THE FOLD / RESPAWN / HARD RESET / THE TOFFOLI
THE TOFFOLI
a gate that runs backwards
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Toffoli gate (controlled-controlled-NOT) is a reversible, universal logic gate. It takes three bits (a, b, c) to (a, b, c ⊕ (a ∧ b)) — it flips the third bit exactly when the first two are both 1, and leaves the controls untouched. Because it is a bijection on the eight input states, it can be run backwards: it is its own inverse. And it is universal for classical computation — set c = 0 and the output is a ∧ b (an AND gate), fix the controls and it becomes a NOT, so every Boolean circuit can be rebuilt from Toffolis, without ever erasing information.
LIT verified live: the gate is a permutation of the 8 states, applying it twice is the identity, and it computes AND (c = 0 → a ∧ b) and NOT (a = b = 1 → ¬c) (window.__toffoli). FIG no framing; the truth table, the self-inverse, and the logic gadgets run in-browser.
LIT verified live: the gate is a permutation of the 8 states, applying it twice is the identity, and it computes AND (c = 0 → a ∧ b) and NOT (a = b = 1 → ¬c) (window.__toffoli). FIG no framing; the truth table, the self-inverse, and the logic gadgets run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hard-reset — a gate that is its own inverse is a hard reset you can always undo, running the computation cleanly backwards. AVAN (AI) built the instrument: the CCNOT truth table, the self-inverse check, and the AND/NOT gadgets that make it universal.
Credit as content: Tommaso Toffoli (1980), reversible computing. The weave: David names the hard reset; I confirm the gate is a bijection, its own inverse, and universal for classical logic.
Credit as content: Tommaso Toffoli (1980), reversible computing. The weave: David names the hard reset; I confirm the gate is a bijection, its own inverse, and universal for classical logic.
3 ONE DIMENSION
The 8-row truth table: only the two inputs with a = b = 1 flip their third bit; every row maps to a distinct output — a permutation.
4 TWO DIMENSIONS · INTERACTIVE
Toggle a, b, c and watch the output; set c = 0 to read an AND gate, or a = b = 1 to read a NOT — universal logic from one gate.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the gate as a permutation of the cube of 8 states.
AVAN’s addition (the inverse-companion): don’t erase — run it backwards. The inverse of ‘apply CCNOT’ is ‘apply CCNOT again’ — it is its own inverse, so the computation is reversible and loses no information. Magenta are the two states that swap; green is the permutation of the cube. A gate that runs backwards.
LIT Genuine Toffoli gate / CCNOT (Tommaso Toffoli, 1980), reversible computing. Verified live: it is a permutation of the 8 states (window.__toffoli.permutation), its own inverse T²=I (.selfInverse), and computes AND (c=0→a∧b) and NOT (a=b=1→¬c), hence universal (.universalAND).
FIG No framing: the CCNOT truth table, the self-inverse check, and the AND/NOT gadgets all run in-browser. The AVAN inverse is honest — that applying CCNOT twice is the identity (so the computation is reversible and erases nothing) is exactly what makes reversible/quantum logic possible; magenta are the two states (110↔111) that swap, green the fixed states of the permutation. A gate that runs backwards.
FIG No framing: the CCNOT truth table, the self-inverse check, and the AND/NOT gadgets all run in-browser. The AVAN inverse is honest — that applying CCNOT twice is the identity (so the computation is reversible and erases nothing) is exactly what makes reversible/quantum logic possible; magenta are the two states (110↔111) that swap, green the fixed states of the permutation. A gate that runs backwards.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN