Amalgamating two sweet models over a common complete subalgebra

The amalgam data are those of [F3]: $O=P_1{\ast }_{P_0}{P}_2$ consists of the pairs admitted by a common positive $B_0$-condition and is ordered coordinatewise. Its canonical dense subset is $D=\{(q_1,q_2)\in O:q_{\mathrm{\ell }}\in D_{\mathrm{\ell }}\text{ for }\mathrm{\ell }=1,2\}\mathrm{.}$ The denseness assertion already includes the synchronization of the two quotient witnesses; it is not inferred from coordinatewise denseness alone.

For $x=(q_1,q_2)\in D$, let $m(x)$ be the least $m$ such that every pair $(q_1^{\mathrm{\prime }},q_2^{\mathrm{\prime }})$ with $q_{\mathrm{\ell }}^{\mathrm{\prime }}<msubsup><mi>E</mi><mi>m</mi><mi\; mathvariant="normal">\ell </mi></msubsup>q_{\mathrm{\ell }}$ is admitted. Existence is the double application of [F2] in the proof of [F3]: a countable directed cover of $P_0$ is used first for $P_1$ and then, after retaining the dense subfamily below the admission witness, for $P_2$. Two reductions in the same directed piece have a common strengthening and hence admit the perturbed pair. The least number $m(x)$ depends only on the two equivalence classes and admission, not on a chosen witness.

No partial-isomorphism extension theorem is needed. The theorem [F3] applies directly to the two named complete embeddings of the arbitrary common complete subalgebra $B_0$. The weak-coordinate maps give complete canonical copies of both factors in the full amalgam, independently of whether those canonical conditions belong to the selected dense presentation $D$.

If $x^{\mathrm{\prime }}=(q_1^{\mathrm{\prime }},q_2^{\mathrm{\prime }})$ has in both coordinates, then the relevant factor classes are unchanged and minimality gives $m(x^{\mathrm{\prime }})=m(x)$. Hence [F3] defines These relations refine with $n$, have countably many classes, and every class is downward directed. In particular every two members of one $E_n$-class are compatible. The diagonal and transfer assertions are the coordinatewise sweetness clauses combined with the same common-admission property; they are not consequences of pairwise compatibility alone.

The canonical embeddings are complete by the exact conclusion of [F3]; no countable-generation hypothesis on $B_0$ is present in that theorem.

Countable chain condition: the amalgam is sweet by [F3] and step 2.1, and a sweet forcing is a countable union of directed sets, hence ccc by [F4].

The steps above exhibit the intrinsic least modulus, the $E_n$-classes, the two canonical complete embeddings and the ccc conclusion for the amalgam over an arbitrary common complete subalgebra, verifying the claimed instance.