Escher did that particular tessellation in both hyperbolic and elliptical geometric—the Poincare disk in "Circle Limit IV" shown in the question, and a spherical wooden sculpture (reproduction shown here: mcescher.com/product/sculpture-sphere-angels-devils ).

Is that supposed to have $\psi[(-\Delta)^{s}]$?

My high school geometry book included axioms that a line contained two points; a plane contained three non-collinear points; and space contained four non-coplanar points. Axioms like these are needed for a rigorous description of Euclidean geometry, but after introducing them, the textbook never mentioned them again.

The exponential mapping always maps provides a diffeomorphism from a sufficiently small ball around the origin in the tangent space (the Lie algebra $\mathfrak{g}$) to a neighborhood of $e$ in $G$. This gives the logarithmic coordinates for $G$ around the identity, which it seems like should be enough to prove this.

Indeed, as somebody who mostly works as a physicist now, my immediate reaction to the question was that the cosine transform formula had to be wrong, simply because it has the wrong units.

@Stef Continuous functions originating from the same point have a relatively natural measure on them, in the form of Brownian motion (Wiener measure). Under that measure, almost all the continuous functions are nowhere differentiable.

Do you mean to be working with partial orders or pre-orders (since the latter is what you have actually defined)?