... and the symmetry break down from SU(2) x SU(2) to SU(2) accompanied by the occurence of the Minkowski space and the well-posedness of the d'Alembert (wave) operator when equipped with the newly proposed distributional Hilbert space containing all "mechanical" distributional Hilbert scales.

(CoR) p. 763: “Families of spherical waves for arbitrarily time-like lines exist only in the case of two and four variables, and then only if the differential equation is equivalent to the wave equation“.

            

The Courant conjecture, Methods of Math. Physics, p. 763.pdf

A potential relationship to the UFT

The Hamiltonian dynamical (energy) operator in the proposed UFT supports the Berry-Keating conjecture (which is in line with the Hilbert-Polya conjecture). The related dynamical energy Hilbert space provides the appropriate framework for well-posed wave/radiation hyperbolic PDE models. An operator is only well-defined in combination with a defined domain (!). In case of the proposed domain (the compact (closed, connected) "unit quaternions") there are no mechanical "cause-action" (initial and boundary value) conditions required. Regarding the Courant conjecture the following related facts may support an appropriate answer:

- the S(1) and S(3) are the only spheres with a "continuous" group structure, (EbH) 7.2

- the spheres S(0), S(1), S(3), and S(7) are the only parallelizable spheres

- „Thurston‘s geometrization theorem stating that every closed connected 3-manifold can be decomposed in a canonical way into eight pieces that each have one of eight types of geometric structure. It is an anlogue of the uniformization theorem for two-dimensional surfaces, which states that every simple connected Riemann surface can be given one of the three geometries, Euclidian, spherical, or hyperbolic", Wikipedia

(CoR) Courant R., Hilbert D., Methods of Mathematical Physics, Volume II, Wiley Classics Edition, 1989

(EbH) Ebbinghaus H.-D., Numbers, Springer Science + Business Media, New York, 1991