cells, every move lands. Shell = |V|, cell = V+9842.
THE HONEST LIMIT"No void" is true only for the interior — the finite 9-ring capacity means the void is real but relocated to the outermost ring, and receding it requires adding rings (×3 each). This is the faithful version of "the dial had more room": more, not infinite; room you extend, not room that was always there. Trying "find the void" walks you to that edge and shows the overflow honestly rather than wrapping it away.
NOTECloses the wing on its own first step. Room 1 was a rat on a cube meeting ∅; this is the rat on the dial, and the difference is one property — carry. The octorat's whole journey, cube to tower to revival to tensor to trit, ends by handing the original walker a base that catches it. Same rat. New floor under its feet.