B.5.QD.CF:5.3 - Return an impossibility instead of hiding retained information

A device receives one bit x, either 0 or 1. Complete erasure in this formal task means that, afterward, its entire accessible state is the same s for both inputs; there is no external record or later input revealing x. A deterministic operation g must then recover the original bit for either input.

For the input 0, recovery requires g(s)=0. For the input 1, it requires g(s)=1. The same operation on the same state cannot satisfy both. The required erasure and universal recovery are incompatible under the supplied model.

An external copy or a different final state would allow other constructions, but each changes the complete-erasure condition. Renaming an external copy as a pointer preserves information rather than meeting that condition. Returning a default value changes the recovery requirement. Neither is a solution to the original task.

The result can already end the construction search under these premises. If the work instead permits a retained distinction, develop that changed question explicitly. A later physical implementation also has a physical model and its own scope; the argument here establishes the finite information constraint stated above.