Theorem: Conditional frame-invariant maximal speed

Let $D\ge 2$. Assume C1–C5 and C7, with Euclidean $\nu$. Every $g\in \mathcal{G}$ is an orthochronous Poincare transformation. The operational maximal speed $c_{\mathrm{cont}}=c$ has the same value in every admitted inertial frame.

Proof. Rescale the time coordinate to $ct$. The cone order is the Minkowski causal order. The Alexandrov–Zeeman theorem identifies its bijective order automorphisms, in spacetime dimension at least three, as

where $\lambda >0$ and $L$ is an orthochronous Lorentz transformation, possibly reversing spatial orientation Alexandrov,Zeeman.

Such a map multiplies spacetime volume by ${\lambda }^{D+1}$. C7(iii) therefore forces $\lambda =1$. C7(iv) connects the distinguished inertial direction with every admitted one. By C7(i), signalling pairs and their law gaps transform with the frame. Poincare transformations preserve the boundary cone, and Theorem thm:contspeed gives its attained maximal speed $c$.