How do octonions manage to dictate the properties of elementary particles?

Answer: Just as the Dirac equation carries much more information than it's square i.e. the Klein-Gordon equation, the octonionic space carries much more information than it's square, the conventional Minkowski space-time. Octonionic space is to Minkowski spacetime what Dirac equation is to the Klein-Gordon equation.

In the space spanned by the octonion O, elementary particles are defined by different directions. This is achieved through spinors of a Clifford algebra constructed from octonionic chains. The symmetries of the octonion algebra coincide with those of the standard model, allowing the identification of quarks and leptons. There is no freedom.

The dynamical variable Q is a matrix valued quantity on this octonionic space.

The norm of O has the properties of a metric [Euclidean / Lorentzian]. We can make the a_i functions of position, and we will get the analog of the usual curved space metric. We can construct a QFT on this metric in the usual way, but if we did not know that this metric has come from an underlying octonionic space we would not realise that properties of the elementary particles have already been fixed by the O-space. This is like Klein-Gordon vs. Dirac. This is how octonionic dynamics knows more than QFT on Minkowski spacetime.

And this is happening at low energies, before the interactions are made strong by going to high energies. The latter is achieved by switching on the Q, which are the analog of [square-root] of the metric. The equivalent of the metric comes from taking the trace of a matrix polynomial - this is the Lagrangian in our theory.

The Q-matrices incorporate the known standard model interactions, and also the precursor of gravitation. The low energy limit is achieved by setting the Q matrices to identity.

In conventional QFT, even if one works with tetrads (i.e. the square-root of the metric) the manifold is still R^4, and hence commutative. Quantum space-time is achieved by replacing R^4 by the octonionic space. This is the completion of quantum theory, where quantumness is extended to spacetime as well. In so doing we realise we no longer have the freedom to choose symmetries and properties of elementary particles. The octonions dictate these.

This can be called lifting of general relativity to the quantum level. In GR spacetime tells matter how to move. Now (octonionic) spacetime is telling (quantum) matter what to be !! Matter still curves spacetime - this is what the Q-matrices do, but now the curving represents not just gravitation, but all the four known forces.