Model specification: Certificates for the conditional theorem

Fix $m\ge 3$, a finite actual predicate list $\mathsf{H}=(H_1,\dots ,H_r)$, and a predeclared single-valued rule ${\mathfrak{F}}_m$. Let

The conditional theorem requires the following certificates:

1. Non-vacuity. A specified witness $D_{\star }\in \mathcal{A}$, or a valid existence proof, is supplied.

2. Representation rule. For every $D\in \mathcal{A}$, the rule supplies $(V_D,{\iota }_D,K_D)$, with $\dim V_D=m$, satisfying Definition def:representation for $(E_D,{⪯}_D)$. A rule returning only a feasibility set does not satisfy this clause.

3. Selecting properties. Each constructed $K_D$ is homogeneous and strictly convex. These properties are either proved from the actual operational premises or explicitly listed as additional assumptions on the rule's output.

4. Matched affine covariance. The domain is invariant under the declared matched equivalences. For each such equivalence with induced $\bar{f}$, an invertible affine map $v\mapsto A_{f}v+b_f$ satisfies

   $${\iota }_{D^{\prime }}(\bar{f}(p))=A_f{\iota }_D(p)+b_f(p\in E_D).$$

   This includes self-equivalences. Mere existence of an unrelated unlabelled affine isomorphism is not this matched covariance.

5. Input and status audit. The actual rule and predicates use only named operational data and the declared non-geometric interface. No independently supplied spacetime metric, light cone, geometric causal relation, or invariant speed is an input. Every assumed property and class restriction is disclosed. The input does not prescribe the cone claimed to be extracted.

6. Additional-content and countermodel audit. The claimed comparison with the scheduler package is justified in a common operational model universe. Relevant countermodel interpretations and exclusions are identified and proved for this rule and carrier.

This is a certificate template. No actual successful $\mathsf{H},{\mathfrak{F}}_m,D_{\star }$ meeting all clauses is furnished here.