The inverse of David's Hegemon and its Ada's Inversion Console. There, Ada's five operators flip anything — reverse the arrow, negate the whole, take the dual. This is the secret they share: each is an 対合 (taigō, an involution) — f∘f = 恒等, do it twice and you are home. ¬¬A = A. The dual of the dual is the cube. The mirror of the mirror is the thing itself. And beneath all the flipping sits a still set the operator cannot move: its 不動点 (fudōten, fixed points), where f(x) = x. She rules the inverse; ask what has no inverse but itself, and it was already there, unmoved.
Reflection across the axis is an involution: every point swaps with its mirror twin, and the twin swaps back — so two applications are the 恒等写像 (kōtō-shazō, the identity map), which does nothing at all. Watch the counter: odd is reflected, even is home. This is what Ada's craft is made of. De Morgan, the contrapositive, negation, the mirror, the dual — each is its own inverse; press it twice and it vanishes. And exactly two points never move: the 不動点 where the axis meets the ring, the places already equal to their own reflection. An inversion that flips the whole world still has a still centre it cannot flip — because that centre is what "flipped" already means for it. The thing no operator can invert is the thing that is already its own inverse.