open-problem-program.md

MHD Open Problem Program

Level A — tractable restricted problems

1. complete the variable-resistivity classification on cylindrical annular families
2. derive first-order perturbative correction formulas for selected nonconstant eta(r) profiles
3. expand the explicit counterexample library for false smooth-closure claims
4. extend the current smooth-axis no-go from radial families to broader separable cylindrical ansätze

Level B — restricted-to-broad problems

1. determine whether any smooth nonconstant-eta exact family survives on domains touching the axis
2. classify broader separable cylindrical families beyond the current radial/z/theta anchors
3. compare local and global closure behavior on annuli versus full disks

Level C — ambitious grounded problems

1. spherical and toroidal closure classification under nonconstant resistivity
2. geometric characterization of trivial versus nontrivial closure
3. closure uniqueness or no-smooth-closure theorems for realistic resistivity profiles

Best next theorem target

Extend the new smooth-axis variable-resistivity no-go from the radial supported families to a broad separable cylindrical family.

Best next falsification target

Any claim that variable resistivity merely perturbs the constant-resistivity exact families without changing the exact-family class.