The frozen target and its acceptance conditions

Equation (eq:faithful) prevents an arbitrary supercone from being called an extracted causal cone. Equation (eq:generated) prevents an independently appended Lorentz cone from supplying the desired answer. Neither equation follows merely from drawing a dependency graph.

-generation check. If $E_D$ is finite, the set of generators in (eq:generated) is finite. Its conical hull is a closed polyhedral cone. By contrast, ${\mathcal{L}}_m$, $m\ge 3$, has the infinitely many extreme rays

Invertible linear maps preserve extreme rays. Therefore a finite event carrier with exactly the generation rule (eq:generated) cannot supply the proposed Lorentz cone.

A later theorem must consequently declare an infinite operational family, an operationally justified extension, or a limiting construction. This is not a requirement of infinitely many parties. It is a requirement that the claimed continuum cone not be substituted for the finite event-difference cone without a connecting construction. A finite tomographically complete list can determine a declared finite-dimensional process; that fact is different from determining a continuum event carrier CDP,Pollock.

Find one named quantum process class $\mathcal{P}$, one named data assignment $\mathcal{D}(\mathcal{P})$, a finite list

and one predeclared extraction rule $\mathfrak{E}$, such that:

1. The admissible data set

   $${\mathcal{D}}_{\mathsf{H}}=\{D\in \mathcal{D}(\mathcal{P}):H_1(D),\dots ,H_r(D)\}$$

   is nonempty.

2. For every $D\in {\mathcal{D}}_{\mathsf{H}}$, the rule $\mathfrak{E}$ constructs a specified order ${⪯}_D$ and a faithful generated affine representation $(V_D,{\iota }_D,K_D)$ as in Definition def:representation.

3. On a declared fixed dimension $m\ge 3$, an invertible linear map $T_D:V_D\to {\mathbb{R}}^m$ exists with

   $$T_D(K_D)={\mathcal{L}}_m.$$

   Consequently the represented event comparisons satisfy

   $$p{⪯}_{D}q⟺T_D({\iota }_D(q)-{\iota }_D(p))\in {\mathcal{L}}_m.$$

4. The extraction uses only the specified operational data, their declared non-geometric interface, and $\mathsf{H}$. No spacetime metric, geometric causal relation, physical light cone, or invariant speed is installed as input.

5. The additional class restrictions and hypotheses are not supplied merely by $\mathsf{S}$. Their status is stated before the extraction is applied.

A comparison with a supplied VII-J-type causal relation is a subsequent embedding problem, not part of the selection.

Research target target:selection is a specification of the result to be established in later stages. It has no proof in this note.