Hypergeometrical Universe

The rest of the paper in fact makes use of some simple geometrical arguments to obtain distances and I have to admit I do not understand what they refer to. Proper null geodesics of the metric must be calculated to be able to say anything about distances.

Answer: Those simple arguments are worth understanding. Within them are :

the concept that light that reaches us here and now, has to “diffract” from hypersphere to hypersphere at 45 degrees.

The current Doppler shift is replaced by a projection of a 4D k-vector onto a local hyperplane (a hyperplane is a local approximation of a very large hypersphere in a small neighborhood). As light travels from hypersphere to hypersphere the 4D k-vector adjust itself until it becomes just a retarded electromagnetic wave always traveling at 45 degrees.

This theory was created using a totally different framework than the one you are used to. That explain the difficulty you faced. The idea that “one needs to use null geodesics of the metric to say anything about distances” show attachment to standard tools. The obvious distinction that the hypersphere has a symmetric and homogeneous distribution of mass and thus will pose a very different metric than the ones the reviewer might be used to was not realized.

* I do not understand the assertion that the speed of light is \sqrt{2}c. What is light for the author? What is the experiment which would give such a result?

No experiment would show that since we live in a 3D Hypersphere.

The proposed article would support the Hyperspherical Geometry and the Lightspeed Expansion and thus indirectly support a dynamical reference frame traveling at the speed of light.

I mentioned in the past answers to this Peer Review. Light is a spatial modulation of dilaton field. Not unlike a modulation on a carrier.