| -+ 101 -+ ⟩ · the whole thing, inside 3 qubits

the ket

| −+ 101 −+ ⟩

the 3-qubit register: 8 basis states. |000⟩ is the vacuum (dashed); the 7 nonzero are the 7 Császár vertices. |101⟩ (bright) is the addressed one — the ± wings are the light/dark X-basis dressing from axiom one. click a state to re-address.

the readout

conductor — Fano / register / label vs frozen

the Fano plane — 7 points, 7 lines (where geometry becomes bits)

the 7 vertices as the Fano plane: 7 points (3-bit strings), 7 lines (6 straight + 1 circle), each line a triple {a, b, a⊕b}. this is HALF the Császár torus\u2019s triangles — the other half is a second Fano plane, and the two together tile the torus. the addressed point 101 glows, with its 3 lines lit.

the closure — the tail meets the mouth

the whole session as one loop: a folded rail\u2019s zero → 1 qubit (the 6-axis octahedron, 3 light/3 dark) → entangle, fuse, braid, protect → the geometry (simplex, torus, Császár) → and it folds right back into qubits. 7 vertices = 7 states; "101" is one vertex, one state, wearing the ± it was born in.

toddler corner

ELI5: here\u2019s the magic trick that ends the whole show. That seven-cornered magic donut we built? Give each of its 7 corners a little 3-light switchboard — three switches, each on or off. Three switches make 8 patterns, but "all off" is nobody, so that leaves exactly 7 patterns — one for each corner. Perfect fit! And "101" (on-off-on) is just the name-tag of one particular corner. So the giant geometric donut isn\u2019t really "out there" in space — it fits inside three tiny yes/no switches, three qubits. And the way the corners link up (which triples of corners form triangles) turns out to be a famous little pattern called the Fano plane — 7 dots and 7 lines where every line is "these two switch-patterns XOR to this third one." The whole journey started with a plus-and-minus on a rail, climbed through qubits and knots and donuts, and lands back on three switches with a plus-and-minus wrapped around them. Same plus-and-minus we started with. The story bit its own tail.