Well this is just an early example of the lorentz breaking string theory niche. You can find a lot more. In the short time frame where opera had this supposedly faster than light neutrinos, a lot of papers were published in that regard.
For example you can have string theories that lead to finsler spacetimes, which were used to explain the opera results.
I am not sure what you are refering to. You absolutely can break Lorentz Invariance in string theory[1]. There is a reason why even some string theory researchers call it the theory of anything.
I agree, that neutrinoless double beta decay would be incredibly interesting, but it is very speculative (and depending on the neutrino mass hierarchy not really falsifiable).
My original point was just that lepton number is not a good symmetry as it is broken by rhe chiral anomaly, which is not speculative at all. Of course, the sphaleron effects are negligible in collider settings, but for cosmology they are crucial and might be indirectly observable.
I kind of disagree on your statement about lepton violation. The standard model predicts lepton violating processes (sphalerons). The true symmetry of the standard model is B-L. Of course you are right, that these topological effects will not lead to majorana mass terms.
Just because they are not mainstream, you definitely can have lorentz violation in string theory [0]. Spontaneous symmetry breaking can lead to induced finsler geometries, which can basically have multiple light cones (when there was the "faster than light neutrinos" result on the table. Some people used finsler spacetime to explain them).
So you made your prediction only by choosing an axiom.
There is a reason that even some proponents of string theory call it the theory of anything.
This comparison is not really fair. On the one hand you have the prediction of a scalar excitation with a lot of restrictions (cosmology and naturalness). On the other hand you have a giant framework that can predict or fit almost anything.
Don't get me wrong, I still regard string theory as a big success. It taught us a lot about mathematics and field theories in the last decades.
However the predictive nature is basically non-existant so far.
I tried to use the extension, but unfortunately i couldn't resolve my problems with it. I always run into errors when i try to execute queries on delta tables.
If you have some knowledge of group theory the 'why' is pretty straight forward.
Quaternions are the generators of SU(2) which is a double covering of SO(3). The latter describes rotations in 3 dimensions. Thus you can express any rotation in 3d with quaternions.
For example you can have string theories that lead to finsler spacetimes, which were used to explain the opera results.