Theorem: Faithful generated Lorentz representation

For every datum in Specification spec:rich, (eq:pilot-embedding) is injective and satisfies (eq:faithful)–(eq:generated), with

The cone is homogeneous and strictly convex in the cone sense. The construction is covariant under trial relabellings carrying their operational parameters and under matched common changes of the declared Pauli frame.

Proof. Within a trial, the positive differences are $d_e,2d_e$, so its operational chain is represented correctly.

Across two trials the offset separation is at least ten. Since $k,l\in \{0,1,2\}$,

Neither directed cross-trial difference lies in ${\mathcal{L}}_4$. The points are also distinct.

Every generating displacement belongs to ${\mathcal{L}}_4$. Conversely, density of the $r_e$ places every $(1,u)$, $u\in S^2$, in the closed generated cone. If $\Vert x\Vert \le t$, write $x/t$ as a convex combination of unit vectors when $t>0$. For $x\ne 0$, the combination of $u=x/\Vert x\Vert$ and $-u$ with weights $(1+\Vert x\Vert /t)/2$ and $(1-\Vert x\Vert /t)/2$ suffices. The zero case uses any antipodal pair. Hence the generated cone is exactly ${\mathcal{L}}_4$. Its generators span $V$. The cross-trial estimate and the within-trial calculation now prove faithfulness for the actual $K$.

Two nonproportional nonzero future-null vectors have a sum in the interior by the strict case of the triangle inequality. This is cone strict convexity.

For homogeneity, rotate an interior vector $(t,x)$ so $x$ lies on the first spatial axis. With $a=\sqrt{t^2-\Vert x{\Vert }^2}>0$, write $t=a\cosh \eta$, $\Vert x\Vert =a\sinh \eta$. A Lorentz boost sends $(a,0)$ to $(t,x)$, and positive scaling relates all such $a$. These invertible maps preserve the cone.

Trial relabelling merely transports $r_e,p_e$. A matched common unitary change of Pauli frame rotates both Bloch vectors and the calibrated first axis by the same orthogonal map. The corresponding spatial rotation, with identity on the first coordinate, intertwines the representations.