This paper shows that a basic spacetime distance function must be a power of a nondegenerate quadratic form if it obeys a few natural symmetry and determinism rules. The authors assume only an “invariant interval” D that assigns a single number to any displacement between events. They do not assume light, electromagnetic theory, observers, or the usual split between space and time.

The argument starts from four physical principles turned into mathematical axioms. Smoothness means D is continuous and differentiable away from zero. Homogeneity means the same rules hold everywhere and at every scale, so D(λv)=λ^pD(v) for some p>0. Isotropy means there is no preferred direction: the symmetry group acts the same way on each level set of D and can reverse transverse directions. Determinism of inertial motion says a freely moving particle is fixed uniquely by its initial point and direction.

From these assumptions the authors build a Lagrangian L(v)=D(v) and define inertial paths (geodesics) by the Euler–Lagrange equations. Homogeneity and the determinism axiom force those geodesics to be straight lines. The isotropy condition then restricts the shape of D on vectors and leads to the form D(v)=C (v^T S v)^{p/2} for a nondegenerate symmetric matrix S and constant C. In the case that S has indefinite signature — the case that allows a lightlike or null cone — the exponent p must equal 2, so D is exactly a quadratic form.

The result explains how Euclidean geometry (S positive definite) or Lorentzian geometry (S indefinite, which includes Minkowski spacetime) can arise from general structural assumptions. It recovers the usual invariant interval of special relativity and the associated symmetry group (the Lorentz transformations) as consequences, not as starting assumptions about light or observer frames.