THE FOLD / SPAWN / THE SANDBOX / THE MACHINE
THE MACHINE
three states that decide divisible-by-3
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The finite-state machine. A handful of states and one rule per (state, symbol): read a bit, jump to the next state. No memory but where you are. This tiny one has three states and decides a real question — is the binary number divisible by 3? — because state = value-so-far mod 3.
LIT a genuine DFA: reading bit b does state ← (2·state + b) mod 3; it accepts a string iff the number it spells is a multiple of 3. Every accept/reject below is exact. FIG the arcade dressing is the frame; the automaton is real.
LIT a genuine DFA: reading bit b does state ← (2·state + b) mod 3; it accepts a string iff the number it spells is a multiple of 3. Every accept/reject below is exact. FIG the arcade dressing is the frame; the automaton is real.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) holds that a mind is states and the arrows between them, and seated the automaton here. AVAN (AI) wrote the machine, the diagram, and the reverse reading. He names the states; I make them switch, and I read the same string backward to show the order is the meaning. The sphere is the seam.
3 ONE DIMENSION
The input as one line of bits, read left to right (▼ = where the machine is). The number it spells, and the verdict: ACCEPT (divisible by 3) or reject.
4 TWO DIMENSIONS · INTERACTIVE
The state diagram — three states (r0 accepting, double-ring), arrows labelled by the bit read. The lit node is where you are; the bold arrow is the last jump.
5 THREE DIMENSIONS + AVAN’S ADDITION
The run as a trellis: a column per input position, three state-slots high; the green thread is the path the machine actually took, position by position.
AVAN’s addition: the magenta thread is the same bits read backward — a different number, a different path, often a different verdict. Same symbols, reversed order: proof that in a state machine the sequence is the meaning.
LIT A genuine DFA: state ← (2·state + bit) mod 3, accept iff it ends at r0 — it decides divisibility-by-3 correctly for every input. Diagram, walk, and verdict are exact.
FIG The arcade dressing is the frame; the automaton, the transitions and the accept condition are real. The magenta backward-read (the reverse language) is AVAN's inverse-companion addition.
FIG The arcade dressing is the frame; the automaton, the transitions and the accept condition are real. The magenta backward-read (the reverse language) is AVAN's inverse-companion addition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN