For a 3D scalar field, does the Frenet-Serret frame always align with the principal curvatures of its isosurfaces?

Question

In a given 3D scalar field that is smooth (in this case, with continuous 1st and 2nd derivatives, i.e. $f ∈ C^2(U)$), its gradient vector field is dual to the infinite set of isosurfaces in the function in the following sense: the gradient vector at any point in the function points perpendicularly to the isosurface at that point.

My question is, to what degree does the Frenet-Serret frame align with the isosurface principal curvature frame? Specifically, at each point, do the normal and binormal vectors of a "gradient path" align with the principal curvature directions of the corresponding isosurface?

If so, I cannot find anything in the literature to this effect, and I don't want to assume it just because it makes sense to me. I would love to see a paper or other source where this "alignment" is specified thoroughly.

To elaborate. I would think they always perfectly align. In such a function, no two gradient paths can cross each other, and by definition, no two isosurfaces can cross one another. With that and the below descriptions in mind, we already know that the tangent vector of a "gradient path" always lies perpendicular to the isosurface at a given point. Then, if I know the principal curvature directions and magnitudes at a point on an isosurface, I should therefore know the direction of the normal vector of the gradient path intersecting the isosurface at that point. This is because I also know that 1) in order for the "upper" and "lower" subsequent isosurfaces to be nested and not overlapping, and 2) in order for the gradient path to pass through all isosurfaces perpendicularly, the gradient path passing through the isosurface at that point must be curving "along with" the isosurface.

I would also expect that the magnitudes of path and isosurface curvature would be related, though not equal. For a counterexample, in a spherical function, all gradient paths are straight lines at every point, i.e. the normal vector is degenerate and has zero magnitude, while the isosurface principal curvature at every point goes as $\frac{1}{R}$ where $R$ is the distance from the "center" of the spherical function. So in this case, the magnitudes of $k_1$ and $k_2$ never match those of the normal and binormal vectors, though both the path normal vectors and surface principal curvatures are pointing in the same (degenerate) directions, so one could say they're still "aligned".

As an example, lets use $f(x,y,z)=cos(x)+cos(y)+cos(z)$. Here's an image depicting the first octant of $f$, with four randomly placed "gradient paths" (red) and 10 isosurfaces placed at regular (linear) intervals of the value of $f$. This is just to help visually convey my point; that the curvature of the gradient paths matches with that of the isosurfaces. The blue box indicates a region where one path is curving sharply, and the nearby isosurface is also curving sharply.

Frenet-Serret frame:

• For paths we have the tangent and normal (and binormal) vectors that describe the shape of the path at each point.
• The tangent (unit) vector $T$ lies tangent to the path, pointing in the direction of path "motion"
• The normal vector $N$ points in the direction that the path is instantaneously bending, and its magnitude corresponds to how sharply it is bending. It is calculated as the derivative of $T$ with respect to path arc-length. $N$ is always normal to $T$
• Path torsion is described by how the normal vector rotates around the path.

(Iso)surface principal curvature:

• At each point on a 2d surface embedded in $R^3$, such as an isosurface in our $f$, the first principal curvature $k_1$ is a vector pointing in the direction where the surface is bending the most (has highest geometric curvature). The sign of the principal curvature indicates whether the surface is bending "inward" or "outward" at that point, and the magnitude indicates the amount of curvature. The principal curvature lies tangent to the surface.
• The second principal curvature $k_2$ is perpendicular to the first, also lies tangent to the surface, and its magnitude describes the amount of surface curvature in that direction.