Model extraction from the completed well-ordering set (issue #12, step 5)

The completion endpoint hands step 5 a Henkin-complete S containing the base diagram Bφ = {φ̂} ∪ {d_q < d_r : q < r} opaquely; the quotient term model of the forward truth lemma then realizes every member of S, hence every member of Bφ.

This file is the relational/countable extraction: [L.IsRelational] is consumed by the quotient term model (the relational-core collapse of closed terms to constants), and [Countable (Σ l, L.Relations l)] by the fair enumeration. Removing both restrictions is the later transport step, not done here.

The rational map through the interpreted constants

The step-5 endpoint (relational/countable form)

theorem FirstOrder.Language.exists_model_relPreserving_relational {L : Language} (φ : L.Sentenceω) (lt : L.Relations 2) [L.IsRelational] [Countable ((l : ℕ) × L.Relations l)] (h : HasWellOrderedChains φ lt) : ∃ (M : Type) (x : L.Structure M) (_ : Nonempty M) (f : ℚ → M), φ.Realize M ∧ RelPreserving lt f

Step 5 endpoint, relational/countable form (Marker, Theorem 4.26 at a relational language with countable relational core): under the well-ordered-chains hypothesis, some nonempty model of φ carries a relation-preserving map f : ℚ → M — the rational map of the extracted expansion, through its reduct L-structure. The raw positive conclusion (D2): no injectivity, no order-embedding packaging; those are derived corollaries tracked for subsequent commits, and the arbitrary-language form is the later transport step.