The Volume Axis, Horizon Equilibrium and Spin as a Relation: What Follows from a Field of Labels

In the label framework matter is a field of comoving labels whose relabellings preserve the label volume. We ask which of its postulates follow from this description; premises are collected in one table. The volume axis: of the fifteen conformal generators exactly the ten Poincaré ones keep the spacetime volume, and on sampled backgrounds a three-label medium with nF''/F'>0 has real speeds in every direction only in signature (1,3). Horizon equilibrium: equilibrium of local horizons gives the trace-free Einstein equation, a positive coupling and Λ as a constant of integration. For minimally coupled scalars in one lattice regulator with step a, the area coefficients of a plane and of a sphere, both 0.0235/a² per field, differ by less than the spread of the fits. Spin as a relation: for a hedgehog in a compact label space, rotations of space and of labels lock, J=I, and with global label rotations and the Finkelstein–Rubinstein sign only half-integer spin is allowed. Colour (within the symmetries considered), fermion triplets (on the hedgehog route) and mass values do not follow; six postulates and thirteen groups of labels remain.