Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut–Magidor Forcing

1. Introduction

Simultaneous stationary traces and branch-cover obstructions interact with level-size constraints in a typed forcing motivated by ladder systems on two-cardinal trees. For a ladder system on a two-cardinal tree, a cofinal branch may meet the ladder cofinally, stationarily, or on a club. Hayut and Magidor proved, relative to a supercompact cardinal, a separation between ${\omega }_1$-cofinal and ${\omega }_1$-club catching at ${\omega }_2$ ([9], Theorem 5.2, p. 1121), and asked whether cofinal catching can be separated from stationary catching ([9], Question 6.1, p. 1128). Their four-coordinate construction combines a binary ordinal tree, coherent designated branches, a generic ladder, and an auxiliary coordinate ([9], Definition 5.5(1)–(4) and Notation 5.6, p. 1123).

The notion of a ${\mathcal{P}}_{\kappa }\lambda$-tree goes back to Jech’s strong-compactness characterization ([10], Section 2), while Magidor characterized supercompactness by the corresponding ineffability principles on ${\mathcal{P}}_{\kappa }\lambda$ for every $\lambda \ge \kappa$ (Magidor1974, p. 282). Subsequent work developed the strong tree property at accessible cardinals, including two successive cardinals, small cardinals, and successors of singular cardinals [4,5,6]. The generalized ineffable and super tree principles were related to the Proper Forcing Axiom and to their motivating large-cardinal characterizations in [19,20]. Their interaction with the singular cardinal hypothesis was analyzed in [7], while their consistency at successors of singular cardinals was studied further in [1,8]. More recent work relates generalized tree properties to guessing models, Kurepa trees, and cardinal arithmetic [13,14]. Two-cardinal Kurepa families give a related bound on the number of restrictions to small sets. For every uncountable strong limit $\kappa$ and $\lambda \ge \kappa$, there is a family of $2^{\lambda }$ subsets of $\lambda$ with fewer than $\kappa$ distinct restrictions to each $x\in {\mathcal{P}}_{\kappa }\lambda$ ([21], Definition 2.1(i)–(ii) and Proposition 2.2(iii), p. 4). Together, these developments place ladder-system catching within the broader theory of two-cardinal branch principles.

Two features of the published presentation require explicit choices in a typed formulation. First, the displayed type of the auxiliary coordinate and the types used in the closure and no-club-catching arguments do not coincide ([9], Definition 5.5(4), p. 1123, and Claims 5.7–5.8, p. 1124). Second, the displayed tree clause imposes no cardinality restriction on a condition tree, although the generic object is described as an $\alpha$-tree and Claim 5.7 states a $2^{<\alpha }$ bound for the size of the forcing ([9], Definition 5.5(1), Notation 5.6, and Claim 5.7, p. 1123). Accordingly, ${\mathbb{S}}^{*}\left(\alpha \right)$ is an explicitly typed variant retaining the displayed unrestricted tree clause. Its auxiliary coordinate is a partial function on countable sets. A non-dummy value at z is a node of the restriction level indexed by z, and ladder coordinates of cofinality at most $\omega$ receive the empty value. No forcing equivalence with the printed presentation is claimed. The precise comparison is given in SubSection 2.1. Unless explicitly stated otherwise, cardinal and regularity assumptions on $\alpha$ refer to the ground model.

The resulting forcing separates the trace and cover phenomena from the width behavior of the unrestricted tree clause. Simultaneous bistationary traces and the countable branch-cover obstruction survive an explicit Kurepa-style level-size bound, whereas the unrestricted clause drives the wide-level phenomenon and the exact strong-limit boundary for the scaled restriction system.

Let $T_{\alpha }$, $L_{\alpha }$, and $B_{\alpha }\left(u\right)$ denote the generic tree, ladder-coordinate set, and designated branches added by ${\mathbb{S}}^{*}\left(\alpha \right)$. For $b{\in }^{ \alpha }2$, put ${Tr}_{{\omega }_1}(b,L_{\alpha })=\{x\in {\mathcal{P}}_{{\omega }_1}\alpha :b\upharpoonright x\in L_{\alpha }\}$, and say that $\mathcal{B}{\subseteq }^{ \alpha }2$ generates $L_{\alpha }$ when each $y\in L_{\alpha }$ is extended by some $b\in \mathcal{B}$.

Theorem 1.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Then α remains a regular cardinal in $V\left[G\right]$, and the following assertions hold.

(i)

for every nonempty countable $U\subseteq T_{\alpha }$, the set

$$\bigcap _{u\in U}{Tr}_{{\omega }_1}(B_{\alpha }\left(u\right),L_{\alpha })$$

is stationary and costationary in ${\mathcal{P}}_{{\omega }_1}\alpha$.

(ii)

no countable family of cofinal branches through $T_{\alpha }$ generates $L_{\alpha }$.

(iii)

for every infinite cardinal $\mu <\alpha$ in V, there are a limit ordinal $\delta <\alpha$ and an injection

$${{(}^{\mu }2)}^V⟶{\left(T_{\alpha }\right)}_{\delta }.$$

(iv)

the endpoint-corrected Hayut–Magidor-style restriction family

$${\mathcal{T}}_{\alpha ,x}^{{\omega }_2}=\{r\upharpoonright x:r\in {\left(T_{\alpha }\right)}_{sup\left(x\right)+1}\}(x\in {\mathcal{P}}_{{\omega }_2}\alpha )$$

has a level of size at least ${\omega }_2$ and hence is not a ${\mathcal{P}}_{{\omega }_2}\alpha$-tree.

For $x\in {\mathcal{P}}_{\alpha }\alpha$, put ${\mathcal{T}}_{\alpha ,x}=\{r\upharpoonright x:r\in {\left(T_{\alpha }\right)}_{sup\left(x\right)+1}\}$, and write ${\mathcal{T}}_{\alpha }$ for this scaled restriction system. Its levels have size less than α if and only if α is strongly inaccessible in V. Under these equivalent conditions, ${\mathcal{T}}_{\alpha }$ is a ${\mathcal{P}}_{\alpha }\alpha$-tree, and no countable family of its branches generates $L_{\alpha }$. In particular, if α is a successor cardinal in V, then the scaled restriction system is not a ${\mathcal{P}}_{\alpha }\alpha$-tree.

The empty-value convention also implies the level-domain failure in Proposition 1, independently of the unrestricted tree clause. Simultaneous stationarity uses a dummy value to reserve a countable domain forced into a named club. Anchored insertion is then applied to countably many designated seeds while the same dummy is retained. $\sigma$-closure then gives a common lower bound putting every corresponding branch restriction into $L_{\alpha }$. Costationarity comes from the complementary non-dummy value and Proposition 4. Thus every nonempty countable intersection of the designated traces is bistationary.

The countable branch-cover obstruction has two proofs. A small-cover argument uses ${\omega }_1$ designated branches with stationary traces, while a forcing argument diagonalizes uniformly against any prescribed countable sequence of branch names. The two core trace and cover conclusions also survive the Kurepa-style level bound of Proposition 5, so they do not depend on unrestricted width. The unrestricted tree-extension construction produces levels containing copies of ${{(}^{\mu }2)}^V$ for every infinite cardinal $\mu <\alpha$ in V. Consequently, the restriction family indexed by ${\mathcal{P}}_{{\omega }_2}\alpha$ is always too wide. For the scaled system indexed by ${\mathcal{P}}_{\alpha }\alpha$, all restriction levels are small precisely when $\alpha$ is strongly inaccessible in V.

The conclusions for ${\mathbb{S}}^{*}\left(\alpha \right)$ concern its direct generic extension. They do not settle the question posed by Hayut and Magidor ([9], Question 6.1, p. 1128). Immediately before that question, Hayut and Magidor note that their witnessing branch already meets the ladder system on a stationary rather than merely unbounded set ([9], p. 1128). Here every nonempty countable intersection of designated traces is bistationary, and the generic ladder-coordinate set cannot be generated by countably many cofinal branches.

The hypotheses on $\alpha$ have distinct roles. Ground-model regularity keeps all strategic limit constructions below $\alpha$, and $\alpha$-strategic closure preserves $\alpha$ as a regular cardinal. For the ${\omega }_1$-length fixed-seed construction, the additional bound $\alpha \ge {\omega }_2$ keeps the supremum of the tree heights below $\alpha$ and leaves room for a ladder coordinate of cofinality ${\omega }_1$. The unrestricted tree clause makes the Hayut–Magidor-style restriction family too wide and gives the exact strong-limit boundary for the scaled ${\mathcal{P}}_{\alpha }\alpha$ system.

2. The Typed Forcing and Branch Covers

2.1. Relation to the Hayut–Magidor Construction

The published presentation motivates the typing choice for the f-coordinate. The displayed Definition 5.5(4) requires pairs $(z,r)$ with $r\in t_{supz}\cup \{-1\}$ ([9], Definition 5.5(4), p. 1123). Such a non-dummy value has ordinal domain. In contrast, the proof of Claim 5.8 uses the assignment $f^q(M\cap \delta )=x$ with $dom\left(x\right)=M\cap \delta$, while the proof of Claim 5.7 writes $f\left(y\right)$ for a ladder restriction y ([9], p. 1124).

A non-dummy value at a countable set z is the restriction to z of an ordinal-tree node. This makes both $range\left(f\right)\cap \ell \left(\xi \right)$ and $f(M\cap \delta )=x$ well typed. Accordingly, the expression $f\left(y\right)$ in that proof is interpreted as $f(dom(y\left)\right)$. At coordinates of cofinality at most $\omega$, this variant imposes the empty value. Definition 5.5(2) prescribes the displayed ladder form only at uncountable cofinality ([9], Definition 5.5(2), p. 1123), while the proof of Claim 5.7 uses the empty value at the countable limit in its closure construction ([9], p. 1124).

The printed order requires only end extension of the f-coordinate ([9], Definition 5.5, p. 1123). The typed variant requires every new f-domain to have supremum above the previous tree height. This convention is used in the support and disjointness arguments at limit stages.

The displayed tree clause in ([9], Definition 5.5(1), p. 1123) imposes no cardinality restriction on a condition tree. At the same time, the surrounding discussion describes the generic object as an $\alpha$-tree in the ${\mathcal{P}}_{{\omega }_2}\alpha$ sense fixed immediately before that definition ([9], p. 1122), and ([9], Claim 5.7, p. 1123) states that the forcing has size $2^{<\alpha }$. If $\alpha$ is strongly inaccessible, every condition level nevertheless has size less than $\alpha$, since $|{\left(t_p\right)}_{\xi }|\le 2^{\left|\xi \right|}<\alpha$. At successor cardinals the displayed clause gives no corresponding automatic bound. The displayed unrestricted tree clause is retained rather than replaced by a size-controlled alternative. The unconditional wide-level and exact strong-limit conclusions in Lemma 6 and Proposition 7 depend essentially on this retained clause. Because of the remaining typing conventions, forcing equivalence with the printed presentation is not claimed.

Cummings’s standard Kurepa-tree forcing at an inaccessible cardinal imposes the level bound $|T_{\xi }|\le |\xi |+\omega$ ([2], Example 6.1, p. 794). Proposition 5 shows that the bistationary-trace and countable branch-cover conclusions remain valid for the companion forcing obtained by restricting the typed conditions to those satisfying this bound. Thus the two core trace and cover conclusions are not consequences of unrestricted width.

The restriction-system passage ([9], p. 1122) is stated for ordinals $\alpha \le {\omega }_2$ and indexes levels by $sup\left(x\right)$. For the regular cardinals considered here, the endpoint-corrected analogue is ${\beta }_x=sup\left(x\right)+1$. The successor is necessary when x has a maximum.

The unchanged b-coordinate produces the designated branches ([9], Definition 5.5(3) and Notation 5.6, p. 1123). At countable limits, a dummy value reserves a new supremum. At uncountable limits, equality between a ladder restriction and a non-dummy f-value forces equality of their domains, and uniqueness of the corresponding supremum gives the required f–ℓ disjointness. The auxiliary coordinate plays complementary roles in Theorem 3 and Proposition 4. A non-dummy value excludes a decided restriction, while a dummy value leaves room for a later insertion.

For an arbitrary ${\mathcal{P}}_{{\omega }_2}\lambda$-tree and ladder system in the relevant intermediate extension, Lemma 5.11 constructs a branch in a lifted model ([9], Lemma 5.11, pp. 1125–1126). Claim 5.12 shows that the further forcing adds no ${\omega }_1$-cofinally catching branch when the original generic extension has none ([9], Claim 5.12, p. 1126). The final paragraph of the proof of Lemma 5.11 ([9], p. 1127) shows that the lifted-model trace contains a club. This does not determine the traces of the designated family considered here.

2.2. Definition, Closure, and Branch Covers

For a cardinal $\rho$ and a set X, let

$${\mathcal{P}}_{\rho }X=\{x\subseteq X:|x|<\rho \}.$$

A subset of ${\mathcal{P}}_{\rho }X$ is club if it is unbounded under inclusion and closed under increasing unions of length less than $cf\left(\rho \right)$. This is the convention stated immediately before Definition 4.1 of ([9], p. 1119). Throughout this paper, the increasing sequences in this closure clause are understood to have positive length. A set is stationary if it meets every club. For background on generalized stationary sets, see ([11], pp. 93–128).

For cardinals $\kappa \le \lambda$, a ${\mathcal{P}}_{\kappa }\lambda$-tree is a system

$$\mathcal{T}=\langle {\mathcal{T}}_x:x\in {\mathcal{P}}_{\kappa }\lambda \rangle$$

such that $⌀\ne {\mathcal{T}}_x{\subseteq }^{ x}2$, restrictions from larger levels belong to smaller levels, and $|{\mathcal{T}}_x|<\kappa$ for every x. A branch is a function $b:\lambda \to 2$ satisfying $b\upharpoonright x\in {\mathcal{T}}_x$ for every $x\in {\mathcal{P}}_{\kappa }\lambda$. This is Jech’s notion ([10], Section 2), in the formulation used in ([9], Definition 4.1, p. 1119).

A ladder system on $\mathcal{T}$ is a set $L\subseteq {\bigcup }_{x\in {\mathcal{P}}_{\kappa }\lambda }{\mathcal{T}}_x$ such that $L\cap {\mathcal{T}}_x\ne ⌀$ for club many x, and whenever $\eta \in L\cap {\mathcal{T}}_x$ and $cf\left(\right|x\cap \kappa \left|\right)>\omega$, there is a club $E_{\eta }\subseteq {\mathcal{P}}_{|x\cap \kappa |}x$ satisfying $\eta \upharpoonright y\in L$ for every $y\in E_{\eta }$ ([9], Definition 4.2, p. 1119).

Let $\rho$ be an infinite cardinal and let $\lambda$ be an ordinal. For a function $b{\in }^{ \lambda }2$ and a set L of partial functions on $\lambda$, put

$${Tr}_{\rho }(b,L)=\{x\in {\mathcal{P}}_{\rho }\lambda :b\upharpoonright x\in L\}.$$

The function b meets L$\rho$-cofinally if this trace is cofinal under inclusion, and meets L on a $\rho$-club if the trace contains a club. Here, club is understood in the sense recalled above. The source notion following Definition 4.2 ([9], p. 1119) is formulated for a regular cardinal $\rho \le \kappa$. In that scope, the cofinal notions agree when $\rho =\kappa$. For $\rho <\kappa$, the definition used here is stronger because the source definition does not require its witness to lie in ${\mathcal{P}}_{\rho }\lambda$. The final paragraph of the proof of Lemma 5.11 ([9], p. 1127) nevertheless proves the relevant ${\mathcal{P}}_{{\omega }_1}\lambda$-trace unbounded, so that application satisfies the small-domain form used here. A trace is bistationary if it and its complement are stationary. In particular, a bistationary trace is stationary but contains no club.

For a set S of ordinals, “nowhere stationary” means that $S\cap \beta$ is nonstationary in $\beta$ for every $\beta$ of uncountable cofinality. This meaning is used for the support of the f-coordinate below.

Throughout, $-1$ denotes a formal symbol that is not a binary function. Binary sequences are ordered by extension. A tree $t{\subseteq }^{ \le \gamma }2$ is normal if it is nonempty, downward closed, and every node has an extension on every higher level below $\gamma +1$.

For such a tree t and $z\in {\mathcal{P}}_{{\omega }_1}\gamma$, let

$$\widehat{z}=sup\{\xi +1:\xi \in z\}\le \gamma ,t\left[z\right]=\{r\upharpoonright z:r\in t_{\widehat{z}}\}.$$

Put ${\beta }_z=sup\left(z\right)+1$. If ${\beta }_z\le \gamma$, then normality and downward closure give

$$t\left[z\right]=\{r\upharpoonright z:r\in t_{{\beta }_z}\}.$$

The only remaining case is that z has no maximum and is cofinal in $\gamma$. Then $\widehat{z}=\gamma$ and ${\beta }_z=\gamma +1$, and every normal end extension $t^{\prime }$ of t whose top height is at least ${\beta }_z$ satisfies

$$t\left[z\right]=\{r\upharpoonright z:r\in t_{{\beta }_z}^{\prime }\}.$$

Thus $t\left[z\right]$ agrees with the corresponding restriction level of the generic tree, while $\widehat{z}$ keeps the definition internal to the present condition.

Definition 1.

Let α be an uncountable regular cardinal. A condition in ${\mathbb{S}}^{*}\left(\alpha \right)$ is a tuple $p=\langle t_p,{\ell }_p,b_p,f_p\rangle$ with tree height ${\gamma }_p+1<\alpha$ satisfying the following clauses.

(1)

$t_p{\subseteq }^{\le {\gamma }_p}2$ is a normal binary tree.

(2)

${\ell }_p$ is a function whose domain is closed in ${\gamma }_p+1$. If $\xi \in dom\left({\ell }_p\right)$ and $cf\left(\xi \right)\le \omega$, then ${\ell }_p\left(\xi \right)=⌀$. If $cf\left(\xi \right)>\omega$, then for some $x\in {\left(t_p\right)}_{\xi }$ and some club $E_x\subseteq {\mathcal{P}}_{{\omega }_1}\xi$,

$${\ell }_p\left(\xi \right)=\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}.$$

(3)

$b_p:t_p\to {\left(t_p\right)}_{{\gamma }_p}$ and $v\subseteq b_p\left(v\right)$ for every $v\in t_p$.

(4)

$f_p$ is a partial function with $dom\left(f_p\right)\subseteq {\mathcal{P}}_{{\omega }_1}{\gamma }_p$. For $z\in dom\left(f_p\right)$,

$$f_p\left(z\right)\in t_p\left[z\right]\cup \{-1\}.$$

Distinct members of $dom\left(f_p\right)$ have distinct suprema, and

$$\{supz:z\in dom\left(f_p\right),f_p\left(z\right)\ne -1\}$$

is nowhere stationary.

(5)

For every $\xi \in dom\left({\ell }_p\right)$,

$$range\left(f_p\right)\cap {\ell }_p\left(\xi \right)=⌀.$$

For conditions p and q, write $q\le p$ when the following clauses hold.

(a)

$t_q{\cap }^{\le {\gamma }_p}2=t_p$.

(b)

${\ell }_q$ extends ${\ell }_p$ and $dom\left({\ell }_q\right)\cap ({\gamma }_p+1)=dom\left({\ell }_p\right)$.

(c)

$b_p\left(v\right)\subseteq b_q\left(v\right)$ for every $v\in t_p$.

(d)

$f_q$ extends $f_p$, and every $z\in dom\left(f_q\right)\setminus dom\left(f_p\right)$ satisfies $supz>{\gamma }_p$.

The t- and b-clauses and the clause for ℓ at coordinates of uncountable cofinality are those of ([9], Definition 5.5(1)–(3), p. 1123). At coordinates of cofinality at most $\omega$, the empty value is imposed. For cofinality $\omega$, this is the value used in the countable-limit construction in the proof of Claim 5.7, whose final calculation evaluates f at the domain of the ladder restriction ([9], p. 1124). Clause (4) gives a type-correct version of the assignment in the proof of Claim 5.8 ([9], p. 1124). In particular, every non-dummy value $f_p\left(z\right)$ has domain exactly z. Consequently, if $z\subseteq dom\left(x\right)$ and $f_p\left(w\right)=x\upharpoonright z$, then $w=z$.

Theorem 2.

The forcing ${\mathbb{S}}^{*}\left(\alpha \right)$ from Definition 1 is σ-closed. After adjoining a formal maximum condition, it is α-strategically closed. Consequently, ${\mathbb{S}}^{*}\left(\alpha \right)$ is $<\alpha$-distributive and preserves α as a regular cardinal.

Proof. 

For $\sigma$-closure, let $\langle p_n:n<\omega \rangle$ be descending. If the tree heights are eventually constant, then the sequence itself is eventually constant by the end-extension clauses. Otherwise, pass to a cofinal subsequence with strictly increasing heights, and let $\delta$ be their supremum. Regularity of $\alpha$ gives $\delta <\alpha$, and $cf\left(\delta \right)=\omega$. Take the union of the old trees and add the coherent b-limits as a new top level. For an old node v, let its new b-value be the corresponding coherent limit, and let each new top node be its own b-value.

End extension makes the unions of the old ℓ- and f-coordinates well defined. The union of the old ℓ-domains is closed below $\delta$. If $\beta <\delta$ is a limit point of this union, choose n whose tree height exceeds $\beta$. The old domain at stage n is already closed and contains every point of the eventual union below $\beta$. If $\delta$ is itself a limit point of the ℓ-domains, put an empty ladder-coordinate value there and add a dummy value $-1$ at a countable set cofinal in $\delta$. If a prescribed $a\in {\mathcal{P}}_{{\omega }_1}\delta$ is cofinal in $\delta$, it may serve as this dummy domain. Every old f-domain has supremum below $\delta =supa$, so the new supremum is fresh and clause (d) of Definition 1 holds. Let S be the union of the nontrivial supports. For every $\beta <\delta$, the set $S\cap \beta$ eventually agrees with the corresponding initial segment of one old support. There is no requirement at $\delta$, whose cofinality is $\omega$, and above $\delta$ the support is bounded. Hence the resulting condition is a common lower bound.

The strategic-closure argument begins with the limit-stage scheme in the proof of Claim 5.7 ([9], p. 1124). There, Even is passive at successor stages ([9], p. 1123). The typed strategy instead uses prepared positive Even successor moves to introduce reserved dummy suprema, which are used in the disjointness verification at limits of uncountable cofinality. Adjoin a formal maximum $\mathbf{1}$ to ${\mathbb{S}}^{*}\left(\alpha \right)$. Since the original forcing is dense in the enlarged poset, the generic extensions are unchanged. Let $G_{\alpha }\left(P\right)$ denote the game from ([2], Definition 5.14, p. 793). Odd plays at odd stages, Even plays at even stages (including all limit stages), and Even plays $\mathbf{1}$ at stage 0. If Odd repeats $\mathbf{1}$ at stage 1, Even first plays any genuine condition of top height c with $cf\left(c\right)=\omega$, $\ell \left(c\right)=⌀$, and $f\left(d_c\right)=-1$ for some countable set $d_c\subseteq c$ cofinal in c. At every other positive successor stage assigned to her, Even end-extends Odd’s preceding condition to a fresh top height c of cofinality $\omega$, puts $\ell \left(c\right)=⌀$, and adds a countable set $d_c\subseteq c$ cofinal in c with $f\left(d_c\right)=-1$. After every positive move, Even maintains the following protection invariant at her top height c. The coordinate c belongs to the ℓ-domain, and either $cf\left(c\right)=\omega$ and the reserved dummy is present, or $cf\left(c\right)>\omega$. The prepared successor move is legal. The new supremum is fresh, the nontrivial support is unchanged, and the dummy contributes no binary function to the range of f.

Let $0<\epsilon <\alpha$ be a limit stage. Along any legal play, stage 2 is genuine and all later moves lie below it. For $2\le \xi <\epsilon$, write $p_{\xi }=\langle t_{\xi },{\ell }_{\xi },b_{\xi },f_{\xi }\rangle$ with tree height ${\gamma }_{\xi }+1$. If stage 1 was genuine, $p_2$ extends it, so starting at stage 2 loses no information. Earlier limit moves make the height sequence continuous, while the prepared positive Even successor moves ensure that the Even heights form a strictly increasing cofinal subsequence. Thus

$$\delta =\underset{2\le \xi <\epsilon }{sup}{\gamma }_{\xi }<\alpha ,cf\left(\delta \right)=cf\left(\epsilon \right).$$

Indeed, a cofinal subset of $\delta$ of size below $cf\left(\epsilon \right)$ would have all of its witnessing stages bounded below $\epsilon$, so the subset itself would be bounded in $\delta$, a contradiction. If $cf\left(\epsilon \right)=\omega$, apply the $\sigma$-closure construction to a countable cofinal subsequence, taking the new top height to be $\delta$. The Even top heights are cofinal in the union of the ℓ-domains, so the construction puts $\ell \left(\delta \right)=⌀$ and a dummy value at a countable cofinal subset of $\delta$. The protection invariant is preserved.

Suppose that $cf\left(\epsilon \right)>\omega$. Let

$$\tilde{t}=\bigcup _{2\le \xi <\epsilon }{t}_{\xi },B\left(v\right)=\bigcup _{{\xi }_{*}<\xi <\epsilon }{b}_{\xi }\left(v\right),$$

where ${\xi }_{*}\ge 2$ is any stage at which v is already present, and put

$$t_{\epsilon }=\tilde{t}\cup \{B\left(v\right):v\in \tilde{t}\}.$$

Define $b_{\epsilon }\left(v\right)=B\left(v\right)$ on old nodes and let every new top node be its own b-value. This gives a normal end extension of height $\delta +1$. The unions of the old ℓ- and f-coordinates are well defined. The union of the old ℓ-domains is closed below $\delta$ by eventual stabilization, and adding the coordinate $\delta$ closes it. Every old f-value remains legal because its countable domain is bounded below some earlier top height, where end extension has already fixed the corresponding restriction set. Distinct domain suprema and all old f–ℓ disjointness requirements are inherited from the descending play.

Let D be the set of top heights occurring at positive Even stages below $\epsilon$. Continuity of the height sequence and the fact that the positive Even stages are cofinal in $\epsilon$ make D a club in $\delta$. No member of D belongs to the nontrivial support. For $c\in D$, all f-domain suprema present before c is introduced are below c. At $cf\left(c\right)=\omega$ the strategy introduces at c only the reserved dummy. At $cf\left(c\right)>\omega$ it introduces no f-domain there. The last extension clause in Definition 1 requires every f-domain added later to have supremum above c. Consequently, the union of the nontrivial supports is nonstationary in $\delta$. At every $\beta <\delta$ of uncountable cofinality, eventual stabilization ensures that the support is nonstationary in $\beta$, while above $\delta$ the support is bounded.

Let $C\subseteq D$ consist of the protected top heights of cofinality $\omega$. The prepared successor moves make C unbounded in $\delta$, and C is closed under increasing $\omega$-sequences. The supremum of the corresponding stages is a limit stage of cofinality $\omega$, where the strategy again introduces a protected top. Choose a top node $x\in {\left(t_{\epsilon }\right)}_{\delta }$ and put

$$E_x=\{z\in {\mathcal{P}}_{{\omega }_1}\delta :supz\in C\}.$$

This set is club. For unboundedness, extend any countable subset of $\delta$ by a countable cofinal subset of a larger member of C. Closure follows from the $\omega$-closure of C and the positive-length convention for increasing unions. Define

$${\ell }_{\epsilon }=\bigcup _{2\le \xi <\epsilon }{\ell }_{\xi }\cup \left\{\langle \delta ,\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}\rangle \right\},f_{\epsilon }=\bigcup _{2\le \xi <\epsilon }{f}_{\xi }.$$

The only new disjointness requirement is at $\delta$. The top node x is not an f-value because every non-dummy f-value has countable domain. If $x\upharpoonright z=f_{\epsilon }\left(w\right)$ for $z\in E_x$, equality of domains gives $w=z$. With $c=supz\in C$, both w and the reserved dummy domain $d_c$ have supremum c. Uniqueness of domain suprema therefore gives $w=d_c$. Hence $f_{\epsilon }\left(w\right)=-1$, contradicting the assumed equality $f_{\epsilon }\left(w\right)=x\upharpoonright z$. The new ladder coordinate and all new tree nodes occur strictly above every earlier top height, and the b-values extend all earlier b-values. Thus every order clause holds and the limit move is a legal condition below the whole play.

This defines Even’s strategy through every stage below $\alpha$. Since $\alpha$-strategic closure implies $<\alpha$-strategic closure, Cummings’s definitions and discussion ([2], Definition 5.8(3), p. 792, Definition 5.15(1)–(2), p. 793, and the discussion on p. 794) imply that ${\mathbb{S}}^{*}\left(\alpha \right)$ adds no sequence of ordinals of length less than $\alpha$. Hence it is $<\alpha$-distributive and preserves $\alpha$ as a regular cardinal. □

By Theorem 2, the forcing adds no new sequences of ordinals of length less than $\alpha$. In particular, it preserves every cardinal $\le \alpha$, and $\alpha$ remains regular.

Let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Write

$$T_{\alpha }=\bigcup _{p\in G}{t}_p,{\ell }_G=\bigcup _{p\in G}{\ell }_p,L_{\alpha }=\bigcup range\left({\ell }_G\right).$$

For $u\in T_{\alpha }$, the designated branch through u is

$$B_{\alpha }\left(u\right)=\bigcup \{b_p\left(u\right):p\in G,u\in t_p\}.$$

Call $L_{\alpha }$ the generic ladder-coordinate set. The trace and covering arguments use this set of partial functions rather than a ladder system in the full two-cardinal sense. The notation follows ([9], Notation 5.6, p. 1123). The explicit expression $\bigcup range\left({\ell }_G\right)$ separates the node set from its level-indexing function. When $T_{\alpha }$, $L_{\alpha }$, or $B_{\alpha }\left(u\right)$ for a ground-model node u occurs inside a forcing statement, the same symbol denotes the corresponding ${\mathbb{S}}^{*}\left(\alpha \right)$-name. Check accents on ground-model objects are occasionally suppressed when no ambiguity can arise.

Proposition 1.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Then, with

$$D_{\alpha }=\{dom\left(y\right):y\in L_{\alpha }\},$$

no ordinal $\xi \in [{\omega }_1,\alpha )$ with ${cf}^{V\left[G\right]}\left(\xi \right)=\omega$ belongs to $D_{\alpha }$. In particular, $D_{\alpha }$ does not contain a club in ${\mathcal{P}}_{\alpha }\alpha$. Consequently, $L_{\alpha }$ does not satisfy the club-many-level requirement in the definition of a ladder system on a ${\mathcal{P}}_{\alpha }\alpha$-tree.

Proof. 

By Theorem 2, $\alpha$ remains regular and no new countable sequence of ordinals is added. If ${\omega }_1\le \xi <\alpha$ and ${cf}^{V\left[G\right]}\left(\xi \right)=\omega$, then ${cf}^V\left(\xi \right)=\omega$. The definition of the forcing assigns the empty value at the coordinate $\xi$. Every other member of $L_{\alpha }$ has either an ordinal domain of uncountable cofinality or a countable restriction domain. Hence $\xi \notin D_{\alpha }$.

Viewed as subsets of $\alpha$, the ordinals in the interval $[{\omega }_1,\alpha )$ form a club in ${\mathcal{P}}_{\alpha }\alpha$. If $D_{\alpha }$ contained a club, then its intersection with this ordinal club would determine a club subset of $\alpha$. Since the ordinals of cofinality $\omega$ are stationary in $\alpha$, that club would contain some $\xi \ge {\omega }_1$ of cofinality $\omega$, contradicting $\xi \notin D_{\alpha }$. □

This level-domain failure comes from the additional empty-value convention at coordinates of cofinality at most $\omega$. It does not use the unrestricted tree clause.

Lemma 1.

Let α be an uncountable regular cardinal and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Then $T_{\alpha }$ is a normal binary tree of height α, and $B_{\alpha }\left(u\right)$ is a cofinal branch through $T_{\alpha }$ for every $u\in T_{\alpha }$.

Proof. 

The union of end extensions is a downward-closed binary tree. Given a condition p and $\theta <\alpha$, choose

$$max\{{\gamma }_p,\theta \}<\gamma <\alpha .$$

End-extend every old top node by a chain to level $\gamma$, extend each old b-value along the corresponding chain, and assign to every new node its top continuation. Leave ${\ell }_p$ and $f_p$ unchanged. The resulting condition has height $\gamma +1$, so conditions of height above $\theta$ are dense. When applied below a condition containing a fixed node, the same construction gives an extension of that node to every prescribed higher level. Hence $T_{\alpha }$ is normal and has height $\alpha$.

For a fixed u, the values $b_p\left(u\right)$ appearing in the generic filter are coherent by the ordering. The height-extension operation can be carried out below every condition containing u, so their union has domain $\alpha$. Each of its initial segments belongs to an old tree level and hence to $T_{\alpha }$. Thus the union is a cofinal branch through $T_{\alpha }$. □

Let $\lambda$ be an ordinal and let L be a set of partial binary functions on $\lambda$. A family $\mathcal{B}{\subseteq }^{ \lambda }2$ generates L if every $y\in L$ is extended by some member of $\mathcal{B}$. If T is a tree whose cofinal branches are functions on $\lambda$, write $Br\left(T\right)$ for the family of those branches and define

$$bcov(T,L)=min\left\{\left|\mathcal{B}\right|:\mathcal{B}\subseteq Br\left(T\right)\mathrm{and}\mathcal{B}\mathrm{generates}L\right\},$$

with value ∞, understood to exceed every cardinal, when there is no such family.

For comparison with two-cardinal trees, if $\kappa \le \lambda$ are cardinals and $\mathcal{B}{\subseteq }^{ \lambda }2$, put

$$L_{\mathcal{B}}^{\kappa }=\{c\upharpoonright x:c\in \mathcal{B},x\in {\mathcal{P}}_{\kappa }\lambda \}.$$

Thus $\mathcal{B}$ generates $L_{\mathcal{B}}^{\kappa }$ in the preceding sense.

Lemma 2.

Let ρ be an infinite cardinal, let λ be an ordinal, and let L be a set of partial binary functions on λ. Suppose that L is generated by $\mathcal{B}{\subseteq }^{ \lambda }2$ with $\left|\mathcal{B}\right|<\rho$. If $b{\in }^{ \lambda }2$ meets Lρ-cofinally, then $b\in \mathcal{B}$.

Proof. 

Assume that $b\notin \mathcal{B}$. For each $c\in \mathcal{B}$, choose ${\beta }_c<\lambda$ such that $b\left({\beta }_c\right)\ne c\left({\beta }_c\right)$. The set

$$a=\{{\beta }_c:c\in \mathcal{B}\}$$

has cardinality at most $\left|\mathcal{B}\right|<\rho$, and hence belongs to ${\mathcal{P}}_{\rho }\lambda$. Choose $z\supseteq a$ with $b\upharpoonright z\in L$. Since $\mathcal{B}$ generates L, some $c\in \mathcal{B}$ extends $b\upharpoonright z$. But ${\beta }_c\in z$, contradicting the choice of ${\beta }_c$. □

Corollary 1.

Let ρ be an infinite cardinal, let λ be an ordinal, let L be a set of partial binary functions on λ, and let T be a tree whose cofinal branches are functions on λ. If $\mathcal{C}\subseteq Br\left(T\right)$ is a family of pairwise distinct branches of cardinality at least ρ such that every member of $\mathcal{C}$ meets L ρ-cofinally, then

$$bcov(T,L)\ge \rho .$$

Proof. 

Suppose toward a contradiction that $bcov(T,L)<\rho$. Then there is a generating family $\mathcal{B}\subseteq Br\left(T\right)$ with $\left|\mathcal{B}\right|<\rho$. By Lemma 2, every member of $\mathcal{C}$ belongs to $\mathcal{B}$, contradicting $\left|\mathcal{C}\right|\ge \rho$. □

The lemma also has a closure interpretation. Suppose that $\rho \le \kappa \le \lambda$ are infinite cardinals. Let ${\tau }_{<\rho }$ be the topology on ^λ2 whose basic neighborhoods prescribe fewer than $\rho$ coordinates. For $\mathcal{B}{\subseteq }^{ \lambda }2$, a function b meets $L_{\mathcal{B}}^{\kappa }$$\rho$-cofinally if and only if $b\in {\overline{\mathcal{B}}}^{{\tau }_{<\rho }}$. Agreement with some member of $\mathcal{B}$ on every prescribed small coordinate set is precisely the closure condition. Thus Lemma 2 is the set-theoretic form of the fact that a subset of size $<\rho$ is closed in this topology. Branch terminology is used because the typed forcing supplies a distinguished family of cofinal branches and because the invariant bcov measures generation of the whole target set, not merely closure of one branch.

The strict cardinal inequality in Lemma 2 is sharp, and a branch may catch the generated set without belonging to the chosen generating family.

Proposition 2.

Let $\rho <\kappa \le \lambda$ be infinite cardinals. There exist a ${\mathcal{P}}_{\kappa }\lambda$-tree $\mathcal{T}$, a family $\mathcal{B}\subseteq Br\left(\mathcal{T}\right)$ of size ρ, a ladder system $L=L_{\mathcal{B}}^{\kappa }$, and a branch $b\notin \mathcal{B}$ such that

$${Tr}_{\rho }(b,L)={\mathcal{P}}_{\rho }\lambda .$$

Proof. 

Regard $\rho$ as a subset of $\lambda$. Let b be constantly zero. For $\xi <\rho$, let $c_{\xi }$ agree with b except at coordinate $\xi$, and put $\mathcal{B}=\{c_{\xi }:\xi <\rho \}$. Define

$${\mathcal{T}}_x=\{b\upharpoonright x\}\cup \{c_{\xi }\upharpoonright x:\xi <\rho \}(x\in {\mathcal{P}}_{\kappa }\lambda ).$$

Every level is nonempty, the system is closed under restrictions, and each level has size at most $\rho <\kappa$. Hence $\mathcal{T}$ is a ${\mathcal{P}}_{\kappa }\lambda$-tree. By construction, b and every $c_{\xi }$ are branches through $\mathcal{T}$. The family $L_{\mathcal{B}}^{\kappa }$ is a ladder system. It meets every level. If $\eta =c_{\xi }\upharpoonright x\in L_{\mathcal{B}}^{\kappa }$ and $cf\left(\right|x\cap \kappa \left|\right)>\omega$, take $E_{\eta }={\mathcal{P}}_{|x\cap \kappa |}x$. Then $E_{\eta }$ is club and $\eta \upharpoonright y=c_{\xi }\upharpoonright y\in L_{\mathcal{B}}^{\kappa }$ for every $y\in E_{\eta }$.

Fix $x\in {\mathcal{P}}_{\rho }\lambda$. Since $\left|x\right|<\rho$, choose $\xi \in \rho \setminus x$. Then

$$b\upharpoonright x=c_{\xi }\upharpoonright x\in L_{\mathcal{B}}^{\kappa }.$$

Thus the trace is all of ${\mathcal{P}}_{\rho }\lambda$, although $b\notin \mathcal{B}$. □

Consequently, small-cover rigidity alone distinguishes covers of size strictly below $\rho$ from covers of size $\rho$. For $\rho ={\omega }_1$, it rules out countable covers but does not by itself determine whether an ${\omega }_1$-sized cover exists.

Proposition 3.

Let α be an uncountable regular cardinal and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Then the family of cofinal branches

$${\mathcal{B}}_{\alpha }=\{B_{\alpha }\left(u\right):u\in T_{\alpha }\}$$

generates $L_{\alpha }$.

Proof. 

By Lemma 1, the displayed functions are cofinal branches through $T_{\alpha }$. Fix $y\in L_{\alpha }$, and choose $p\in G$ and $\xi \in dom\left({\ell }_p\right)$ with $y\in {\ell }_p\left(\xi \right)$. Values at coordinates of cofinality at most $\omega$ are empty. At a coordinate of uncountable cofinality, the definition of ${\mathbb{S}}^{*}\left(\alpha \right)$ gives either $y=x$ or $y=x\upharpoonright z$ for some $x\in {\left(t_p\right)}_{\xi }$. In both cases the designated branch $B_{\alpha }\left(x\right)$ extends y. Hence ${\mathcal{B}}_{\alpha }$ generates $L_{\alpha }$. □

3. Bistationary Traces, Wide Levels, and Branch-Cover Rigidity

3.1. Fixed-seed Ladder Insertion

The limit construction in the proof of Claim 5.7 ([9], p. 1124) specifies the tree and b-coordinates. It admits a fixed-seed refinement in which a prescribed old node is extended coherently to a distinguished new top node.

Lemma 3.

Let $\alpha \ge {\omega }_2$ be regular, $p\in {\mathbb{S}}^{*}\left(\alpha \right)$, $u\in t_p$, and $\theta <\alpha$. There exist $q\le p$, an ordinal $\theta <\delta <\alpha$ with $cf\left(\delta \right)={\omega }_1$, a node

$$x=b_q\left(u\right)\in {\left(t_q\right)}_{\delta }$$

extending u, and a club $E_x\subseteq {\mathcal{P}}_{{\omega }_1}\delta$ such that

$$\delta \in dom\left({\ell }_q\right)\mathrm{and}{\ell }_q\left(\delta \right)=\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}.$$

Proof. 

Apply the limit construction from the proof of Claim 5.7 ([9], p. 1124), keeping the seed u fixed. First strengthen p to a condition $p_0$ whose tree has fresh top height ${\gamma }_0>max\{\theta ,{\gamma }_p\}$ of cofinality $\omega$. Put ${\gamma }_0$ into $dom\left({\ell }_{p_0}\right)$ with empty value, choose $z_0\in {\mathcal{P}}_{{\omega }_1}{\gamma }_0$ cofinal in ${\gamma }_0$, and set $f_{p_0}\left(z_0\right)=-1$.

Construct a descending sequence

$$\langle p_{\xi }=\langle t_{\xi },{\ell }_{\xi },b_{\xi },f_{\xi }\rangle :\xi <{\omega }_1\rangle .$$

Write ${\gamma }_{\xi }+1$ for the height of $t_{\xi }$. Arrange that $\langle {\gamma }_{\xi }:\xi <{\omega }_1\rangle$ is continuous and strictly increasing, every ${\gamma }_{\xi }$ has cofinality $\omega$, and

$$maxdom\left({\ell }_{\xi }\right)={\gamma }_{\xi },{\ell }_{\xi }\left({\gamma }_{\xi }\right)=⌀.$$

The fixed-seed invariant is

$$u_{\xi }=b_{\xi }\left(u\right)\in {\left(t_{\xi }\right)}_{{\gamma }_{\xi }}.$$

At a successor stage, end-extend the tree to a fresh top height of cofinality $\omega$. Extend every old b-value to that top and choose $u_{\xi +1}=b_{\xi +1}\left(u\right)$ extending $u_{\xi }$. Put the new top height into the domain of ℓ with empty value, choose a countable cofinal set $z_{\xi +1}$ at that height, and set $f_{\xi +1}\left(z_{\xi +1}\right)=-1$. Above every old top node, choose enough continuations so that every new node has a top extension, and use those top extensions to define the remaining new b-values.

At a nonzero countable limit $\lambda <{\omega }_1$, choose an increasing cofinal sequence $\langle {\xi }_n:n<\omega \rangle$ in $\lambda$ and use the countable-limit construction from Theorem 2 for $\langle p_{{\xi }_n}:n<\omega \rangle$, taking its lower bound as $p_{\lambda }$. Since the condition sequence is descending, $p_{\lambda }$ lies below every $p_{\xi }$ for $\xi <\lambda$. If a node v has appeared by stage ${\xi }_{*}\left(v\right)$, its new top extension is

$$B_{\lambda }\left(v\right)=\bigcup _{{\xi }_{*}\left(v\right)\le \xi <\lambda }{b}_{\xi }\left(v\right),b_{\lambda }\left(v\right)=B_{\lambda }\left(v\right).$$

Since u is present from stage 0,

$$u_{\lambda }=b_{\lambda }\left(u\right)=\bigcup _{\xi <\lambda }{u}_{\xi }.$$

Thus the fixed-seed union is one of the top nodes of the limit condition. The countable-limit construction also puts the new top height into the domain of ℓ with empty value and adds a dummy value whose domain, denoted by $z_{\lambda }$, has that supremum.

Let

$$\delta =\underset{\xi <{\omega }_1}{sup}{\gamma }_{\xi }.$$

Since $\alpha$ is regular and $\alpha \ge {\omega }_2$, $cf\left(\alpha \right)=\alpha >{\omega }_1$, so $\delta <\alpha$. Since the height sequence is strictly increasing and ${\omega }_1$ is regular, $cf\left(\delta \right)={\omega }_1$. Put

$$\tilde{t}=\bigcup _{\xi <{\omega }_1}{t}_{\xi },B\left(v\right)=\bigcup _{{\xi }_{*}<\xi <{\omega }_1}{b}_{\xi }\left(v\right),$$

where ${\xi }_{*}$ is any stage after v first appears. Define

$$t_q=\tilde{t}\cup \{B\left(v\right):v\in \tilde{t}\},x=B\left(u\right)=\bigcup _{\xi <{\omega }_1}{u}_{\xi }.$$

The invariant shows that $x\in {\left(t_q\right)}_{\delta }$, extends u, and equals $b_q\left(u\right)$.

Set

$$E_x=\{z\in {\mathcal{P}}_{{\omega }_1}\delta :supz\in \{{\gamma }_{\xi }:\xi <{\omega }_1\}\}.$$

This set is unbounded. Given $a\in {\mathcal{P}}_{{\omega }_1}\delta$, choose $\xi$ with $supa<{\gamma }_{\xi }$ and adjoin to a a countable cofinal subset of ${\gamma }_{\xi }$. It is closed under increasing unions of positive length below ${\omega }_1$ by continuity of the height sequence. Hence $E_x$ is club.

Let

$${\ell }_q=\bigcup _{\xi <{\omega }_1}{\ell }_{\xi }\cup \left\{\langle \delta ,\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}\rangle \right\},f_q=\bigcup _{\xi <{\omega }_1}{f}_{\xi }.$$

Define $b_q\left(v\right)=B\left(v\right)$ for $v\in \tilde{t}$ and $b_q\left(w\right)=w$ for every new top node w. The tree is normal, the ℓ-domain is closed, and every old b-value is extended. All f-values introduced during the recursion are dummies, so the nontrivial support remains bounded below $\delta$ and is nowhere stationary.

It remains to verify $range\left(f_q\right)\cap {\ell }_q\left(\delta \right)=⌀$. The top node x is not in the range because every non-dummy value $f_q\left(w\right)$ has the countable domain w, whereas x has domain $\delta$. Suppose that $x\upharpoonright z=f_q\left(w\right)$ for some $z\in E_x$ and $w\in dom\left(f_q\right)$. Equality of domains gives $z=w$. Choose $\xi$ with $supz={\gamma }_{\xi }$. The dummy domain $z_{\xi }$ also has supremum ${\gamma }_{\xi }$. Because $w,z_{\xi }\in dom\left(f_q\right)$ have the same supremum, uniqueness of domain suprema gives $w=z_{\xi }$. Hence $f_q\left(w\right)=-1$, contradicting $f_q\left(w\right)=x\upharpoonright z$. The old ladder-coordinate values remain disjoint from the range because all f-values introduced during the recursion are dummies. Thus q is the required condition. □

The fixed seed is needed for this construction. The conclusion does not follow from the mere existence of an increasing sequence of nodes above u. At a countable limit, the construction in the proof of Claim 5.7 ([9], p. 1124) adds top nodes that are unions of coherent b-approximations associated with a single previously existing node. The displayed identity

$$b_{\lambda }\left(u\right)=\bigcup _{\xi <\lambda }{b}_{\xi }\left(u\right)$$

therefore shows that $b_{\lambda }\left(u\right)$ belongs to the new top level. Without the fixed seed, an arbitrary union may fail to be one of the nodes inserted by the limit condition. This is also why the uniform diagonal argument first decides the branch names and only then chooses the seed to follow.

An anchored refinement places one prescribed countable set among the restrictions at the new ladder level.

Lemma 4.

Let $\alpha \ge {\omega }_2$ be regular, let $p\in {\mathbb{S}}^{*}\left(\alpha \right)$, and let $u\in t_p$. Suppose that $a\in dom\left(f_p\right)$ and $f_p\left(a\right)=-1$. There exist $q\le p$, an ordinal $\delta <\alpha$ of cofinality ${\omega }_1$, a node $x=b_q\left(u\right)\in {\left(t_q\right)}_{\delta }$, and a club $E_x\subseteq {\mathcal{P}}_{{\omega }_1}\delta$ such that

$$a\in E_x\mathrm{and}{\ell }_q\left(\delta \right)=\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}.$$

Proof. 

Apply Lemma 3 with $\theta ={\gamma }_p$, obtaining $q^0\le p$, $\delta$, x, and $E_x^0$. Then $f_{q^0}\left(a\right)=-1$ and $a\subseteq {\gamma }_p<\delta$, so $a\in {\mathcal{P}}_{{\omega }_1}\delta$. Put $E_x=E_x^0\cup \left\{a\right\}$. Then $E_x$ is still club. For successor-length sequences, closure is immediate. For a limit-length sequence, either cofinally many terms belong to $E_x^0$ or the sequence is eventually constant at a.

Let q agree with $q^0$ except that ${\ell }_q\left(\delta \right)=\left\{x\right\}\cup \{x\upharpoonright z:z\in E_x\}$. The only possible new node is $x\upharpoonright a$. If $x\upharpoonright a=f_q\left(w\right)$, equality of domains gives $w=a$, contradicting $f_q\left(a\right)=-1$. Hence q is a condition, and $\delta >{\gamma }_p$ gives $q\le p$. □

3.2. Bistationary Traces

A countable set in a named club can be chosen with its supremum reserved by a dummy value.

Lemma 5.

Let α be an uncountable regular cardinal, let $p\in {\mathbb{S}}^{*}\left(\alpha \right)$, and suppose that $p⊩\dot{C}$ is club in ${\mathcal{P}}_{{\omega }_1}\alpha$. There exist $s\le p$ and $a\in {\mathcal{P}}_{{\omega }_1}\alpha$ such that, with $\delta =supa$,

$$s⊩\check{a}\in \dot{C},{\ell }_s\left(\delta \right)=⌀,f_s\left(a\right)=-1.$$

The top height of $t_s$ may be taken to be δ.

Proof. 

Choose a sufficiently large regular cardinal $\chi$ and a countable $M\prec H\left(\chi \right)$ containing $p,\dot{C},{\mathbb{S}}^{*}\left(\alpha \right)$, and $\alpha$. Put $a=M\cap \alpha$ and $\delta =supa$. The set a has no maximum. If $\beta \in M\cap \alpha$, then $\beta +1\in M\cap \alpha$ by elementarity. Hence $a\subseteq \delta$ and $cf\left(\delta \right)=\omega$. Enumerate $a=\{{\eta }_n:n<\omega \}$.

By Theorem 2, every name forced to belong to ${\mathcal{P}}_{{\omega }_1}\alpha$ can be decided as a ground-model countable set. First strengthen the condition to decide whether the name is empty. In the nonempty case, use the maximum principle to choose a name for a surjection from $\omega$ onto the named set, decide its values coordinate by coordinate, and take a common lower bound. Using elementarity at each finite stage, construct a descending sequence $\langle p_n:n<\omega \rangle$ with $p_n\in M$ and $p_0=p$, and an increasing sequence $\langle y_n:n<\omega \rangle$ in $M\cap {\mathcal{P}}_{{\omega }_1}\alpha$. Choose $q_n\in M$ and $y_n$ such that $q_n\le p_n$ and

$$q_n⊩{\check{y}}_n\in \dot{C}\mathrm{and}{y}_{n-1}\cup \{{\eta }_0,\dots ,{\eta }_n\}\subseteq y_n,$$

where $y_{-1}=⌀$. End-extend $q_n$ to a condition $p_{n+1}$ with fresh top height ${\gamma }_n\in M\cap \alpha$ of cofinality $\omega$, chosen above ${\eta }_0,\dots ,{\eta }_n$ and, when $n>0$, above ${\gamma }_{n-1}$. Then $\langle {\gamma }_n:n<\omega \rangle$ is increasing and cofinal in $\delta$. Put ${\ell }_{p_{n+1}}\left({\gamma }_n\right)=⌀$, choose $z_n\in M\cap {\mathcal{P}}_{{\omega }_1}{\gamma }_n$ cofinal in ${\gamma }_n$, and set $f_{p_{n+1}}\left(z_n\right)=-1$. Extend every old b-value to the new top. Freshness of ${\gamma }_n$ preserves the uniqueness of domain suprema and all old f–ℓ clauses.

Since $y_n\in M$ is countable, $y_n\subseteq M\cap \alpha =a$. The displayed inclusions give ${\bigcup }_n{y}_n=a$. Use the countable-limit construction from Theorem 2 with top $\delta$, ${\ell }_s\left(\delta \right)=⌀$, and dummy $f_s\left(a\right)=-1$. No earlier member of $dom\left(f_s\right)$ has supremum $\delta$.

The condition s forces $y_n\in \dot{C}$ for every n. Since $\dot{C}$ is closed under increasing $\omega$-unions and $a={\bigcup }_{n<\omega }{y}_n$ in the ground model,

$$s⊩\check{a}\in \dot{C}.$$

□

The club point supplied by the lemma can be combined with countably many anchored insertions to handle any countable collection of designated branches.

Theorem 3.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. For every nonempty countable $U\subseteq T_{\alpha }$,

$$\bigcap _{u\in U}{Tr}_{{\omega }_1}(B_{\alpha }\left(u\right),L_{\alpha })$$

is stationary in ${\mathcal{P}}_{{\omega }_1}\alpha$.

Proof. 

It is enough to prove the corresponding forcing statement for a countable sequence of nodes. Indeed, by the maximum principle, every nonempty countable set of nodes in an extension has an $\omega$-enumeration name, with repetitions when the set is finite. Let $\langle {\dot{u}}_n:n<\omega \rangle$ be a sequence of ${\mathbb{S}}^{*}\left(\alpha \right)$-names, let $p\in {\mathbb{S}}^{*}\left(\alpha \right)$ force that ${\dot{u}}_n\in T_{\alpha }$ for every $n<\omega$ and that $\dot{C}$ is club in ${\mathcal{P}}_{{\omega }_1}\alpha$, and fix $r\le p$.

Recursively strengthen the condition to decide each name ${\dot{u}}_n={\check{u}}_n$ and then raise the tree height above $dom\left(u_n\right)$. Since the strengthened condition still forces $u_n\in T_{\alpha }$ and later end extensions cannot alter that level, the node $u_n$ already belongs to the tree of that condition. By Theorem 2, there exist $r^{*}\le r$ and nodes $u_n\in t_{r^{*}}$ such that

$$r^{*}⊩{\dot{u}}_n={\check{u}}_n(n<\omega ).$$

Apply Lemma 5 below $r^{*}$ to obtain $s_0\le r^{*}$ and $a\in {\mathcal{P}}_{{\omega }_1}\alpha$ such that

$$s_0⊩\check{a}\in \dot{C},f_{s_0}\left(a\right)=-1.$$

Construct a descending sequence $\langle s_n:n<\omega \rangle$. Given $s_n$, apply Lemma 4 below $s_n$ with seed $u_n$ and obtain $s_{n+1}\le s_n$. Put

$$x_n=b_{s_{n+1}}\left(u_n\right).$$

The anchored insertion preserves the dummy value at a and ensures that $x_n\upharpoonright a$ belongs to a ladder-coordinate value of $s_{n+1}$. Use Theorem 2 once more to obtain a common lower bound q for the sequence. Every later b-value at $u_n$ extends $x_n$, and every ladder-coordinate value already inserted is preserved by end extension. Hence

$$q⊩B_{\alpha }\left(u_n\right)\upharpoonright a=x_n\upharpoonright a\in L_{\alpha }(n<\omega ).$$

Since $q\le s_0$, it also forces $a\in \dot{C}$. Thus every condition below p has an extension forcing that the named club meets all the designated traces simultaneously. □

The proof gives the following dense formulation. Let $\alpha \ge {\omega }_2$ be regular, let $p\in {\mathbb{S}}^{*}\left(\alpha \right)$, and let $\langle u_n:n<\omega \rangle$ be a sequence in $t_p$. Suppose that p forces $\dot{C}$ to be club in ${\mathcal{P}}_{{\omega }_1}\alpha$. For every $r\le p$ there exist $q\le r$ and $a\in {\mathcal{P}}_{{\omega }_1}\alpha$ in the ground model such that

$$q⊩\check{a}\in \dot{C}\mathrm{and}q⊩B_{\alpha }\left(u_n\right)\upharpoonright a\in L_{\alpha }(n<\omega ).$$

The same reserved domain a is preserved through all anchored insertions, so q forces a to belong to every designated trace in the sequence. No cardinal-arithmetic or large-cardinal hypothesis is used beyond the stated assumptions on $\alpha$.

The typed analogue of ([9], Claim 5.8, p. 1124) replaces the dummy value in the countable-limit construction by a non-dummy value.

Proposition 4.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. No cofinal branch through $T_{\alpha }$ meets $L_{\alpha }$ on an ${\omega }_1$-club.

Proof. 

Let p force that $\dot{c}$ is a cofinal branch through $T_{\alpha }$ and that $\dot{C}$ is a club in ${\mathcal{P}}_{{\omega }_1}\alpha$. Fix $r\le p$ and choose a sufficiently large regular cardinal $\chi$ and a countable $M\prec H\left(\chi \right)$ containing $r,\dot{c},\dot{C},{\mathbb{S}}^{*}\left(\alpha \right)$, and $\alpha$. Put $a=M\cap \alpha$ and $\delta =supa$. If $\beta \in a$, then $\beta +1\in a$ by elementarity, so a has no maximum. Hence $a\subseteq \delta$, $cf\left(\delta \right)=\omega$, and $\widehat{a}=\delta$. Enumerate $a=\{{\eta }_n:n<\omega \}$.

Using elementarity at each finite stage, construct a descending sequence $\langle p_n:n<\omega \rangle$ with $p_n\in M$ and $p_0=r$, an increasing sequence $\langle y_n:n<\omega \rangle$ in $M\cap {\mathcal{P}}_{{\omega }_1}\alpha$, a sequence of ordinals $\langle {\gamma }_n:n<\omega \rangle$, and nodes $v_n\in {\left(t_{p_{n+1}}\right)}_{{\gamma }_n}$. Given $p_n$, first choose $r_n\le p_n$ and $y_n$ such that

$$r_n⊩{\check{y}}_n\in \dot{C},y_{n-1}\cup \{{\eta }_0,\dots ,{\eta }_n\}\subseteq y_n,$$

where $y_{-1}=⌀$. End-extend $r_n$ to a condition $s_n$ with a fresh top height ${\gamma }_n$ of cofinality $\omega$, chosen above ${\eta }_0,\dots ,{\eta }_n$ and, when $n>0$, above ${\gamma }_{n-1}$. Put an empty ladder-coordinate value there and a dummy f-value at a cofinal subset of ${\gamma }_n$. The resulting sequence of heights is increasing and cofinal in $\delta$. Using the $<\alpha$-distributivity from Theorem 2, strengthen $s_n$ to $p_{n+1}$ and decide

$$p_{n+1}⊩\dot{c}\upharpoonright {\gamma }_n={\check{v}}_n$$

for some $v_n\in {\left(t_{p_{n+1}}\right)}_{{\gamma }_n}$. By elementarity, these choices can be made in M. The nodes $v_n$ are coherent because the conditions are descending and the heights are increasing.

Let $y={\bigcup }_n{v}_n{\in }^{ \delta }2$ and $x=y\upharpoonright a$. Use the tree, ladder, and b-coordinate part of the countable-limit construction from Theorem 2, adding y to the new top level alongside the standard coherent b-limits. Every initial segment of y already belongs to one of the earlier trees, so the enlarged tree remains normal. Retain the standard b-values and set $b_q\left(y\right)=y$. Put ${\ell }_q\left(\delta \right)=⌀$. At the new supremum, use the non-dummy value

$$f_q\left(a\right)=x$$

in place of the dummy value used in that construction. This is a legal non-dummy value because $\widehat{a}=\delta$ and $x=y\upharpoonright a\in t_q\left[a\right]$. No earlier f-domain has supremum $\delta$. End extension stabilizes the old nontrivial support below each bounded ordinal, and adjoining the single point $\delta$ therefore preserves nowhere stationarity. The new value is not a member of any old ladder-coordinate value because its domain a is cofinal in $\delta$, whereas every node occurring in an old ladder-coordinate value has domain bounded below $\delta$. Thus q is a condition below every $p_n$.

Since each $y_n\in M\cap {\mathcal{P}}_{{\omega }_1}\alpha$ is countable, $y_n\subseteq M\cap \alpha =a$. The displayed inclusions therefore give $a={\bigcup }_n{y}_n$. Closure of $\dot{C}$ under increasing $\omega$-unions then gives $q⊩\check{a}\in \dot{C}$. Moreover, for every $\beta \in a$, choose n with $\beta <{\gamma }_n$. Since $q\le p_{n+1}$ and $p_{n+1}⊩\dot{c}\upharpoonright {\gamma }_n={\check{v}}_n$, while $y\upharpoonright {\gamma }_n=v_n$, the condition q forces $\dot{c}\upharpoonright a=x$. Since $x\in range\left(f_q\right)$ and every stronger condition must keep its ladder-coordinate values disjoint from the range of its f-coordinate, the condition q forces $x\notin L_{\alpha }$. Hence the trace of $\dot{c}$ misses the member a of the named club. Since $r\le p$ was arbitrary, no branch meets $L_{\alpha }$ on a club. □

Corollary 2.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. Every nonempty countable subfamily of

$$\left\{{Tr}_{{\omega }_1}(B_{\alpha }\left(u\right),L_{\alpha }):u\in T_{\alpha }\right\}$$

has bistationary intersection. In particular, every designated branch meets $L_{\alpha }$${\omega }_1$-cofinally, but no designated branch meets it on an ${\omega }_1$-club.

Proof. 

Stationarity of every nonempty countable intersection follows from Theorem 3. Fix such a subfamily and one of its members. By Proposition 4, that trace contains no club. If the complement of the intersection were nonstationary, some club would be contained in the intersection and hence in the chosen member, a contradiction. Thus the intersection is also costationary. Every stationary subset of ${\mathcal{P}}_{{\omega }_1}\alpha$ is cofinal under inclusion. □

3.3. Wide Levels and Branch-Cover Rigidity

The designated branch cover supplied by Proposition 3 cannot be replaced by a countable family, even when arbitrary cofinal branches are permitted.

Lemma 6.

Let α be an uncountable regular cardinal, let $\mu <\alpha$ be an infinite cardinal in V, and let $p\in {\mathbb{S}}^{*}\left(\alpha \right)$. There are $q\le p$, a limit ordinal $\delta <\alpha$, and a node $u\in {\left(t_p\right)}_{{\gamma }_p}$ such that

$$\{u^{⌢}d:d{\in }^{ \mu }2\}\subseteq {\left(t_q\right)}_{\delta }.$$

Consequently, in every ${\mathbb{S}}^{*}\left(\alpha \right)$-generic extension, some level of $T_{\alpha }$ contains a copy of ${{(}^{\mu }2)}^V$.

Proof. 

Put $\eta ={\gamma }_p$ and fix $u\in {\left(t_p\right)}_{\eta }$. Above u, attach a copy of ^<μ2. For $s{\in }^{ \xi }2$ with $\xi <\mu$, the corresponding node $u^{⌢}s$ lies on level $\eta +\xi$. At the limit level

$$\delta =\eta +\mu$$

add every node $u^{⌢}d$ with $d{\in }^{ \mu }2$. Above every other old top node, add a single chain to level $\delta$. Since $\alpha$ is a cardinal and $\eta ,\mu <\alpha$, the ordinal $\delta +1$ is below $\alpha$.

Every node in the binary subtree above u has a continuation on the new top level, and every node on one of the other chains has its chain continuation. Extend each old b-value to a compatible new top node and assign a top continuation to every new node. The resulting tree is normal and end-extends $t_p$, while the new b-coordinate coherently extends $b_p$.

Leave ${\ell }_p$ and $f_p$ unchanged. Their domains are bounded by the old top height. Every limit point of $dom\left({\ell }_p\right)$ in $\delta +1$ is therefore at most $\eta$ and belongs to the old closed domain. Moreover, for every $z\in dom\left(f_p\right)$, $\widehat{z}\le \eta$, and hence the end extension gives $t_q\left[z\right]=t_p\left[z\right]$. Thus all old non-dummy values remain legal. Below or at $\eta$, nowhere stationarity is inherited from p. Above $\eta$, the old support is bounded. Every old f–ℓ disjointness clause is also preserved. The resulting tuple is a condition $q\le p$ with the displayed top-level copy of ^μ2.

For fixed $\mu$, the conditions supplied by the first part form a dense set. A generic filter meets it, and end extension preserves the displayed level. □

Corollary 3.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. For $x\in {\mathcal{P}}_{{\omega }_2}\alpha$, put

$${\mathcal{T}}_{\alpha ,x}^{{\omega }_2}=\{r\upharpoonright x:r\in {\left(T_{\alpha }\right)}_{sup\left(x\right)+1}\}.$$

Then some level of this restriction family has size at least ${\omega }_2$. Consequently, it is not a ${\mathcal{P}}_{{\omega }_2}\alpha$-tree.

Proof. 

Use the construction from Lemma 6 with $\mu ={\omega }_1$. If $\eta$ is the old top height, that construction takes $\delta =\eta +{\omega }_1$. Put

$$x=[\eta ,\delta )=\{\eta +\xi :\xi <{\omega }_1\}.$$

Then $x\in {\mathcal{P}}_{{\omega }_2}\alpha$. The nodes $u^{⌢}d$ for $d\in {{(}^{{\omega }_1}2)}^V$ lie on level $\delta$ and have pairwise distinct restrictions to x. By normality, extend them to level $\delta +1$. Their restrictions to x are unchanged. Hence there is an injection

$${{(}^{{\omega }_1}2)}^V⟶{\mathcal{T}}_{\alpha ,x}^{{\omega }_2}.$$

In V, fix an injection ${\omega }_2⟶{{(}^{{\omega }_1}2)}^V$. By Theorem 2, ${\omega }_2$ is preserved, so in $V\left[G\right]$ the composition of these injections witnesses that the displayed level has size at least ${\omega }_2$. □

The construction in Lemma 6 modifies only t and b. It leaves ℓ and f unchanged and fixes all old tree levels. Hence, for $\alpha ={\omega }_2$, this construction is compatible with the displayed tree and branch clauses ([9], Definition 5.5(1) and (3), p. 1123), independently of the f-typing discrepancy ([9], Definition 5.5(4), p. 1123, and Claims 5.7–5.8, p. 1124). In the case $\mu ={\omega }_1$, let $\eta$ denote the old top height and put $\delta =\eta +{\omega }_1$ and $x=[\eta ,\delta )$. Already in V, the restriction level at x defined by the formula in ([9], p. 1122) contains a copy of ${{(}^{{\omega }_1}2)}^V$, so its size is at least ${\omega }_2^V$. Thus the local width obstruction lies in the displayed tree and branch clauses themselves.

Theorem 4.

Let $\alpha \ge {\omega }_2$ be regular. In every ${\mathbb{S}}^{*}\left(\alpha \right)$-generic extension, $L_{\alpha }$ is not generated by any countable family of cofinal branches through $T_{\alpha }$. Equivalently,

$$bcov(T_{\alpha },L_{\alpha })\ge {\omega }_1$$

for the ordinal-tree branch-covering number.

Proof. 

Apply Lemma 6 with $\mu =\omega$. Since ${\mathbb{S}}^{*}\left(\alpha \right)$ preserves ${\omega }_1$ and ${\left(2^{\omega }\right)}^V\ge {\omega }_1$, fix a level of the generic tree containing ${\omega }_1$ distinct nodes and write

$$U=\{u_{\xi }:\xi <{\omega }_1\}.$$

The designated branches $B_{\alpha }\left(u_{\xi }\right)$ are pairwise distinct because they pass through distinct nodes on the same level. By Corollary 2, every one of these branches meets $L_{\alpha }$ ${\omega }_1$-cofinally.

Applying Corollary 1 with $\rho ={\omega }_1$ to this family gives $bcov(T_{\alpha },L_{\alpha })\ge {\omega }_1$. □

Proposition 5.

Let $\alpha \ge {\omega }_2$ be regular, and let ${\mathbb{S}}_{\mathrm{K}}^{*}\left(\alpha \right)$ be the forcing obtained by restricting ${\mathbb{S}}^{*}\left(\alpha \right)$ to conditions satisfying

$$|{\left(t_p\right)}_{\xi }|\le |\xi |+\omega (\xi \le {\gamma }_p),$$

with the inherited order. The forcing ${\mathbb{S}}_{\mathrm{K}}^{*}\left(\alpha \right)$ is σ-closed. After adjoining a formal maximum, it is α-strategically closed. Write $T_{\alpha }^{\mathrm{K}}$, $L_{\alpha }^{\mathrm{K}}$, and $B_{\alpha }^{\mathrm{K}}\left(u\right)$ for its generic tree, ladder-coordinate set, and designated branches. Then, for every nonempty countable $U\subseteq T_{\alpha }^{\mathrm{K}}$, the intersection

$$\bigcap _{u\in U}{Tr}_{{\omega }_1}(B_{\alpha }^{\mathrm{K}}\left(u\right),L_{\alpha }^{\mathrm{K}})$$

is bistationary, and no countable family of cofinal branches through $T_{\alpha }^{\mathrm{K}}$ generates $L_{\alpha }^{\mathrm{K}}$.

Proof. 

It remains to verify the level bound throughout these constructions. Every genuine Odd move satisfies the bound by definition, and Even maintains it. Successor extensions in these constructions use one chain above each old top node, so every new level has size at most that of the old top level. At a nonzero limit height $\delta$, the constructions add at most one coherent top limit for each node already present below $\delta$. If t is the union of the old trees below $\delta$, then

$$\left|t\right|\le \sum _{\xi <\delta }\left(\right|\xi |+\omega )\le \left|\delta \right|+\omega .$$

Adding the single extra top node in the no-club argument does not change the bound. The basic height extension used to obtain the generic tree and designated branches also preserves the bound. Thus the $\sigma$-closure lower bounds, Even’s strategy, and the insertion and no-club constructions all remain inside ${\mathbb{S}}_{\mathrm{K}}^{*}\left(\alpha \right)$. As in Theorem 2, strategic closure gives $<\alpha$-distributivity and preserves ${\omega }_1$ and $\alpha$. The stationary-trace and no-club arguments therefore give the stated bistationarity conclusion for the companion forcing.

The full width conclusion of Lemma 6 is unnecessary for the branch-cover argument. Given $p\in {\mathbb{S}}_{\mathrm{K}}^{*}\left(\alpha \right)$, first end-extend to a top height $\eta >{\gamma }_p$ with $\eta <\alpha$ and $\left|\eta \right|\ge {\omega }_1$. Regularity of $\alpha$ gives $\eta +\omega <\alpha$. Choose $D{\subseteq }^{ \omega }2$ of size ${\omega }_1$. Above one old top node u, place the restrictions

$$\{u^{⌢}(d\upharpoonright n):d\in D\}$$

on level $\eta +n$ for $n<\omega$, and place $\{u^{⌢}d:d\in D\}$ on level $\eta +\omega$. Above every other old top node, add a single chain. The split part contributes at most ${\omega }_1$ nodes to any new level and the remaining chains at most $\left|\eta \right|+\omega$, by the bound at level $\eta$. Hence every new level has size at most $\left|\eta \right|+\omega$. Extend every old b-value to a compatible new top node. For each split node, choose a member of D extending the node’s displayed finite tail and use the corresponding top node as the b-value. Leave ℓ and f unchanged. Their domains lie below the old top height, so end extension fixes all restriction sets relevant to them and preserves the old f–ℓ disjointness clauses. The resulting tuple is a legal stronger condition.

Hence conditions producing a level with at least ${\omega }_1$ distinct nodes are dense in ${\mathbb{S}}_{\mathrm{K}}^{*}\left(\alpha \right)$. Such a level gives ${\omega }_1$ pairwise distinct designated branches. By the bistationarity conclusion, their traces are stationary and hence ${\omega }_1$-cofinal. Applying Corollary 1 with $\rho ={\omega }_1$ rules out a countable generating family. □

Under the Kurepa bound, every condition level has cardinality less than $\alpha$, and the two core trace and cover conclusions remain valid. The companion forcing imposes only this ordinal-level bound and leaves the low-cofinality ladder-coordinate convention unchanged. For $\alpha >{\omega }_2$, this bound also does not by itself ensure that restriction levels have cardinality less than ${\omega }_2$, as required for a ${\mathcal{P}}_{{\omega }_2}\alpha$-tree.

Proposition 6 gives a forcing-local argument for Theorem 4 that is independent of the stationary-trace argument. By the maximum principle, any nonempty countable generating family has an $\omega$-enumeration name. Applying the proposition densely to such a name produces a member of $L_{\alpha }$ that no branch in the generating family extends. The empty family is ruled out by Lemma 3.

Proposition 6.

Let $\alpha \ge {\omega }_2$ be regular. Suppose that $p\in {\mathbb{S}}^{*}\left(\alpha \right)$ and $\langle {\dot{c}}_n:n<\omega \rangle$ is a sequence of ${\mathbb{S}}^{*}\left(\alpha \right)$-names such that

$$p⊩``{\dot{c}}_n\mathrm{is}\mathrm{a}\mathrm{cofinal}\mathrm{branch}\mathrm{through}{T}_{\alpha }^{\prime \prime }(n<\omega ).$$

There exist $q\le p$, $\delta \in dom\left({\ell }_q\right)$, and $y\in {\ell }_q\left(\delta \right)$ such that

$$q⊩\check{y}\neg \subseteq {\dot{c}}_n(n<\omega ).$$

Proof. 

Apply Lemma 6 with $\mu =\omega$ to strengthen p to $p_0$ and fix a level $\gamma$ with

$$U=\{u_{\xi }:\xi <{\omega }_1\}\subseteq {\left(t_{p_0}\right)}_{\gamma }$$

consisting of distinct nodes. Using the $<\alpha$-distributivity from Theorem 2, recursively choose $p_{n+1}\le p_n$ and $v_n\in {\left(t_{p_0}\right)}_{\gamma }$ such that

$$p_{n+1}⊩{\dot{c}}_n\upharpoonright \gamma ={\check{v}}_n.$$

End extensions do not alter the $\gamma$-th level, and $\sigma$-closure supplies a common lower bound r of the decision sequence. After all decisions have been made, choose

$$u_{*}\in U\setminus \{v_n:n<\omega \}.$$

Apply Lemma 3 below r with seed $u_{*}$ and $\theta =\gamma$. Obtain $q\le r$, a top height $\delta$, a top node $x=b_q\left(u_{*}\right)$, and a club $E_x\subseteq {\mathcal{P}}_{{\omega }_1}\delta$. For every $n<\omega$, choose ${\beta }_n<\gamma$ with $u_{*}\left({\beta }_n\right)\ne v_n\left({\beta }_n\right)$, and put $A=\{{\beta }_n:n<\omega \}$. Choose $z\in E_x$ with $A\subseteq z$ and set

$$y=x\upharpoonright z\in {\ell }_q\left(\delta \right).$$

For every $n<\omega$,

$$q⊩{\dot{c}}_n\left({\beta }_n\right)=v_n\left({\beta }_n\right)\ne u_{*}\left({\beta }_n\right)=x\left({\beta }_n\right)=y\left({\beta }_n\right).$$

Hence the single ladder-coordinate node y is not extended by any ${\dot{c}}_n$. □

Corollary 4.

Let $\alpha \ge {\omega }_2$ be regular. In the ${\mathbb{S}}^{*}\left(\alpha \right)$-generic extension,

$${\omega }_1\le bcov(T_{\alpha },L_{\alpha })\le \left|T_{\alpha }\right|.$$

Proof. 

The lower bound is Theorem 4. The upper bound follows from the designated generating family in Proposition 3. □

The scaled two-cardinal analysis rests on a comparison between cofinal branches of the ordinal tree and branches of its restriction system. More generally, suppose that $\alpha$ is a regular cardinal in the ambient universe and that $T{\subseteq }^{ <\alpha }2$ is a normal binary tree of height $\alpha$. For $x\in {\mathcal{P}}_{\alpha }\alpha$, put

$${\beta }_x=sup\left(x\right)+1,{\mathcal{T}}_x=\{r\upharpoonright x:r\in T_{{\beta }_x}\}.$$

Regularity of $\alpha$ ensures ${\beta }_x<\alpha$. The successor is included so that $x\subseteq {\beta }_x$ also when x has a maximum. Normality of T gives nonempty levels, and downward closure gives coherent restriction maps. Thus the displayed system is a ${\mathcal{P}}_{\alpha }\alpha$-tree exactly when each level has size less than $\alpha$.

For the generic tree, Theorem 2 ensures that $\alpha$ is a regular cardinal in $V\left[G\right]$. Every domain occurring in $L_{\alpha }$ therefore belongs to ${\mathcal{P}}_{\alpha }\alpha$. If $y\in L_{\alpha }$ has ordinal domain $\xi$, then $y\in {\left(T_{\alpha }\right)}_{\xi }$. The node y itself witnesses $y\in {\mathcal{T}}_{\xi }$ when ${\beta }_{\xi }=\xi$, and normality supplies an extension on level ${\beta }_{\xi }$ when ${\beta }_{\xi }>\xi$. If $y=x\upharpoonright z$ arises from a ladder coordinate $\xi$ of uncountable cofinality, then $z\in {\mathcal{P}}_{{\omega }_1}\xi$ and hence ${\beta }_z<\xi$. Downward closure gives $x\upharpoonright {\beta }_z\in {\left(T_{\alpha }\right)}_{{\beta }_z}$, which witnesses $y\in {\mathcal{T}}_z$. Thus $L_{\alpha }$ is a set of nodes of the restriction system. By Proposition 1, it is not a ladder system on that restriction system in the full sense of ([9], Definition 4.2, p. 1119), even when the level-size condition holds.

Lemma 7.

Let $\mathcal{T}$ be the restriction system just defined from T. A function $c{\in }^{ \alpha }2$ is a cofinal branch through the ordinal tree T if and only if

$$c\upharpoonright x\in {\mathcal{T}}_x\mathrm{for}\mathrm{every}x\in {\mathcal{P}}_{\alpha }\alpha .$$

Consequently, whenever the level-size condition holds, so that $\mathcal{T}$ is a ${\mathcal{P}}_{\alpha }\alpha$-tree, its branches are exactly the cofinal branches through T.

Proof. 

If every ordinal initial segment of c belongs to T, then for $x\in {\mathcal{P}}_{\alpha }\alpha$ the node $c\upharpoonright {\beta }_x$ belongs to $T_{{\beta }_x}$ and witnesses $c\upharpoonright x\in {\mathcal{T}}_x$. Conversely, suppose that c is a branch through $\mathcal{T}$. For every ordinal $\eta <\alpha$, $\eta \in {\mathcal{P}}_{\alpha }\alpha$. The branch condition at the level $x=\eta$ gives $r\in T_{{\beta }_{\eta }}$ with $c\upharpoonright \eta =r\upharpoonright \eta$. Since $\eta \le {\beta }_{\eta }$ and T is downward closed, it follows that $c\upharpoonright \eta \in T_{\eta }$. Hence c is a cofinal branch through T. □

Proposition 7.

Let $\alpha \ge {\omega }_2$ be regular and let $G\subseteq {\mathbb{S}}^{*}\left(\alpha \right)$ be generic. For the scaled restriction system ${\mathcal{T}}_{\alpha }$ indexed by ${\mathcal{P}}_{\alpha }\alpha$ and induced by $T_{\alpha }$, the following conditions are equivalent.

(a)

every level of ${\mathcal{T}}_{\alpha }$ has size less than α.

(b)

α is a strong limit cardinal in V.

(c)

α is strongly inaccessible in V.

When these conditions hold, ${\mathcal{T}}_{\alpha }$ is a ${\mathcal{P}}_{\alpha }\alpha$-tree and

$$bcov({\mathcal{T}}_{\alpha },L_{\alpha })\ge {\omega }_1.$$

In particular, if α is a successor cardinal in V, then ${\mathcal{T}}_{\alpha }$ is not a ${\mathcal{P}}_{\alpha }\alpha$-tree.

Proof. 

Assume first that every level of ${\mathcal{T}}_{\alpha }$ has size less than $\alpha$. Let $\mu <\alpha$ be an infinite cardinal in V. By Lemma 6 and genericity, there are a limit ordinal $\delta <\alpha$ and a ground-model injection

$${{(}^{\mu }2)}^V⟶{\left(T_{\alpha }\right)}_{\delta }.$$

Since $\delta$ is a limit ordinal, ${\beta }_{\delta }=\delta +1$. Normality and downward closure of $T_{\alpha }$ give

$${\mathcal{T}}_{\delta }=\{r\upharpoonright \delta :r\in {\left(T_{\alpha }\right)}_{\delta +1}\}={\left(T_{\alpha }\right)}_{\delta }.$$

The ground-model injection remains an injection in $V\left[G\right]$. Because $\alpha$ remains a cardinal there, the inequality ${\left(2^{\mu }\right)}^V\ge \alpha$ would contradict $|{\mathcal{T}}_{\delta }|<\alpha$. Hence

$${\left(2^{\mu }\right)}^V<\alpha \mathrm{for}\mathrm{every}\mathrm{infinite}\mathrm{cardinal}\mu <\alpha \mathrm{in}V,$$

so $\alpha$ is a strong limit cardinal in V.

Conversely, suppose that $\alpha$ is a strong limit cardinal in V. Let $x\in {\mathcal{P}}_{\alpha }\alpha$ in $V\left[G\right]$. Regularity of $\alpha$ in the extension gives ${\beta }_x<\alpha$. Choose $p\in G$ whose tree height is above ${\beta }_x$. End extension fixes that level, so

$${\left(T_{\alpha }\right)}_{{\beta }_x}={\left(t_p\right)}_{{\beta }_x}.$$

The ordinal ${\beta }_x$ belongs to V, and $|{\beta }_x{|}^V<\alpha$. In the ground model put

$$\kappa ={{|}^{{\beta }_x}2|}^V={\left(2^{|{\beta }_x{|}^V}\right)}^V<\alpha$$

and fix an injection

$${\left(t_p\right)}_{{\beta }_x}⟶\kappa .$$

The same injection exists in $V\left[G\right]$. Since $\kappa <\alpha$ and $\alpha$ is a cardinal there, ${\left|\kappa \right|}^{V\left[G\right]}<\alpha$. The level ${\mathcal{T}}_x$ is the image of ${\left(t_p\right)}_{{\beta }_x}$ under restriction to x, and hence $|{\mathcal{T}}_x|<\alpha$. This proves the equivalence of (a) and (b). Since $\alpha$ is regular and uncountable in V, conditions (b) and (c) are equivalent.

Under these conditions, the restriction system is a ${\mathcal{P}}_{\alpha }\alpha$-tree. By Lemma 7, its branches are exactly the cofinal branches through $T_{\alpha }$, so the lower bound follows from Theorem 4. A successor cardinal in V is not a strong limit cardinal in V, which gives the final assertion. □

Proof of Theorem 1.

The preservation assertion is Theorem 2. Part (i) is Corollary 2, part (ii) is Theorem 4, part (iii) follows from Lemma 6 and genericity, and part (iv) is Corollary 3. Proposition 7 gives the exact scaled restriction-system boundary and the accompanying branch-cover lower bound. □

For every regular $\alpha \ge {\omega }_2$, every nonempty countable family of designated generic branches has a bistationary common trace on $L_{\alpha }$, and no countable family of cofinal branches generates $L_{\alpha }$. The corresponding trace and branch-cover conclusions hold for the Kurepa-style companion forcing, so neither conclusion depends on unrestricted width. The unrestricted tree clause has a different effect. In the unrestricted forcing, for every infinite ground-model cardinal $\mu <\alpha$, some level of the generic tree contains a copy of ${{(}^{\mu }2)}^V$. Hence the endpoint-corrected ${\mathcal{P}}_{{\omega }_2}\alpha$ restriction family is too wide. For the scaled ${\mathcal{P}}_{\alpha }\alpha$ system, every level has size less than $\alpha$ exactly when $\alpha$ is strongly inaccessible in V, and in that case the branch-covering number of $L_{\alpha }$ relative to that system is at least ${\omega }_1$. A separate failure comes from the low-cofinality empty-value convention. It prevents the set of domains of $L_{\alpha }$ from containing a club, so $L_{\alpha }$ is not a ladder system in the full two-cardinal sense. The forcing ${\mathbb{S}}^{*}\left(\alpha \right)$ is $\sigma$-closed and, after adjoining a formal maximum, $\alpha$-strategically closed. It is therefore $<\alpha$-distributive and preserves $\alpha$ as a regular cardinal. The conclusions stated here for ${\mathbb{S}}^{*}\left(\alpha \right)$ concern its direct generic extension. No forcing equivalence with the printed Hayut–Magidor presentation is claimed, and the results do not settle Hayut and Magidor’s question of whether cofinal catching can be separated from stationary catching.

Several questions remain. Is the lower bound in Corollary 4 sharp under additional cardinal-arithmetic hypotheses, or can the generic ladder-coordinate set require strictly more than ${\omega }_1$ branches?

Proposition 5 shows that the two core trace and cover conclusions survive a Kurepa-style level bound. At $\alpha ={\omega }_2$, a distinct problem is to find a typed, size-controlled modification that also treats the low-cofinality ladder coordinates and produces a genuine ${\mathcal{P}}_{{\omega }_2}{\omega }_2$-tree with a full ladder system while preserving the trace and cover conclusions.

A further question is whether the master-condition argument in the proof of Lemma 5.11 ([9], p. 1126) admits a typed counterpart. Relevant forcing background includes Mitchell’s original tree-property forcing and later iterated tree-property constructions [3,12,16]. For lifting and strong master conditions, see ([2], Proposition 9.1, p. 805, and Definition 12.2, p. 814). A separate problem is to determine which subsequent forcing notions preserve the trace conclusions. This is related to the fragility and indestructibility of tree properties [17,18].