It is commonly stated that the low energy theory of some non-relativistic fermion band structure with a linear dispersion $E = vk$ implies the
low energy effective theory is a relativistic fermion theory.
there is an emergent Lorentz symmetry at or near the node
Of course $E = vk$ is necessary condition, but it does not seem to be sufficient to show 1 and 2. Also the Lorentz spinor degrees of freedom is unclear.
why 1 and 2 are true? What conditions guarantee those? (Sketch a physics proof?)
How to determine which relativistic fermions they have at low energy: Dirac, Majorana, Weyl or Majorana-Weyl spinors? as Lorentz spinors?
why are they Lorentz spinors? (they can be approximate spatial rotational symmetries, but not the Lorentz spacetime boost + rotational symmetries ?)