The answer was anisotropy

The measurements on brain MRI pointed somewhere specific. If a continuous field earns its keep where sampling is the binding constraint, then the place to look is data where sampling is genuinely poor along some axis. That is not exotic. It is most of clinical MRI outside neuroimaging.

A typical abdominal MR volume might be 1.5 mm in plane and 7 mm between slices. That is close to five to one anisotropy. A voxel lattice trying to represent a smooth invertible deformation across slices that far apart is being asked to do something genuinely hard.

Turning anisotropy into a dial

The CHAOS abdominal benchmark is unusually well suited to this, because it contains MR at roughly five to one and CT at roughly 1.3 to 1, cleanly separated with no overlap, inside one dataset with one label protocol. That turns anisotropy from a confound between datasets into a dial inside a single benchmark.

The primary endpoint is pre-specified, and deliberately it is not accuracy. It is the fold free rate: the proportion of tuned configurations producing a deformation with essentially no negative Jacobian determinants. Accuracy is secondary and only comparable at matched folding, because any model can buy overlap by letting the warp tear.

What the early runs suggest

Early search level runs on the anisotropic arm point the way the mechanism predicts, and the near isotropic arm looks flat, which is the pattern a dose response would produce.

They are run at reduced fidelity, so they are enough to justify the full study and not enough to be it. The paper says so, and lists input resolution under scope rather than results.

A flat arm where the mechanism predicts flat is doing as much work as a moving one where it predicts movement. That pairing is what turns an observation into a dose response, and it is why the design matters more than any single number.

Why the design matters more than the number

A single contrast, continuous beats lattice on one dataset, invites an obvious objection: something else about those two datasets could explain it. A dose response along an axis inside one dataset does not have that problem in the same way. The statistical model carries an explicit anisotropy by representation interaction term, so any confound that stays constant across the dial can shift the intercept but cannot manufacture the interaction.

The per axis decomposition is the encouraging part. Essentially all of the continuous field measured deviation from a lattice concentrates on the coarsely sampled axis rather than the two fine ones, which is exactly where the hypothesis says it should appear.

What has to happen next

One confound is named and queued. The continuous model carries a displacement clamp and the lattice control does not, and a clamp bears on folding directly, so an unclamped arm runs before any claim is made. On isotropic brain data the unclamped lattice already folds less rather than more, which is suggestive, and suggestive is not the same as settled.

Naming that in the paper before a reviewer does costs one paragraph and buys a great deal of credibility.