The Quantum Huzita-Justin (Huzita-Hatori) Axioms, Algorithmic Reachability in Superposed Folding

The Huzita-Justin axioms, often termed the Huzita-Hatori axioms, give an operational foundation for classical origami constructions by specifying which single-crease operations are permitted when points and lines are treated as exact geometric primitives. This paper proposes and formalizes a quantum analogue in which the choice of crease is not a classical branch but a coherent register, and a fold is implemented as a unitary reflection controlled on that register. The resulting model defines a “quantum folder” as a finite sequence of controlled fold unitaries and coherent axiom operators, together with a measurement rule that recovers classical folding as a commuting, decohered limit. Within this framework we introduce a notion of algorithmic reachability for geometric targets, defined by the probability of producing designated intersection events or alignment relations after interference over crease histories. The comparison protocol is stated on matched discretizations and resource measures, so that any change in reachability can be attributed to an explicit mechanism, whether generic amplitude amplification against a verifier or geometric interference specific to crease superposition.