Everybody claims continuity. Almost nobody measures it.
Implicit representation papers all assert their field is continuous. I built the instruments to check, and the answer was more interesting than a yes.
Read enough papers on implicit neural representations for registration and you notice a pattern. Every one of them states that the field is continuous, that it can be queried at arbitrary resolution, that this is the central advantage. And not one of them measures it.
It is not that anybody is being dishonest. Continuity feels like a property you get by construction: you wrote a function, functions are continuous, what is there to check? But "the field is a continuous function" and "the field carries information a grid cannot" are completely different claims, and only the first one follows from the architecture.
Building the instruments
So I built measurements. Field continuity divergence asks how far the native field is from a trilinear resampling of itself, which is exactly the thing a lattice could have represented. Super Nyquist spectral energy asks how much of the field power sits above the voxel Nyquist frequency, which is the content a grid provably cannot carry. And a Jacobian convergence certificate compares the finite difference Jacobian against the automatic differentiation one as the step size shrinks: if the field is genuinely differentiable, those two must converge.
The convergence certificate came out clean. Relative error falls from 0.26 to 0.023 as the step shrinks by two orders of magnitude, which is what a differentiable field is supposed to do and what a staircase would not.
The result that reframed the project
The divergence measurement was more interesting. On 1 mm isotropic brain MRI the continuous field is measurably distinct from a lattice, where every grid baseline scores exactly zero by construction. But the size of that difference is small, and its effect on anatomical accuracy is smaller still.
The spectral measurement explained why. Almost all of the trained velocity field power sits below the voxel Nyquist frequency. The field is smooth. It is not secretly carrying high frequency structure that a grid was missing, because at 1 mm isotropic there is very little such structure left in the data to carry.
That is not a failure of the representation. It is a statement about the sampling regime, and it told me exactly where to look next.
Why this was worth doing
The honest version of the claim is narrower and much more useful than the one I started with. A continuous field is not automatically better than a lattice. It is better where sampling is the binding constraint. On isotropic brain MRI sampling is generous, so the two representations converge in what they can express, and the advantage shows up in topology and parameter efficiency rather than in raw overlap.
Having an instrument rather than an assertion is what let me say that precisely instead of hand waving. It also meant that when I found the regime where continuity does pay, I could show the mechanism rather than just the outcome.