A long chain with club continuity below aleph omega

If $\mathcal{H}\subseteq P$ has size $\rho <\mu$, set $b(n)=\sup \{u(n)+1:u\in \mathcal{H}\}$ when ${\mathrm{\aleph }}_n>\rho$, and $b(n)=0$ otherwise. At a large coordinate, each successor is below the infinite cardinal ${\mathrm{\aleph }}_n$, and F2–F3 place the supremum below ${\mathrm{\aleph }}_n$. Only finitely many coordinates have ${\mathrm{\aleph }}_n\le \rho$, since ${\sup }_n{\mathrm{\aleph }}_n=\mu$. Thus $b\in P$ and $u{<}^{\ast }b$ for each $u\in \mathcal{H}$. The empty family gives $b=0$ and imposes no comparisons. If instead $\mathrm{\mid }\mathcal{H}\mathrm{\mid }=\mu$, take a bijection $q:\mu \to \mathcal{H}$ and let ${\mathcal{H}}_j=q[{\mathrm{\aleph }}_j]$, so $\mathcal{H}={\bigcup }_{j<\omega }{\mathcal{H}}_j$ and each $\mathrm{\mid }{\mathcal{H}}_j\mathrm{\mid }<\mu$. The preceding formula gives a bound $b_j$ for each subfamily and a bound $b$ for the countable family $\{b_j:j<\omega \}$. For $u\in {\mathcal{H}}_j$, compose $u{<}^{\ast }{b}_j{<}^{\ast }b$ outside the union of the two finite failure sets. AC ensures these cardinalities and the indicated enumeration; thus every family of size less than ${\mu }^{+}$ has a strict bound.

For each nonzero limit $\delta <\lambda$, F2 gives a cofinal sequence of length $\theta =\mathrm{cf}(\delta )$. Turn it into a continuous increasing sequence in $\delta$: at successor stages choose a value above both the preceding value and the next cofinal-sequence term, and at nonzero limits take the supremum of earlier values. Each intermediate supremum is below $\delta$ because fewer than $\mathrm{cf}(\delta )$ terms were used, by F2. Successor choices are possible because $\delta$ is a limit. The range is unbounded and closed in $\delta$, hence is a club $E_{\delta }$ of order type $\theta$; closure follows because every nonzero limit point below $\delta$ occurs at a limit index of the continuous sequence. F4 supplies the recursion, and A1 chooses these clubs simultaneously from their nonempty witness sets. Also fix by A1 a choice function on the nonempty subsets of the set $P$ for use in choosing bounds.

Recursively set $f_0=0$. At every $0<\delta <\lambda$, the previous functions form a family of size at most $\mathrm{\mid }\delta \mathrm{\mid }<\lambda$, so step 1.1 and the choice function of step 1.2 specify a strict eventual bound $u_{\delta }\in P$. If $\mathrm{cf}(\delta )={\kappa }^{++}$ for an uncountable regular $\kappa <\mu$, put $v_{\delta }(n)={\sup }_{\xi \in E_{\delta }}{f}_{\xi }(n)$ at coordinates with ${\mathrm{\aleph }}_n>{\kappa }^{++}$, and put $v_{\delta }(n)=0$ at the others; then set $f_{\delta }(n)=\max \{u_{\delta }(n),v_{\delta }(n)\}+1$. The large-coordinate supremum is below ${\mathrm{\aleph }}_n$ because $\mathrm{\mid }{E}_{\delta }\mathrm{\mid }={\kappa }^{++}<\mathrm{cf}({\mathrm{\aleph }}_n)$ by F2–F3. Its successor is still below that infinite cardinal. At all other stages put $f_{\delta }=u_{\delta }$. Distinct cardinals have distinct double successors, so the special rule is unambiguous. Every value lies in $P$, and F4 implements this specified recursion through $\lambda$. For every $\xi <\delta$ we have $f_{\xi }{<}^{\ast }{u}_{\delta }\le f_{\delta }$, proving strict increase.

Fix any uncountable regular $\kappa <\mu$. By F3 it is ${\mathrm{\aleph }}_j$ for some positive finite $j$, so ${\kappa }^{++}={\mathrm{\aleph }}_{j+2}<\mu <\lambda$, and $\lambda$ is regular by F3. At each index $\delta$ of that cofinality, step 2.1 used the special rule, giving ${\sup }_{\xi \in E_{\delta }}{f}_{\xi }(n)=v_{\delta }(n)<f_{\delta }(n)$ whenever $n>j+2$. The other coordinates form a finite set. This is the required club-continuity premise with bounding index exactly $\delta$. The countably infinite coordinate set and strict chain from step 2.1 meet F1's remaining hypotheses. F1 gives $(\ast {)}_{\kappa }$ for this same chain. As $\kappa$ was arbitrary, all stated properties hold simultaneously. QED.