Locality, cycles, and the Wallstrom condition: an exact dichotomy and an undefinability theorem

The hydrodynamic (Madelung) formulation of quantum mechanics must be supplemented by the circulation-quantization condition of Wallstrom. We ask which parts of this global datum can be fixed by conditions tied to a region, and answer exactly. For conditions on the germ of the data along a compact region K, the separating power is computed in closed form: the strongest such condition is the coset test that the period vector, taken modulo the subspace V_K = im(H¹_c(Q∖K) → H¹(Q)), lie in the image of the lattice (2πℤ)^b₁. Every direction of the circulation datum is therefore either invisible to K or functionally determined by the germ along K, with no middle ground — the Aharonov–Bohm phenomenon at the level of axiomatization (Theorem 4.1, Corollary 4.2). The endpoint of the same scale is a no-go theorem: the carrier of a jet-local law covers no cycles, so no finite conjunction of jet-local axioms — pointwise conditions of arbitrary finite differential order, with arbitrary predicates — can replace the Wallstrom condition, not even one-sidedly (Theorem 3.4). A second and orthogonal blindness is identified as well: the entire jet language — arbitrary quantifiers over points, arbitrary predicates on jets, arbitrary Boolean structure — sees only the support of the jet occupation measure, whereas the circulation integrates against that measure; the condition is therefore not definable in that language at all (Theorem 3.7). The two negative results and the dichotomy are one principle, a ladder of locality: what decides the separating power of a class of conditions is the cycle coverage of its carrier — not differential order, not the amount of information the carrier holds, and not spatial extent. The proofs are constructive, based on explicit bump realizations of jets; the constructive steps are verified symbolically in an accompanying SymPy script. Extensions cover non-compact spaces, punctured domains and nodal sets, where vortex filaments source the datum and the known node-wise conditions are boundary rather than interior-local. A role-accounting survey shows that each proposed fix of the Wallstrom problem relocates, rather than eliminates, the condition. These negative results complement the recent momentum-map resolution of Khesin and Modin: it identifies where the condition lives; we prove it cannot be traded for local laws.