GLOBAL SQUARE SEQUENCES IN EXTENDER MODELS

MARTIN ZEMAN

Abstract. We present a construction of a global square sequence in extender models with λ-indexing.

AMS Subject Classification: 03E05, 03E45, 03E55

Keywords: Global quare sequence, fine structure, extender model.

August 19, 2009

1 1. Introduction

2 The purpose of this paper is to present a construction of sequences related to 3 most commonly used global square principles in an extender model with Jensen’s 4 λ-indexing introduced in [6]. Straightforward, but technical modifications of these 5 constructions give the same results in extender models with Mitchell-Steel indexing 6 introduced in [10] (see also [15]). The main advantage of using λ-indexing is the 7 relative simplicity and cleanness of the all constructions. 8 The basic result on global square sequences in extender models with λ-indexing 9 was announced in [13] (Theorem 21). Constructions of global square sequences in 10 lower level extender models were given by Welch [16] in Dodd-Jensen core models, 11 Wylie [17] in extender models for measures of Mitchell order 0 and Zeman [20] 12 in models up to one strong cardinal. Jensen-Zeman [7] gives a construction of a 13 condensation-coherent global square sequence in models for measures of order 0, 14 which actually goes through in any extender model whose extenders are generated 15 by their normal measures^{1}; this construction is based on certain ideas from [5]. 16 The construction in this paper builds on a combination of techniques from [20],[14] 17 and [19]. Extender models with λ-indexing were introduced by Jensen in [6], and 18 basic facts about these models can also be found in [18]. For some interesting 19 applications of global square sequences coming from L[E]-models, see [2, 8, 12]. 20 The former two papers focus on the use of fine structural global square sequences 21 to obtain lower bounds for consistency strengths of various stationary reflection 22 principles. In [12] fine structural global square sequences are used to determine 23 a lower bound for the consistency strength of the restricted proper forcing axiom 24 PFA(𝔠^{+} − linked) in the following sense: If PFA(𝔠^{+} − linked) holds in a generic 25 extension via proper forcing over a fine structural model M then M must contain 26 the so-called Σ_{1}^{2}-indescribable cardinal 1-gap, which is stronger than the existence 27 of many subcompact cardinals. This is a remarkable result, as it was proved earlier 28 in [11] show that a Σ_{1}^{2}-indescribable 1-gap suffices for obtaining a proper forcing 29 extension of a model satisfying GCH where PFA(𝔠^{+} − linked) holds. Paper [21] is 1 a sequel to this paper where the current methods are further extended to give a 2 characterization of stationary reflection at inaccessible cardinals similar to that in 3 L, that is, in terms of coherent club sequences. Paper [9] is a sequel to both the 4 current paper, [21] and [14] where the methods developed in these three papers are 5 further extended to constructions of nonthreadable square sequences at successor 6 cardinals in extender models. 7 Recall that given a class 𝒮 of singular ordinals, a sequence ⟨C_{α}; α ∈ 𝒮⟩ is a global 8 square sequence with domain 𝒮, briefly a □𝒮-sequence, just in case that each C_{α} 9 is a closed unbounded subset of α of order type strictly smaller than α such that 10 lim(C_{α}) ⊆ 𝒮 and the sets C_{α} are coherent in the sense that Cᾱ = C_{α} ∩ ᾱ whenever 11 ᾱ ∈ lim(C_{α}). The principle □𝒮 postulates the existence of a global square sequence 12 whose domain is 𝒮. We write briefly □ if 𝒮 is the class of all singular ordinals. Given 13 a class A of singular ordinals, a □𝒮(A)-sequence is a □𝒮-sequence which satisfies 14 the additional condition that lim(C_{α}) ∩ A = ∅ for all α ∈ 𝒮. By the result of 15 Burke and Jensen (proved independently) [1, 14], the principle □_{κ} fails whenever 16 κ is subcompact. Since any global square sequence whose domain are all singular 17 ordinals yields a □_{κ}-sequence for each cardinal κ, it is not possible to have such a 18 global square sequence in an extender model in general. However, it turns out that 19 subcompact cardinals constitute the only limitation on domains of global square 20 sequences in extender models. Our main theorem can be formulated as follows.

Research partially supported by the NSF grants DMS-0204728 and DMS-0500799.

1Equivalently, in any extender model that satisfies the anti-large cardinal requirement that there is no extender with two generators.

21 Theorem 1.1 (Main theorem). Let M be an extender model with λ-indexing and

𝒮^{∗} = Singular Ordinals of M − ⋃{(κ, κ^{+}); κ is subcompact in M}

22 Then in M, there is a global square sequence with domain 𝒮^{∗}. In fact, for any class 23 A ⊆ On there is a class A′ ⊆ A such that for every regular κ,

A ∩ κ is stationary ⟹ A′ ∩ κ is stationary

24 and □^{𝒮^{∗}}(A′) holds. Consequently, any extender model with λ-indexing which con-25 tains no subcompact cardinals satisfies the principle □.

26 Constructions of canonical global square sequences in extender models are all 27 based on Jensen’s original construction in L, and are carried out on two disjoint 28 classes: the class of all singular cardinals, and that of all ordinals which fail to be 29 cardinals. Notice that ordinals in the latter class are elements of intervals (κ, κ^{+}) 30 where κ is a cardinal, so in the case of this class such constructions give rise to □_{κ}- 31 sequences. A typical construction of this kind was presented in [14]. Thus, to obtain 32 a global square sequence, it suffices to focus on the class of all singular cardinals. 33 Our construction will give rise to a version of □^{𝒮}(A′)-sequence where the sets are 34 fully coherent; the precise formulation is stated in the theorem below. Theorem 1.1 35 is a consequence of the fact that □_{κ}(A′)-sequences can be combined with a □^{𝒮}(A′)- 36 sequence, where 𝒮 is the class of all singular cardinals, into a single global square 37 sequence □(A′); this is the easier implication in Jensen’s result that (∀κ)□_{κ} together 38 with □^{𝒮} is equivalent to □. To obtain A′ ∩ (κ, κ^{+}) from a given class A, we can 39 simply choose A′ ∩ (κ, κ^{+}) to be a stationary subset of A ∩ (κ, κ^{+}) such that the 40 order types of sets in the □_{κ}-sequence for α ∈ A′ are the same, so our construction 41 will only focus on the less obvious task of obtaining A′ ∩ Singular Cardinals. We 42 are now ready to state Theorem 1.2. This theorem is the actual result we are 1 going to prove; Theorem 1.1 is a direct consequence of Theorem 1.2 and the above 2 considerations.

3 Theorem 1.2. The following is true in any extender model with λ-indexing. Let 𝒮 4 be the class of all singular cardinals and A ⊆ 𝒮 be a class. There are a class A′ ⊆ A 5 such that for all inaccessible κ,

A ∩ κ is stationary ⟹ A′ ∩ κ is stationary

6 and a sequence ⟨C_{α}; α ∈ 𝒮⟩ satisfying the following. 7 (a) Each C_{α} is a closed subset of α ∩ 𝒮; if cf(α) > ω then C_{α} is unbounded in 8 α. (If α is ω-cofinal, it may happen that C_{α} = ∅.) 9 (b) Cᾱ = C_{α} ∩ ᾱ whenever ᾱ ∈ C_{α}. (That is, we have full coherency.) 10 (c) otp(C_{α}) < α. 11 (d) lim(C_{α}) ∩ A′ = ∅.

12 Section 3 is devoted to the technical tools used in the construction of a sequence 13 as in the previous lemma and Section 4 to the actual construction. The big picture 14 here is similar to that in the construction of a □_{κ}-sequence, but the details differ, 15 in some instances quite significantly. For this reason, several basic notions and 16 technical lemmata have to be amended to the current context. In order to keep 17 the account brief, we state several technical lemmata in Section 3 without proofs 18 whenever these proofs are straightforward amendments of proofs in [14, 18]. How-19 ever we do present proofs of lemmata where the amendments are not negligible. In 20 order to keep this paper self-contained, we also state those technical lemmata whose 21 formulations alone do not require any substantial amendment; it is however still 22 the case that the proofs of these lemmata may require amendments. The sources 23 [14, 19] and [18] should be considered a prerequisite here. We will be using notions 24 and results from these sources without further notice. Let us stress that throughout 25 the proof, we work in a fixed extender model L[E] with λ-indexing of extenders.

26 2. Outline of the construction

27 In this section we attempt to give a big picture of the construction and explain 28 the important points in possibly general terms. It is mentioned in the introduction 29 that the construction builds on that of □_{κ}-sequence presented in [14], so we begin 30 with refreshing the main points here as well as the main points in the construction 31 of a global square sequence in L. We then isolate the key issues that need to be 32 addressed in order to modify the ideas from [14] to obtain tools for construction of 33 a global square sequence on singular cardinals in an extender model. 34 The initial idea, due to Jensen, is to construct a square sequence ⟨C_{τ} ⟩_{τ} (no 35 matter whether local or a global) so that the sets C_{τ} consist of critical points of 36 certain canonical embeddings. One of the key tasks is then the choice of a structure 37 N_{τ} to each τ in a canonical way that would allow to extract C_{τ} with all requisite 38 properties as a set of critical points of suitable elementary maps σ_{τ̄,τ} : Nτ̄ → N_{τ}. 39 In the construction of a □_{κ}-sequence in L, we consider ordinals τ ∈ (κ, κ^{+}) which 40 look like local successor cardinals, that is, κ is the largest cardinal in J_{τ}, and the 41 structures N_{τ} are taken to be collapsing levels for τ in L. Thus τ is regular in 42 N_{τ}, but is definably (in parameters) singularized, and therefore collapsed, over N_{τ}. 43 From now on assume for simplicity that there is a singularizing/collapsing partial 44 function f : κ → τ that is Σ_{1}-definable over N_{τ} in parameters. This is equivalent to 1 saying that κ is the first projectum of N_{τ} and to the fact that the Skolem hull of the 2 first projectum together with the standard parameter h̃_{N_{τ}}^{1} (κ ∪ {p_{N_{τ}} }) is the entire 3 N_{τ}; here p_{N_{τ}} is the standard parameter. We then define D_{τ} to be the set of critical 4 point of maps σ_{τ̄,τ} as above with maximal possible degree of elementarity that are 5 the identity on κ and preserve standard parameters. This degree of elementarity is 6 Σ_{0}, as it follows easily that the only Σ_{1}-preserving map that is the identity on κ 7 and has p_{N_{τ}} in its range is the identity map. The sets D_{τ} are are closed, unbounded 8 whenever τ has uncountable cofinality, and satisfy the strong form of coherency. 9 This can be quite easily verified due to the uniform definition of D_{τ}. However, 10 these sets may have order type larger than κ. In order to obtain C_{τ} we “thin out” 11 the sets D_{τ} by a careful analysis of countable elementary substructures of N_{τ}. This 12 can be done in a sufficiently uniform way that preserves closeness and coherency, 13 and additionally guarantees that the order type of C_{τ} does not exceed κ. 14 We now outline how to extend Jensen’s argument to an arbitrary extender model 15 L[E], as is done in detail in [14]. Analogously as before we consider ordinals τ that 16 are local cardinal successors in L[E] and attempt to define N_{τ} to be a singulariz-17 ing/collapsing level of L[E] for τ. There is a minor issue concerning preservation of 18 standard parameters under maps σ_{τ̄,τ} which is resolved by additional requirement 19 that rng(σ_{τ̄,τ}) contains solidity witnesses for elements of the standard parameter of 20 N_{τ}. This introduces some non-uniformity into the construction which causes that 21 the sets D_{τ} are not fully coherent; here a simple combinatorial manipulation is used 22 to turn the sets D_{τ} into fully coherent sets B_{τ}. We then use the same thinning 23 out procedure as before to produce sets C_{τ}; this time thinning out the sets B_{τ} 24 instead of D_{τ}. The just described scenario can really be made to work, up to one 25 important point, namely the proof that the sets B_{τ}, hence D_{τ}, are cofinal whenever 26 τ has uncountable cofinality. Here is an explanation why. We actually show that 27 even proving that D_{τ} is nonempty is highly nontrivial. Assume that N_{τ} is an active 28 premouse, the critical point of the top extender F of N_{τ} is strictly smaller than κ 29 and there is a function collapsing τ to κ that is Σ_{1}-definable over N_{τ} in parameters. 30 If D_{τ} were nonempty, there would be a Σ_{0}-preserving map σ_{τ̄,τ} : Nτ̄ → N_{τ} whose 31 critical point is τ̄ and which preserves standard parameters. As we have seen in 32 the previous paragraph, such a map cannot be Σ_{1}-preserving, and therefore cannot 33 be cofinal. On the other hand σ_{τ̄,τ} will turn out to be sufficiently elementary to 34 guarantee that N_{τ} is of the form ⟨J E ; ν̄, F̄⟩ where F̄ is an extender. Since σ_{τ̄,τ} is 35 the identity below κ, critical points of F and F̄ agree; denote the common critical 36 point by μ. As σ_{τ̄,τ} is not cofinal, there is some set a ⊆ μ such that a ∈ N_{τ} but 37 F (a) ∉ rng(σ_{τ̄,τ}). Clearly a ∈ Nτ̄, as N_{τ} and Nτ̄ agree up to κ and κ is a cardinal 38 in L[E]. But the value F̄(a) cannot be defined, as the degree of elementarity of σ_{τ̄,τ} 39 would force that F (a) = σ_{τ̄,τ} (F̄(a)) and we have seen that F (a) is not in the range 40 of σ_{τ̄,τ}. We thus conclude that F̄ is an extender on J E ; ν̄ that fails to measure all 41 subsets of μ that are elements of J E ; ν̄. As a consequence, Nτ̄ is not the right choice 42 of the singularizing structure for τ̄, since its top extender measures all subsets of its 43 critical point that are elements of Nτ̄. In fact, we have just seen that any structure 44 that would qualify as a candidate for the canonical singularizing structure for τ̄ has 45 top extender that is not total, which means that such structure is not an initial 46 segment of any extender model. Structures of this kind are called protomice. It is 47 now clear that in order to generalize Jensen’s construction to extender models, it 48 is necessary to introduce protomice in the construction.

1 A closer look at protomice reveals that protomice constitute a form of encoding 2 singularizing/collapsing L[E]-levels: Given a protomouse M = ⟨J_{ν}^{E}, F ⟩, over which 3 τ is Σ_{1}-definably singularized (in parameters), let M^{∗} be the longest initial segment 4 of M on which F is total. The fine structural ultrapower N of M^{∗} by F is precisely 5 the singularizing level of L[E] for τ. On the other hand, the ultrapower map π 6 is the inverse of the Mostowski collapsing isomorphism obtained by collapsing the 7 fine structural Skolem hull h̃_{N_{τ}}^{n+1}(μ ∪ {r}) where μ = cr(F), r is the image of the 8 standard parameter of M^{∗} under π and n is largest such that the n-th projectum 9 of N_{τ} is strictly above κ. In other words, we form the fine structural Skolem 10 hull of μ ∪ {r} whose degree of elementarity is Σ_{1}^{(n)}. It can be proved that r 11 is a top segment of the standard parameter of N. It is then obvious that the 12 extender derived from π is F, and is fully determined by the pair (μ, r). Thus 13 there is a simple way of obtaining the singularizing L[E]-level from a singularizing 14 protomouse and vice versa. Moreover, there is a nice translation procedure which 15 enables to translate all fine structural information between the two structures. All 16 of this makes it plausible that replacing L[E] levels with protomice should be a 17 successful approach, but this approach involves solving two crucial issues. First, 18 given τ we have to decide whether the canonical singularizing structure for τ will 19 be an L[E]-level or a protomouse. This decision has to be made before we start the 20 construction, that is, before we see what kind of structures and embeddings we have 21 to consider when defining D_{τ}. Second, whereas the way from a protomouse to the 22 corresponding L[E]-level is unique, that is, each protomouse uniquely determines 23 the corresponding L[E]-level, the converse is not true in general. It is possible to 24 have an L[E]-level which can be converted into many protomice, as there may be 25 many pairs (μ, r) as above. Such pairs are called divisors of N. To be more precise, 26 in our official definition of a divisor we require the divisors to have the form (μ, q) 27 where q is the bottom part of the standard parameter, that is, q = p_{N_{τ}} − r as this 28 definition is more suitable for computations. Of course both (μ, r) and (μ, q) carry 29 the same amount of information, so for the purpose of this informal outline we will 30 consider both to be divisors. To summarize, if we decide to choose a protomouse as 31 the canonical singularizing structure for τ, we face the question which protomouse 32 should be the canonical one. Equivalently, we have to make a choice of a divisor. In 33 order to keep the coherency of the sequence of sets we are constructing, the choice 34 of divisors must be made so that it is preserved under the embeddings σ_{τ̄,τ} which 35 we know have low degree of elementarity. 36 A careful examination of divisors reveals that for each singularizing L[E]-level N_{τ} 37 there is a collection of divisors that are easily identified and are preserved under 38 weakly preserving embeddings. We will call these divisors strong. We omit the 39 definition in this outline, as it is not helpful in understanding the big picture. We 40 then show that if N_{τ} has a strong divisor at all then it has one with largest possible 41 μ, and this divisor is unique for N_{τ}. We will consider this divisor canonical for N_{τ}. 42 It will turn out that although canonical divisors are not literally preserved under 43 Σ_{0}-elementary maps, such maps preserve enough information about them that our 44 construction can be carried out. The construction has the following format. Given 45 τ, we check whether the singularizing L[E]-level N_{τ} has a strong divisor. If the 46 answer is negative, we let the canonical singularizing structure be N_{τ}. If the answer 47 is affirmative, we let the canonical singularizing structure M_{τ} be the protomouse 48 determined by the canonical divisor for N_{τ}. This splits the set of the ordinals τ 1 we are considering into two disjoint subsets 𝒮^{0} and 𝒮^{1}. If our extender model L[E] 2 contains enough large cardinals, both these sets may be stationary. On each of 3 these sets we then carry out the constructions of D_{τ} and B_{τ}. We will be able to 4 show that D_{τ}, and therefore also B_{τ} is unbounded in τ whenever τ has uncountable 5 cofinality, closed on a tail-end, and contained in 𝒮^{i} whenever τ ∈ 𝒮^{i}. We then show 6 that the tail-ends can be chosen in a way that they constitute a coherent sequence, 7 which will make it possible to run the thinning out procedure in a manner similar 8 to that in L to produce the square sequence C_{τ}. 9 Proving the three properties of D_{τ} mentioned in the previous paragraph is the 10 heart of the argument. The proof actually shows that for sufficiently large τ̄ such 11 that there is a Σ_{0}-preserving map σ : M̄ → M_{τ} where M̄ is a singularizing proto-12 mouse for τ̄ we necessarily have M̄ = Mτ̄. That is, the only protomouse that can 13 be embedded into the canonical protomouse M_{τ} are those determined by canonical 14 divisors for Nτ̄. Let us also mention the special case where N_{τ} is as in the example 15 above, that is N_{τ} is active, the critical point μ of its top extender is strictly smaller 16 than τ and there is a Σ_{1}-definable function over N_{τ} in parameters that singularizes 17 τ. If moreover N_{τ} does not have any strong divisors, following our choice of singu-18 larizing structures, we let N_{τ} be the canonical singularizing structure for τ. In the 19 example above we saw that for each τ̄ ∈ C_{τ} the canonical singularizing structure for 20 τ̄ is the protomouse Mτ̄. Hence the maps σ_{τ̄,τ} are maps between protomice and a 21 premouse. Fine structural parameters of protomice ⟨J_{τ}^{E}, F ⟩ are computed relative 22 to predicates E and F, whereas those of premice ⟨J_{τ}^{E}, F ⟩ are computed relative to 23 predicates E, F and possibly some additional constants. To ensure coherency, in 24 situations like this we have to treat N_{τ} as a protomouse; we call it “pluripotent 25 premouse”. In particular, such situations require to work with the Dodd parameter 26 of N_{τ} instead of the standard parameter, as the Dodd parameter is precisely the 27 standard parameter computed relative to E and F. We then we have to prove fine 28 structural lemmata which show that switching to Dodd parameters does not do any 29 harm to the coherency of sets we construct. 30 We are now ready to discuss the construction of a global square sequence on 31 singular cardinals and give a description of the main technical issues that arise 32 when trying to adapt the ideas described above to this construction. As τ, being 33 now a singular cardinal in L[E], is a limit cardinal in any initial L[E]-segment J_{β}^{E} 34 where β ≥ τ, it will look like an inaccessible cardinal in the singularizing L[E]-level 35 N_{τ} for τ. We again split the class of all singular cardinals into two disjoint classes 36 𝒮^{0} and 𝒮^{1} where the former consists of those ordinals for which we let the canonical 37 singularizing structure be N_{τ}, whereas the latter consists of all those ordinals for 38 which we let the canonical singularizing structure be a canonical protomouse. From 39 the point of view of N_{τ}, the inaccessible cardinal τ is definably (in parameters) 40 singularized over N_{τ}. In the construction of a □_{κ}-sequence in [14] κ was the largest 41 cardinal below τ, which was a very handy fact on which we heavily relied, even in 42 the definition of a divisor. Hence the first task in the present construction is to 43 modify the definition of a divisor in a way that would make sense even if no cardinal 44 predecessor of τ exists. Of course, this has the consequence that all of the lemmata 45 concerning divisors and relationship between protomice and the associated L[E]- 46 levels have to be reformulated and the proofs adjusted appropriately. In particular, 47 we have to make sure that we have a meaningful notion of strong divisor in the 48 present context and that the auxiliary lemmata can be appropriately modified as 1 well. The condensation lemma for protomice requires a more careful reformulation 2 and we actually include a proof of the lemma in detail, as it contains nontrivial 3 new technical aspects. There are of course numerous minor points that require 4 some work beyond that included in [14]; those which can be treated in a standard 5 way are left to the reader without further comments. However, our construction 6 involves there three points in that need to be addressed a substantially new way; 7 we would like to say more about these points below. 8 It was explained above that given a divisor (μ, r) for N_{τ} we can determine the 9 top extender of the associated protomouse by forming the fine structural Σ_{1}^{(n)}- 10 elementary hull h̃_{N_{τ}}^{n+1}(μ∪{r}) and subsequent deriving the extender from the inverse 11 of the collapsing isomorphism π. We also said that there is a natural translation 12 procedure that enables to transfer fine structural information between N_{τ} and the 13 associated protomouse M. The heart of the issue is that the complexity degree of 14 facts that are possible to translate using this procedure must match the degree of 15 elementarity of the Skolem hull we form, or equivalently to the degree of elementar-16 ity of the associated ultrapower map. Thus, in our case the translation procedure 17 translates Σ_{1}^{(n)}-facts over N_{τ} into Σ_{1}-facts over M and vice versa. If τ were a 18 local successor (as it happens in the construction in [14]) then for n as above we 19 would have a singularizing function for τ that is Σ_{1}^{(n)}-definable in parameters over 20 N_{τ}, so the translation procedure tells us that the same function is Σ_{1}-definable in 21 parameters over M. In the present case τ is a limit cardinal, so it may happen that 22 even though n is largest such that the n-th projectum of N_{τ} is strictly above τ, the 23 smallest complexity degree of a singularizing function for τ over N_{τ} is much larger 24 than n. Thus, in this case the protomouse M would not be a singularizing proto-25 mouse for τ, as any function that is Σ_{1}-definable over M would be Σ_{1}^{(n)}-definable 26 over N_{τ}, so such function would not singularize τ. In order to fix this issue one 27 might attempt to increase the elementarity degree of the Skolem hull of μ ∪ {r}, 28 but this will clash with the fact that N_{τ} is obtained as a fine structural ultrapower, 29 and it is a general fact about such ultrapowers that the elementarity degree of the 30 ultrapower map is always the largest n such that the n-th projectum of N_{τ} is above 31 π(μ). Given these two observations, the only feasible approach to deal with the 32 issue seems to be imposing an additional restriction the class 𝒮^{1} that every τ ∈ 𝒮^{1} 33 must satisfy the following requirement: If n is largest such that the n-th projectum 34 of N_{τ} is strictly above τ then there is a partial singularizing function for τ that 35 is Σ_{1}^{(n)}-definable over N_{τ} in parameters. Such L[E]-levels N_{τ} will be called exact. 36 Notice that in the case where τ was a local successor N_{τ} was automatically exact, 37 so this restriction is actually quite natural. However, whereas in the construction 38 of □_{κ} the property of being exact was always automatically granted and we did 39 not even have to consider it explicitly, in our current situation we have to develop 40 effort to keep the property of being exact preserved at all sensitive stages in the 41 construction. Moreover, we also have to verify that there is no loss of generality, 42 more precisely, that despite our restriction of 𝒮^{1} the sets C_{τ} are defined for every 43 singular cardinal τ. 44 The two other major issues that arise in the construction are of more technical 45 nature than that discussed above. Both concern the proof that the sets C_{τ} are 46 closed. In the construction of a □_{κ}-sequence where the ordinals τ were local cardinal 47 successors we were able to show that B_{τ} were closed on a tail-end, and a part of 48 the argument was an easy observation that if τ̄ is a limit point of B_{τ} and we have 1 a diagram of structures N_{τ^{∗}} together with the canonical maps σ_{τ^{∗},τ′} between them 2 for τ^{∗}, τ′ ∈ B_{τ} ∩τ̄ then the direct limit structure N̄ is a singularizing structure for τ̄. 3 This is the case since there is a common bound on the domains of the singularizing 4 functions of τ^{∗}, namely κ. In the present situation where all ordinals τ, τ̄ and τ^{∗} 5 are singular cardinals no such bound exists, and actually B_{τ} fails to be closed on 6 a tail-end in general if τ ∈ 𝒮^{0}. We however prove that the sets C_{τ} are closed, by a 7 careful examination of the thinning out procedure that is used to extract C_{τ} from 8 B_{τ}. This is done in Lemma 4.11. The main idea of the proof appears already in [20], 9 although here we use a more elaborate version. It should be noted that a variant of 10 the same argument applied to L is implicit in Jensen’s original proof, although it 11 has a different form, as the entire setting of his construction is different from that 12 chosen in this paper. Let us finally note that in the remark following the proof 13 of Lemma 4.11 we also discuss a scenario where the layout of entire construction 14 makes it possible to prove that the sets B_{τ} can be proved to be closed on a tail-end. 15 However, the author believes the layout chosen in this paper makes the construction 16 somewhat simpler. 17 The remaining major issue we would like to discuss is even more technical. Al-18 ready the construction of a global square sequence in L requires an extra technical 19 tool in order to make the techniques from the construction of a □_{κ}-sequence work. 20 The critical issue is the observation that the maps σ_{τ̄,τ} have maximal possible de-21 gree of elementarity, but are not cofinal in the relevant projectum (for instance, if 22 τ can be singularized via a function that is Σ_{1}-definable over N_{τ} then σ_{τ̄,τ} is not 23 cofinal in the 0-th projectum, that is, not cofinal in the usual sense). One way of 24 arranging this is imposing the additional requirement on the maps σ_{τ̄,τ} that these 25 maps preserve “semi-cofinalities” of τ̄. The “semi-cofinality” of τ is denoted by 26 α_{τ} and is the largest ordinal α < τ such that τ ∩ h̃_{N_{τ}}^{n+1}(α ∪ {p_{N_{τ}} }) = α. The use 27 of “semi-cofinalities” already appears in Jensen’s construction in L. The reason 28 for considering “semi-cofinalities” instead of cofinalities is that the maps σ_{τ̄,τ} do 29 not, in general preserve definable cofinalities, but they are sufficiently elementary 30 to preserve “semi-cofinalities”. The construction in L[E] relies on preservation of 31 “semi-cofinalities” more heavily than that in L in several respects. We will now 32 focus on the most important of them which also goes substantially beyond the 33 construction from [14]. This point is the proof that for τ ∈ 𝒮^{1} the sets B_{τ} are 34 closed on a tail-end (in contrast to the case τ ∈ 𝒮^{0} discussed in the previous para-35 graph). Superficially, the reason why B_{τ} are closed on a tail-end is the existence 36 of a common bound on cofinalities of the kind described in the previous paragraph 37 in connection with the construction in [14], and this bound is μ^{+} where we recall 38 that μ is the critical point of the top extender of the protomouse M_{τ}. The ma-39 jor issue we would like to describe arises in verification that for sufficiently large 40 τ̄, if there is an embedding σ_{τ̄,τ} : M̄ → M_{τ} between two protomice and M̄ is a 41 singularizing protomouse for τ̄ then necessarily M̄ = Mτ̄. The argument is done 42 by contradiction assuming that we have a strictly increasing sequence ⟨τ_{ξ} | ξ < γ⟩ 43 cofinal in τ for which we chose the canonical divisor incorrectly. Letting (μ_{ξ}, q_{ξ}) 44 be the the corresponding correct canonical divisors, necessarily μ_{ξ} > μ where (μ, q) 45 is the canonical divisor of N_{τ}. Without loss of generality we may assume that the 46 ordinals μ_{ξ} constitute an increasing (not necessarily strictly) sequence converging 47 to some ordinal μ′. If μ′ < τ we will draw the contradiction by arguing that the 48 choice of (μ, q) was incorrect, as N_{τ} must have a strong divisor of the form (μ′, q′)

1 where μ′ > μ. This is a kind of reflection argument. However, if μ′ = τ then this 2 argument cannot be carried out. The issue is resolved by setting up the construc-3 tion so that, roughly speaking, the “semi-cofinalities” of elements of B_{τ} computed 4 relative to their corresponding canonical protomice are all equal to 0. It is then 5 proved that this setting will rule out the possibility μ′ = τ.

6 3. Technical tools

7 We begin with a version of the condensation lemma which is useful for combi-8 natorial constructions; this version is slightly more general than that formulated 9 in [14]. This more general formulation helps to better organize the material in the 10 current section. Moreover we obtain formulations of technical lemmata that can be 11 directly applied to broader area of problems than those discussed in our paper. 12 Recall that if N is an acceptable structure and ν is an ordinal then core_{ν}(N) is the 13 transitive collapse of the hull h̃_{N}^{n+1}(ν ∪ {p_{N}}) where p_{N} is the standard parameter 14 of N, h̃_{N}^{n+1} is the Σ_{1}^{(n)}-Skolem function for N and n is such that ωϱ_{N}^{n+1} ≤ ν < 15 ωϱ_{N}^{n}. In Jensen’s fine structure theory these notions can be defined whenever N 16 is an acceptable structure whose parameters can be lengthened; so N need not 17 be a premouse. The associated core map σ : core_{ν}(N) → N is the inverse to 18 the Mostowski collapsing isomorphism. This map is always Σ^{∗}-preserving. We 19 intend to work with initial segments of the model L[E], however for development 20 of technical lemmata in this section it will be convenient to work with the broader 21 class of premice weakly embeddable into L[E]-levels which still enjoy many of the 22 properties of L[E]-levels.

23 Definition 3.1. We say that a premouse N is weakly embeddable into a pre-24 mouse N′ just in case that there is a finite sequence ⟨(N_{i}, σ_{i}, k_{i}) | i < m + 1⟩ such 25 that each N_{i} is a premouse, N_{0} = N, N_{m+1} = N′ and for each i < m + 1

26 • either N_{i+1} is of same type as N_{i} and σ_{i} : N_{i} → N_{i+1} is a Σ_{0}^{(k_{i})}-preserving 27 map such that σ ↾ ωϱ_{N_{i}}^{k_{i}+1} = id 28 • or else N_{i} is an initial segment of N_{i+1}, σ_{i} = id and k_{i} = 0.

29 Obviously, if N is weakly embeddable into N′, N̄ is a premouse of the same type 30 as N and σ : N̄ → N is a Σ_{0}^{(k)}-preserving embedding such that σ ↾ ωϱk+1 ; N̄ = id 31 then N̄ is weakly embeddable into N′. Also any initial segment of a premouse 32 weakly embeddable into N′ is itself weakly embeddable into N′. The following 33 condensation lemma holds for premice weakly embeddable into L[E]-levels, see [18] 34 for its proof. The lemma can actually be proved for premice which satisfy sufficient 35 iterability hypothesis, however in this paper we try to avoid any direct reference to 36 iterability.

37 Lemma 3.2 (Condensation lemma). Assume N is a premouse weakly embeddable 38 into an L[E]-level. Let N̄ be a premouse of the same type as N and σ : N̄ → N be 39 a Σ_{0}^{(n)}-preserving embedding such that σ ↾ ωϱn+1 ; N̄ = id. Then N̄ is solid and pN̄ is 40 k-universal for every k ∈ ω (briefly universal). Furthermore, if N̄ is sound above 41 ν = cr(σ) then one of the following holds: 42 (a) N̄ = core_{ν}(N) and σ is the associated core map. So if N is sound above ν 43 then N̄ = N and σ = id. 44 (b) N̄ is a proper initial segment of N.

1 (c) N̄ = Ult^{∗}(N || η, E_{α}^{N}) where α ≤ ωη and η < ht(N) is largest such that ν 2 is a cardinal in N || η; moreover, κ = cr(E_{α}^{N}) is a cardinal predecessor of ν 3 in N || η and a single generator of E_{α}^{N}. 4 (d) N̄ is a proper initial segment of Ult(N, E_{ν}^{N}).

5 It follows immediately from the above definition that any premouse weakly em-6 beddable into an L[E]-level is solid and its standard parameter is universal. 7 We now introduce some notation that will help us to economize the text. Given 8 an acceptable J-structure M and an ordinal τ ∈ M, we let n^{∗}(τ, M) be the largest 9 n ∈ ω +1 such that ωϱ_{M}^{n} > τ. Given a singular ordinal τ, we say that an acceptable 10 structure M is a singularizing structure for τ just in case that τ is regular in 11 M and there is some n ∈ ω and a good Σ_{1}^{(n)}(M) map that partially maps some 12 ordinal δ < τ cofinally into τ. We denote the least such n by n(τ, M). Obviously 13 n^{∗}(τ, M) ≤ n(τ, M). We will often say briefly that M singularizes τ. If there is a 14 singularizing partial map for τ that is Σ_{1}^{(m)}(M) in a parameter q then obviously 15 τ ∩ h̃_{M}^{m+1}(δ ∪ {q}) is cofinal in τ for a suitably chosen δ < τ; here h̃_{M}^{m+1} is the 16 canonical Σ_{1}^{(m)}-Skolem function for M. If R_{M}^{n^{∗}+1} ≠ ∅ then of course q can be 17 chosen from R_{M}^{n^{∗}+1}. In particular, if M is sound then we can let q = p_{M}. 18 It may of course happen that n^{∗}(τ, M) < n(τ, M), i.e. that n(τ, M) is not neces-19 sarily the least n such that ωϱ_{M}^{n+1} ≤ τ. This leads to serious difficulties in attempts 20 to translate – in a sufficiently uniform way – fine structural information between 21 premice N and protomice (see below) associated with N, an issue which does not 22 occur in constructions of □_{κ}-sequences in [14] where the focus is on singularizing 23 premice for local successors. Fortunately, premice N with n^{∗}(τ, N) < n(τ, N) can 24 be avoided in the analysis of extender fragments that arise in the construction of 25 the canonical global □ sequence.

26 Definition 3.3. Let M be a singularizing structure for τ. We say that M is exact 27 for τ just in case that n^{∗}(τ, M) = n(τ, M).

28 Since the definition of exactness is restricted to singularizing structures, when-29 ever we say that M is exact for τ, we implicitly require that M is a singularizing 30 structure for τ. Notice that if n(τ, M) = 0 then M is automatically exact. This 31 simple fact will be crucial in dealing with the issue sketched above. It implies that 32 among all protomice, only those associated with exact levels of the extender model 33 are relevant for the construction of a global □ sequence. 34 Recall that forming ultrapowers of premice may give rise to coherent structures 35 whose top predicates are extender fragments. This is typical in interpolation argu-36 ments if the target premouse is so-called pluripotent. For our present purposes, we 37 will use the following definition of pluripotency.

38 Definition 3.4. Let N be a premouse and τ ∈ N be an ordinal. We say that 39 the pair (N, τ) is pluripotent just in case that N is active with cr(E_{top}^{N}) < τ and 40 n^{∗}(τ, N) = 0. If τ is clear from the context, we will briefly say that N is pluripotent.

41 We now introduce the notion of a divisor; again, we just adapt the notion from 42 [14] to the current context. Recall that if N is active then h_{N}^{∗} is the Σ_{1}-Skolem 43 function for N computed in the language for coherent structures. Obviously h_{N}^{∗} = 44 h̃_{N}^{1} whenever N is a premouse of type A or C, but the two Skolem functions may 45 differ for type B premice. For α ≥ max{ωϱ_{N}^{1}, cr(E_{top}^{N})^{+N}} we let d_{N}^{α} be the <^{∗}- 46 least finite set of ordinals d with h_{N}^{∗}(α ∪ {d}) = N if such a d exists; here <^{∗} is 1 the canonical well-ordering of finite sets of ordinals. If d_{N}^{α} exists, we say that N is 2 Dodd sound above α. It is proved in [19] that if d_{N}^{α} exists then also d_{N}^{β} exists for 3 all β ≥ α and d_{N}^{β} = d_{N}^{α} − β. We let d_{N} = d_{N}^{θ} where θ = max{ωϱ_{N}^{1}, cr(E_{top}^{N})^{+N}}. 4 This notation slightly diverges from that in [19], but the two notations agree if 5 cr(E_{top}^{N}) < ωϱ_{N}^{1} which will be the case in our applications. In general, d_{N} need 6 not exist, but it is proved in [19] that d_{N} exists whenever N is a sound premouse 7 weakly embeddable into a level of L[E].

8 Definition 3.5. Let N be a sound premouse weakly embeddable into an L[E]-level, 9 τ ∈ N be a limit cardinal in N and n = n^{∗}(τ, N) ∈ ω. Granting that (N, τ) is not 10 pluripotent, a pair (μ, q) is a divisor for (N, τ) if and only if there is an ordinal λ 11 such that, setting r = p_{N} − q − τ and Y = h̃_{N}^{n+1}(μ ∪ {r}), the following holds: 12 (a) μ < τ ≤ λ < ωϱ_{N}^{n}; 13 (b) q = (p_{N} ∩ λ) − τ; 14 (c) Y ∩ ωϱ^{n} is cofinal in ωϱ_{N}^{n}; 15 (d) λ = min (On ∩ Y − μ). 16 Granting that (N, τ) is pluripotent and cr(E_{top}^{N}) < ωϱ_{N}^{1}, a pair (μ, q) is a divisor 17 for (N, τ) if and only if there is an ordinal λ such that, setting r = d_{N} − q − τ and 18 Y = h_{N}^{∗}(μ ∪ {r}), clauses (a) – (d) above hold with the current λ, r, Y and with d_{N} 19 in place of p_{N} in (b).

20 Throughout the paper, if we say that (μ, q) is a divisor for (N, τ) we implicitly 21 assume that N is sound premouse weakly embeddable into a level of L[E] and τ is 22 a limit cardinal in N. Given a pair (N, τ) and a divisor (μ, q) for (N, τ), the hull 23 Y from the above definition collapses to a premouse N^{∗} = N^{∗}(μ, q), giving rise to 24 the Σ_{1}^{(n)}-preserving map π = π_{N}(μ, q) : N^{∗} → N and the extender fragment

(1)  F = F_{N}(μ, q) =^{def} π ↾ (𝒫(μ) ∩ N^{∗}).

25 The notation is analogous to that in [14] in that we expose the pair N and the 26 divisor (μ, q) with any object whose dependency on N and (μ, q) we want to stress. 27 For instance, we write r_{N}(μ, q) for r and λ_{N}(μ, q) for λ. We can drop τ here since 28 once we know that (μ, q) is a divisor for (N, τ) then λ, r, Y, F and all other objects of 29 interest do not depend on τ. Recall also that, by definition, ν_{N}(μ, q) = λ_{N}(μ, q)^{+N} 30 and ϑ_{N}(μ, q) = μ^{+N^{∗}}. 31 The proofs of Lemmata 2.1 – 2.3 in [14] go through with the present definition of 32 a divisor, so N^{∗} is a proper initial segment of N (and thus a proper level of N || μ^{+}), 33 N = Ult^{∗}(N^{∗}, F), and π is the associated fine ultrapower map. The structure

N(μ, q) =^{def}⟨J_{ν}^{E^{N}} , F ⟩

34 is the protomouse associated with (N, τ) and the divisor (μ, q). 35 If (N, τ) is pluripotent, the above definition of a divisor makes use of the Dodd 36 parameter and the Σ_{1}-Skolem function h_{N}^{∗} in place of p_{N} and h̃_{N}^{n+1}. This may 37 look unnatural at first glance, since the definition using the language of premice 38 makes sense also for pluripotent pairs (N, τ). The reason why we use the language 39 of coherent structures here is that if N is a type B premouse then the definition 40 using the language of premice does not cover all situations we encounter in our 41 main construction. In particular, it does not cover the case where λ_{N}(μ, q) = λ_{N}. 42 Notice also that this case is peculiar since it can happen that N and N^{∗}(μ, d_{N}) are 43 premice of different types, an issue that does not occur in the remaining case where 1 λ_{N}(μ, q) < λ_{N} or, equivalently, μ < λ_{N^{∗}}. If μ < λ_{N^{∗}}, it is irrelevant which language 2 we choose for the definition of a divisor, because the proofs of Lemmata 2.4 and 2.5 3 in [14] go through in the current setting. The fact that the choice of language does 4 not affect the definition of a divisor if μ < λ_{N^{∗}} will be essential in the proof of 5 coherency of the square sequences we construct in the next section. 6 We now state a fact concerning the relationship between the Dodd parameter 7 and standard parameter that will be useful in our main construction and which 8 gives a rigorous background to the remarks from the previous paragraph. If (N, τ) 9 is pluripotent where N is a sound mouse embeddable into a level of L[E] with 10 cr(E_{top}^{N}) < ωϱ_{N}^{1} then d_{N} = p_{N}^{1} ∪ e_{N} where, letting λ_{N}^{∗} be the largest cutpoint of 11 E_{top}^{N} and γ_{N} be the index of E_{top}^{N} | λ_{N}^{∗}, the parameter e_{N} is the <^{∗}-least finite set 12 of ordinals e such that γ_{N} ∈ h_{N}^{∗}(ωϱ_{N}^{1} ∪ {p_{N}^{1} ∪ e}), see [19]. From the above remarks 13 we get the following lemma, whose proof can be extracted from the the proofs of 14 Lemmata 2.4 and 2.5 in [14].

15 Lemma 3.6. If (N, τ) is pluripotent where N is a sound premouse embeddable into 16 an L[E]-level and (μ, q) is a divisor for (N, τ) with λ_{N^{∗}(μ,q)} > μ then 17 • e_{N} ∈ h_{N}^{∗}(μ ∪ {p_{N} − q − τ, γ_{N}}), 18 • γ_{N} ∈ h_{N}^{∗}(μ ∪ {d_{N} − q − τ)}) and 19 • (p_{N}^{1} ∩ λ) − τ = q = (d_{N} ∩ λ) − τ. 20 The same is also true of pairs (μ, q) such that μ = τ and (a) – (d) in Definition 3.5 21 are met with λ > τ. (Such pairs will not be considered divisors in the rest of the 22 text, but we will need to deal with them occasionally.)

23 We will consider coherent structures whose initial segments are premice. Given 24 such a coherent structure M = ⟨J_{ν}^{E^{M}} , F ⟩ we let μ_{M} = cr(F) and λ_{M} = λ(F), so F 25 is an extender at (μ_{M}, λ_{M}). We also let ϑ_{M} be the largest ordinal ϑ ≤ μ_{M}^{+} such that 26 F measures all sets in 𝒫(μ_{M})∩J_{ϑ}^{E^{M}} and μ_{M} is the largest cardinal in J_{ϑ}^{E^{M}}. Finally 27 N^{∗}(M) is the collapsing level of J_{ν}^{E} for ϑ_{M} if ϑ_{M} < μ_{M}^{+M}. Granting that N^{∗}(M) 28 is defined, we let N(M) = Ult^{∗}(N^{∗}(M), F) and π_{M} be the associated ultrapower 29 embedding. In general, it is not clear that this ultrapower is well-founded, but if it 30 is, we consider N(M) to be transitive. We say that M is a potential protomouse 31 if ϑ_{M} < μ_{M}^{+} and N(M) is transitive. M is a protomouse just in case that M 32 is a potential protomouse and N(M) is a premouse weakly embeddable into an 33 L[E]-level. We do not require that N(M) is sound, but it is straightforward to 34 verify that N(M) is automatically sound above λ_{M}. If μ_{M} < τ ≤ λ_{M} then, letting 35 r_{M} = π_{M}(p_{N^{∗}(M)}), n = n^{∗}(μ_{M}, N^{∗}(M)) and Y_{M} = h̃_{N(M)}^{n+1}(μ_{M} ∪{r_{M}}), clauses (a) 36 – (d) in Definition 3.5 hold with λ_{M}, μ_{M}, r_{M}, Y_{M} and q_{M} = (p_{N(M)} ∩ λ_{M}) − τ = 37 p_{N(M)} − r_{M} − τ in place of λ, μ, r, Y and q. For sound N(M) this means that 38 M = N(M)(μ_{M}, q). The requirement on the soundness of N(M) is in a sense 39 superfluous here since we could easily generalize the definition of a divisor to premice 40 that are not sound. This would however, cause a non-uniformity in the definition of 41 a divisor, which relies on the notion of Dodd parameter, so in order to generalize the 42 definition of a divisor we would first need to generalize the notion of Dodd parameter 43 appropriately and develop some of its properties. Since this will not be needed in our 44 application and would make the text unnecessary complicated we prefer to restrict 45 the definition of a divisor for sound premice. In the construction of the canonical 46 global □ sequence we will often work with singularizing protomice for ordinals τ. 47 Only protomice M with n(τ, M) = 0 will be relevant for our construction; this 1 situation is parallel to that in the construction of □_{κ} in [14]. By the remarks made 2 above, such protomice are automatically exact. 3 The method of translation between the definability over protomice and the de-4 finability over their associated premice is analogous to that discussed in Section 2.2 5 of [14]. The following is our basic conversion lemma.

6 Lemma 3.7. Let M be a potential protomouse and N = N(M). Let further 7 n = n^{∗}(μ_{M}, N^{∗}(M)), p ∈ R_{N^{∗}(M)}^{n+1}, r = π_{M}(p) and q ⊂ λ_{M} be finite.

8 (a) If A is Σ_{1}^{(n)}(N) in r ∪ q then there is some A^{∗} that is Σ_{1}(M) in q and ϑ_{M} 9 satisfying A ∩ λ = A^{∗} ∩ λ. 10 (b) If A is Σ_{1}(M) in c ∈ M then there is some ^{∗}A that is Σ_{1}^{(n)}(N) in c, r and 11 ϑ_{M} satisfying A ∩ λ = ^{∗}A ∩ λ.

12 For N we use the language for premice unless N^{∗}(M) is active with λ_{N^{∗}(M)} = μ, 13 in which case we use the language for coherent structures. For M we always use 14 the language for coherent structures.

15 The proof of the above lemma can easily be extracted from the proofs of Lem-16 mata and Corollaries 2.7 – 2.12 of [14]. This lemma, as well as the conversion 17 lemmata stated below, can be formulated in a broader context, not just for poten-18 tial protomice. The current formulations, however, suffice for our later applications. 19 Parallel to Lemmata 2.13 and 2.14 we also have the following conversion concerning 20 definable singletons. 21 We first give some details concerning the abuse of notation we will use to simplify 22 the notation. Let M be a coherent structure with top extender F and μ = cr(F). 23 We write F (f) for π_{M}(f) where f : μ → μ or f : μ → 𝒫(μ). In either case f can be 24 viewed as a subset of μ × μ, so coding by Gödel tuples ⟨η, η′⟩ ↦ ≺η, η′≻ makes it 25 possible to replace f by its natural code a_{f} ⊆ μ. Since obviously a_{π_{M} (f)} = F (a_{f}), 26 it is harmless to trade the correct notation “F (a_{f})” for somewhat sloppy, but more 27 intuitive “F (f)”. Since coding by Gödel tuples allows us to code finite sets of 28 ordinals by ordinals, we usually write f(x), resp. F (f)(x) where f is as above 29 and x is a finite set of ordinals. The correct way of writing this would be f(η_{x}), 30 resp. F (f)(η_{x}) where η_{x} = ≺η_{0}, ⋯, η_{ℓ−1}≻ and ⟨η_{0}, ⋯, η_{ℓ−1}⟩ is the descending 31 enumeration of x. Finally we write f(x, η), resp. F (f)(x, η) for f(≺η_{x}, η≻), resp. 32 F (f)(≺η_{x}, η≻).

33 Lemma 3.8. Let M, N, n and r be as in the previous lemma. Let further μ = μ_{M}, 34 q ⊂ λ_{M} be finite, ζ < λ_{M} and y ⊆ λ_{M}.

35 (a) If ζ (resp. y) is Σ_{1}^{(n)}(N)-definable from r and q as a singleton then there 36 is some f : μ → μ (resp. f : μ → 𝒫(μ)) in N^{∗}(M) such that ζ = F (f)(q) 37 (resp. y = F (f)(q)). 38 (b) If ζ = F (f)(q) for some f : μ → μ in N^{∗}(M) (resp. y = F (f)(q)) for some 39 f : μ → 𝒫(μ) in N^{∗}(M) then ζ (resp. y) is Σ_{1}^{(n)}(N) definable from r, q and 40 some ξ < μ as a singleton.

41 The languages are chosen the same way as in the previous lemma.

42 Lemma 3.9. Let M = ⟨J_{ν}^{E}, F ⟩ be a coherent structure such that F is a total 43 extender on M. Let μ = μ_{M}, q ⊂ λ_{M} be finite, ζ < λ_{M} and y ⊆ λ_{M}.

1 (a) If ζ (resp. y) is Σ_{1}(M)-definable from q as a singleton then there is some 2 f : μ → μ (resp. f : μ → 𝒫(μ)) in J_{ϑ_{M}}^{E^{M}} such that ζ = F (f)(q, μ) (resp. 3 y = F (f)(q, μ)). 4 (b) If ζ = F (f)(q) for some f : μ → μ in J_{ϑ_{M}}^{E^{M}} (resp. y = F (f)(q)) for some 5 f : μ → 𝒫(μ) in J_{ϑ_{M}}^{E^{M}}) then ζ (resp. y) is Σ_{1}(M) definable from q and f as 6 a singleton.

7 The previous two lemmata give us a tool for identifying divisors for premice over 8 their associated protomice; this tool corresponds to Lemma 2.15 of [14]. Notice 9 that the characterization of definability in Lemma 3.8 has always the same form 10 over M, no matter whether μ_{M} < λ_{N^{∗}(M)} or μ_{M} = λ_{N^{∗}(M)}. The same is true of 11 the characterization over N(μ, q) of divisors for N below. This obviously brings 12 many advantages in applications.

13 Lemma 3.10. Assume (μ, q) is a divisor for (N, τ), n = n^{∗}(τ, N) ∈ ω, and F = 14 F_{N}(μ, q). Let μ ≤ μ′ < τ and q′ be a bottom part of q, that is, q′ = q ∩ λ′ for 15 some λ′ ≤ max(q) + 1 where max(∅) = τ. Finally let r′ = q − q′. Then (μ′, q′) 16 is a divisor for (N, τ) if and only if the following condition is satisfied for every 17 f : μ → μ in N^{∗}(μ, q) and every ξ < μ′:

F (f)(r′, ξ) ≤ max(q′) ⟹ F (f)(r′, ξ) < μ′.

18 Recall that solidity witnesses (no matter whether generalized or standard) are 19 computed in the language for premice in the case of premice and in the language 20 for coherent structures in the case of protomice. In certain arguments we will 21 need to consider structures associated with coherent structures M that are defined 22 analogously as standard witnesses W_{M}^{α,q} with the only exception that α is smaller 23 than the ultimate projectum of M and q is allowed to contain ordinals smaller than 24 this projectum. Given an acceptable structure M, a number n ∈ ω, a finite set 25 q of ordinals in M and an ordinal α ∈ M, the structure W_{M}^{n,α,q} is the transitive 26 collapse of the hull h̃_{N}^{k+1}(α∪{q−(α+1)}) where k is such that ωϱ_{M}^{k+1} ≤ α < ωϱ_{M}^{n} if 27 ωϱ_{M}^{n+1} ≤ α and k = n if α < ωϱ_{M}^{k+1}. We call this structure the standard n-witness 28 for α with respect to M and q. If (N, τ) is pluripotent, standard Dodd witnesses 29 are defined in the same manner, but in the language for coherent structures instead 30 of that for premice, and with n = 0. Standard Dodd witnesses for N are denoted 31 by ^{∗}W _{N}^{α,q}. If (N, τ) is pluripotent, we get the following consequence of Lemma 3.6.

(2)

If N and (μ, q) are as in Lemma 3.6, μ ≤ α < min(d_{N} − q) and p
is a parameter such that d_{N} − q ⊆ p then ^{∗}W _{N}^{α,p} = W_{N}^{0,α,p}.

32 The next lemma summarizes conversions between fine structural characteristics 33 of N and its associated protomice. These conversions correspond to Lemma 2.16 34 and Corollary 2.17 in [14]. They easily follow from Lemmata 3.7 – 3.9.

35 Lemma 3.11. Let M be a potential protomouse and n = n^{∗}(μ_{M}, N^{∗}(M)). Then 36 (a) ωϱ_{N}^{n+1} = ωϱ_{M}^{1} 37 where N = N(M). Assume τ is an ordinal satisfying max{μ_{M}^{+M}, ωϱ_{M}^{1}} ≤ τ. Then 38 the following hold. 39 (b) N is a singularizing structure for τ with n(τ, N) = n just in case that M is 40 a singularizing structure for τ with n(τ, M) = 0. Consequently, if M and 41 N are singularizing structures for τ then N is exact for τ if and only if M 42 is exact for τ.

1 (c) p_{M}^{1} − τ = (p_{N}^{n} ∩ λ_{M}) − τ if N^{∗}(M) is either passive or active such that 2 λ_{N^{∗}(M)} > μ_{M}, and p_{M}^{1} −τ = d_{N} −τ if N^{∗}(M) is active with λ_{N^{∗}(M)} = μ_{M}. 3 (d) N is sound above τ if and only if M is sound above τ, granting that N^{∗}(M) 4 is either passive or active with λ_{N^{∗}(M)} > μ_{M}. N is Dodd sound above τ 5 if and only if M is sound above τ, granting that N^{∗}(M) is active with 6 λ_{N^{∗}(M)} = μ_{M}. 7 (e) Assume that ϑ_{M} ≤ α < λ_{M}, p ∈ R_{N^{∗}(M)}^{n+1}, r = π_{M}(p) and q ⊂ λ_{M} is finite. 8 If N^{∗}(M) is either passive or active with λ_{N^{∗}(M)} > μ_{M} then W_{N}^{n,α,r∪q} = 9 Ult^{∗}(N^{∗}(M), F_{M}^{0,α,q}) where F_{M}^{0,α,q} is the top extender of W_{M}^{0,α,q}. Letting 10 W = W_{N}^{0,α,q}, the associated ultrapower embedding π̄ : N^{∗}(M) → W is the 11 inverse to the Mostowski collapsing isomorphism that comes from collapsing 12 the hull h̃_{W}^{n+1}(μ_{M} ∪ {π̄(p}) and F_{M}^{0,α,q} is the extender derived from this 13 embedding. It follows that W_{N}^{n,α,r∪q} ∈ N if and only if W_{M}^{0,α,q} ∈ M. 14 If N^{∗}(M) is active with λ_{N^{∗}(M)} = μ_{M}, the same is true with standard 15 Dodd witnesses for N in place of standard 0-witnesses for N. 16 (f) N is solid above τ if and only if M is solid above τ, granting that N^{∗}(M) 17 is either passive or active with λ_{N^{∗}(M)} > μ_{M}. N is Dodd solid above 18 τ if and only if M is solid above τ, granting that N^{∗}(M) is active with 19 λ_{N^{∗}(M)} = μ_{M}.

20 Recall that the notion of soundness has a definition in Jensen’s fine structure 21 theory that does not depend on the notion of solidity and vice versa, so the way 22 clauses (d) and (f) in the above lemma are formulated does make sense. An ac-23 ceptable structure M is sound above τ just in case that h̃_{M}^{n}(ωϱ_{M}^{n} ∪ {p_{M}^{n} − τ}) = M 24 where n is such that ωϱ_{M}^{n+1} ≤ τ < ωϱ_{M}^{n}; M is solid above τ just in case that 25 W_{M}^{β,p_{M}^{n}} ∈ M for each β ∈ p_{M}^{n} − τ. If M is solid then p_{M}^{n} is a top segment of p_{M}, as 26 can be easily seen. 27 The following two lemmata correspond to Lemma 2.18 in [14]. They make use of 28 “generalized” versions of witnesses W_{M}^{n,α,q} which are defined in similarly as general-29 ized solidity witnesses. If M, α and q are as in the definition of a standard n-witness 30 above, say that a pair ⟨Q, t⟩ is a generalized n-witness for α with respect to M 31 and q just in case that Q is an acceptable structure, t is a finite set of ordinals 32 in Q with |t| = |q − (α + 1)|, and for every Σ_{1}^{(n)}-formula φ(v_{0}, v_{1}, …, v_{ℓ}) and any 33 ξ_{1}, …, ξ_{ℓ} < α we have

M ⊧ φ[q − (α + 1), ξ_{1}, …, ξ_{ℓ}] ⟹ Q ⊧ φ[t, ξ_{1}, …, ξ_{ℓ}].

34 For pluripotent premice, generalized Dodd witnesses are defined in the same man-35 ner, but in the language for coherent structures instead of that for premice, and 36 with n = 0. 37 Given an acceptable structure M, n-witnesses for M (no matter whether stan-38 dard or generalized) do not provide us with any information about solidity of M 39 below ωϱ_{M}^{n+1}. We introduced them for a different purpose — namely, for iden-40 tifying canonical divisors. The next two lemmata, as well as the notion of an 41 n-witness/generalized n-witness can be formulated in a more general setting, but 42 we restrict ourselves to a formulation that suffices for our applications. There is a 43 slight overlap in the conclusions of the lemmata, which might be seen as a deficiency 44 in the way the paper is organized. The reason we chose these formulations is that 45 applications of the lemmata in the arguments to come (especially in the proof of 1 Lemma 4.9) become more natural. The proofs of both Lemma 3.12and Lemma 3.13 2 follow from Lemma 3.11 in a straightforward manner.

3 Lemma 3.12. Let M be a potential protomouse, n = n^{∗}(τ, N) and N = N(M). 4 Let further p ∈ R_{M}^{n+1}, r = π_{M}(p) and q ⊂ λ_{M} be finite. Finally let ϑ_{M} ≤ α < λ_{M}. 5 Assume N^{∗}(M) is either passive or active with λ_{N^{∗}(M)} > μ, and ⟨Q, t⟩ is a 6 generalized n-witness for α with respect to N and r ∪ q. If H is any transitive 7 admissible structure such that ⟨Q, t⟩ ∈ H then W_{N}^{n,α,r∪q} ∈ H and is Σ_{0}-definable 8 in H from ⟨Q, t⟩ and α. It follows that also W_{M}^{0,α,q} ∈ H and is Σ_{0}-definable in H 9 from the parameters ⟨Q, t⟩, α and ϑ_{M}. 10 Assume N^{∗}(M) is active with λ_{N^{∗}(M)} = μ and ⟨Q, t⟩ is a generalized Dodd 11 witness for α with respect to N and r ∪ q. If H is a transitive admissible structure 12 such that ⟨Q, t⟩ ∈ H then also ^{∗}W _{N}^{α,r∪q} is an element of H and is Σ_{0}-definable in H 13 from ⟨Q, t⟩ and α. It follows that W_{M}^{0,α,q} is and element of H and is Σ_{0}-definable 14 in H from ⟨Q, t⟩, α and ϑ_{M}. 15 In either case, the definition of the standard witness is uniform, that is, there 16 is a single formula φ that serves as a definition of the standard witness from the 17 parameters named above. This formula depends neither on these parameters nor 18 on H.

19 Lemma 3.13. Let M be a coherent structure, ϑ_{M} ≤ α < λ_{M} and q ⊂ λ_{M} be finite. 20 Assume M is a potential protomouse and ⟨Q, t⟩ is a generalized 0-witness for α 21 with respect to M and q. If H is a transitive admissible structure and ⟨Q, t⟩ ∈ H 22 then also the standard 0-witness W_{M}^{0,α,q} is an element of H and is Σ_{0}-definable in 23 H from ⟨Q, t⟩, α and ϑ_{M}. 24 Assume is a premouse and ⟨Q, t⟩ is a generalized Dodd witness for α with respect 25 to M and q. If H is a transitive admissible structure and ⟨Q, t⟩ ∈ H then also the 26 standard Dodd witness ^{∗}W _{M}^{α,q} is an element of H and is Σ_{0}-definable in H from the 27 parameters ⟨Q, t⟩, α and ϑ_{M}.

28 We will make use of the following condensation lemma for protomice, which 29 corresponds to the conjunction of Lemmata 2.19 and 2.20 in [14]. The proof of this 30 lemma follows the same strategy as the proofs of the above named lemmata in [14]. 31 There are however differences in the verification of the initial segment condition for 32 certain extenders which arise in the argument, as well as in the application of the 33 condensation lemma; for this reason we give a sketch of the proof that focuses on 34 these new issues.

35 Lemma 3.14. Let M be either a protomouse or an active premouse weakly em-36 beddable into an L[E]-level and let τ be inaccessible in M such that μ_{M} < τ ≤ λ_{M}. 37 Suppose further that M̄ is a coherent structure, τ̄ ∈ M̄ is an ordinal, σ : M̄ → M 38 and the following requirements are met:

39 (a) M̄ is sound and solid with ωϱ1 ; M̄ ≤ τ̄. 40 (b) σ is Σ_{0}-preserving and non-cofinal. 41 (c) cr(σ) = τ̄ and σ(τ̄) = τ.

42 Letting N̄ = N(M̄) and μ = μ_{M} = μM̄, the structure N̄ is a proper level of M, the 43 pair (μ, pM̄) is a divisor for N̄ and M̄ = N̄(μ, pM̄). 44 If M̄ is a singularizing structure for τ̄ with n(τ̄, M̄) = 0 then N̄ is the level of 45 M singularizing τ̄, and is exact for τ̄.

1 Proof. Obviously τ̄ is an inaccessible cardinal in the sense of M̄. The non-cofinality 2 of σ yields ϑM̄ < ϑ_{M}. Since τ̄ is a limit cardinal in M̄, ϑM̄ < μ^{+M̄}, so N^{∗}(M̄) 3 is defined and, being a proper level of M || ϑ_{M}, is in the domain of π_{M}. Then 4 N′ = π_{M}(N^{∗}(M̄)) is a proper level of M; see the discussion of protomice above. 5 Letting F̄ be the top extender of M̄ and N̄ = N(M̄) (this ultrapower is well-6 founded by standard considerations), the canonical embedding σ̃ : N̄ → N′ defined 7 by πM̄(f)(α) ↦ π_{M}(f)(σ(α)) extends σ and is Σ_{0}^{(n)}-preserving where n is such 8 that ωϱ_{N^{∗}(M̄)}^{n+1} ≤ μ < ωϱ_{N^{∗}(M̄)}^{n}. The structure N̄ is a potential premouse; the 9 weak amenability of its top predicate (granting that N̄ is active) follows from the 10 fact that the ultrapower embedding πM̄ : N^{∗}(M̄) → N̄ is the identity on the 11 power set of cr(E_{top}^{N^{∗}(M)}) whenever cr(E_{top}^{N^{∗}(M̄)}) < μ. To see that N̄ is a premouse 12 we need to check the initial segment condition, which can be done by standard 13 considerations unless λ_{N^{∗}(M̄)} = μ. In order to discuss this case, we need some 14 additional information about N̄. Requirement (a) of the current lemma together 15 with Lemma 3.11(a,c,d,f) yield ωϱn+1 ; N̄ ≤ τ̄ < ωϱn ; N̄ and N̄ is sound and solid above 16 τ̄, unless λ_{N^{∗}(M̄)} = μ in which case N̄ is Dodd sound and Dodd solid above τ̄. This

17 is true regardless of the behavior of E_{top}^{N^{∗}(M̄)}. 18 We now complete the verification of the initial segment condition for N̄. Recall 19 that we focus on the case where λ_{N^{∗}(M̄)} = μ. Recall also that pM̄ − τ̄ = dN̄ − τ̄ in 20 this case, again by Lemma 3.11(c). If dN̄ −τ̄ ≠ ∅ then the initial segment condition 21 follows from the fact that the standard Dodd solidity witness W for max(dN̄) is an 22 element of N̄; this is true because either all cutpoints of EN̄ ; top are also cutpoints of 23 E_{top}^{W} or else λ(E_{top}^{W}) is the largest cutpoint of EN̄ ; top. If dN̄ − τ̄ = ∅, we first observe 24 that all cutpoints of EN̄ ; top are strictly smaller than τ̄. This is trivial if λ(EN̄ ; top) = τ̄. If 25 λ(EN̄ ; top) > τ̄, this is a consequence of the soundness of N̄ above τ̄, which guarantees 26 that every ordinal in N̄ is of the form EN̄ ; top(f)(δ) for some f : cr(EN̄ ; top) → cr(EN̄ ; top) 27 and some δ < τ̄. In this case the cutpoints of EN̄ ; top are actually bounded below 28 τ̄; otherwise τ̄ would also be a cutpoint. In either of the above cases, the initial 29 segment condition for N̄ then follows from the initial segment condition for N′ and 30 from the fact that σ̃ ↾ τ̄ = id ↾ τ̄. 31 We would like to apply the condensation lemma to the map σ̃ : N̄ → N′. The 32 discussion in the previous paragraphs guarantees that the assumptions of the con-33 densation lemma are met unless N^{∗}(M̄) is active with λ_{N^{∗}(M̄)} = μ. This remaining 34 case requires two additional steps. First, it might happen that N′ a type C pre-35 mouse (this is the case if N^{∗}(M̄) is of type C), whereas we know that N̄ is of 36 type B whenever λ(EN̄ ; top) > τ̄. The issue can be resolved the same way as in [14] by 37 replacing N′ by N′ || ν′ where, letting λ′ = sup(σ̃′′λ(N̄)), ν′ is the index of E_{top}^{N′} | λ′. 38 From now on assume that N′ is the result of such a replacement; then σ̃ is a Σ_{0}- 39 preserving embedding between two type B premice. The second issue to clear up is 40 the verification of the soundness of N̄ above τ̄. Here we employ Theorem 1.2 from 41 [19]. Although that theorem is formulated for weakly iterable premice, no weak 42 iterability of N̄ is needed for the proof that N̄ is sound; what suffices is the Dodd 43 soundness and Dodd solidity of N̄ above τ̄ which we verified above. 44 The application of the condensation lemma now reduces to ruling out the options 45 (c) and (d). But this is immediate, as both would require that τ̄ is a successor 46 cardinal in N̄. It follows that N̄ is a level of N′, and therefore a proper level of M.

1 In particular, N̄ is sound and solid. By construction, (μ, pM̄ − τ̄) is a divisor for N̄ 2 and M̄ = N̄(μ, pM̄ − τ̄). Regarding the last conclusion of the current lemma, if M̄ 3 is a singularizing structure for τ̄, requirement (a) guarantees the existence of some 4 δ < τ̄ and some partial function mapping δ cofinally into τ̄ that is Σ_{1}(M̄)-definable 5 in the parameter pM̄. Applying Lemma 3.7, this partial function (or at least its 6 part below τ̄) is Σ_{1}^{(n)}(N̄)-definable in the parameters pM̄, πM̄(p_{N^{∗}(M̄)}) and ϑM̄; 7 here n = n^{∗}(μ, N(M̄)). Since τ̄ is obviously regular in N̄, the premouse N̄ is a 8 singularizing structure for τ̄ and n = n(τ̄, N̄), which proves that N̄ is exact for τ̄. 9 The remaining parts of the lemma are then clear. ⊓ ⊔(Lemma 3.14)

10 Finally we summarize the relevant facts about strong divisors reformulated for 11 the present purposes. We will be only considering strong divisors for pairs (N, τ) 12 where N is an initial level of L[E] and τ is a limit cardinal in L[E], as this somewhat 13 simplifies the matters and is sufficient for our application. In particular, if (μ, q) 14 is a divisor for such a pair (N, τ) then q is a bottom part of the standard/Dodd 15 parameter for N, as τ is the ultimate projectum of N. Hence subtracting τ from the 16 standard/Dodd parameter becomes superfluous; for instance (b) in Definition 3.5 17 reads q = p_{N} − λ in the present situation. Recall the notion of a strong divisor 18 first.

19 Definition 3.15. Let N be an L[E]-level, τ ∈ N be a limit cardinal in L[E], 20 n = n^{∗}(τ, N) ∈ ω and (μ, q) be a divisor for (N, τ). We say that (μ, q) is strong 21 just in case that one of the following holds. 22 (a) N is either passive or active with λ_{N^{∗}(μ,q)} > μ, and

𝒫(μ) ∩ N′(μ) = 𝒫(μ) ∩ N^{∗}(μ, q)

23 where N′(μ) is the transitive collapse of h̃_{N}^{n+1}(μ ∪ {p_{N}}). 24 (b) N is active with λ_{N^{∗}(μ,q)} = μ, and 𝒫(μ) ∩ N′(μ) = 𝒫(μ) ∩ N^{∗}(μ, q) where 25 N′(μ) is the transitive collapse of h_{N}^{∗}(μ ∪ {d_{N}}).

26 Of course, in the latter case we have q = d_{N} and N^{∗}(μ, q) = N || ν where ν is 27 the index of the extender E_{top}^{N} | μ. The following characterization, corresponding to 28 [14], Lemma 2.22 and Lemma 2.23 is essential for further analysis of strong divisors, 29 and is also directly used in the main construction.

30 Lemma 3.16. Let N be an L[E]-level with n^{∗}(τ, N) ∈ ω, τ ∈ N be a limit cardinal 31 in L[E] and (μ, q) be a divisor for (N, τ). If N is either passive or active with 32 λ_{N^{∗}(μ,q)} > μ then the following are equivalent. 33 (a) (μ, q) is strong. 34 (b) N^{∗}(μ, q) is the core of N′(μ). 35 (c) | p_{N^{∗}(μ,q)} | = | p_{N′(μ)} | = r_{N}(μ, q). 36 If N is active and λ_{N^{∗}(μ,q)} = μ then the following are equivalent. 37 (e) (μ, q) is strong. 38 (f) E_{top}^{N′(μ)} ∉ N′(μ).

39 The next general fact about the structure N′(μ) is simple, but very useful in 40 applications.

41 Lemma 3.17. Let N be an L[E]-level with n^{∗}(τ, N) ∈ ω, τ ∈ N be a limit cardinal 42 in L[E] and (μ, q) be a divisor for (N, τ). Let π′ : N′(μ) → N be the inverse to the 43 collapsing map. Then π′^{−1}(q) is a top segment of p_{N′(μ)}.

1 The following notion of closeness will be crucial in identifying strong divisors.

2 Definition 3.18. Let M = ⟨J_{ν}^{E}, F ⟩ be a coherent structure. An ordinal ϑ ≤ ϑ_{M} 3 is closed in M relative to q ∈ [λ_{M}]^{<ω} just in case that F (f)(q, ξ) ∩ μ_{M} ∈ J_{ϑ}^{E} for 4 every f : μ_{M} → 𝒫(μ_{M}) in J_{ϑ}^{E} and every ξ < μ_{M}.

5 We obtain a characterization of strong divisors that corresponds to Lemma 2.24 6 in [14]. As in the case of Lemma 3.10, the advantage of this characterization is that 7 it does not depend on the relationship between λ_{N^{∗}}(μ, q) and μ.

8 Lemma 3.19. Let N be an L[E]-level, τ ∈ N be a limit cardinal in L[E] and (μ, q) 9 be a divisor for (N, τ). Then (μ, q) is strong just in case that ϑ_{N(μ,q)} is closed in 10 N(μ, q) relative to q = p_{N(μ,q)}.

11 The two lemmata that provide us with the key for the choice of a canonical 12 divisor correspond to Lemma 2.25 and 2.26 in [14]. Given an L[E]-level N, a limit 13 cardinal τ ∈ N in L[E] and a bottom part q of the standard/Dodd parameter of 14 N, we let 𝒟_{q}(N, τ) be the set of all μ < τ such that (μ, q) is a divisor for (N, τ), 15 and 𝒟_{q}^{∗}(N, τ) be the set of μ < τ such that (μ, q) is a strong divisor for (N, τ).

16 Lemma 3.20. Let N be an L[E]-level, τ ∈ N be a limit cardinal in L[E] and q be 17 a bottom part of the standard/Dodd parameter of N. Then the following holds. 18 (a) 𝒟_{q}^{∗}(N, τ) is closed in τ. 19 (b) If τ is inaccessible in N and N is exact for τ then 𝒟_{q}^{∗}(N, τ) is bounded 20 in τ.

21 Proof. (Sketch.) The proof that 𝒟_{q}^{∗}(N, τ) is closed in τ is the same as that of 22 Lemma 2.25 in [14]. The heart of the proof is showing that if μ is a limit point of 23 𝒟_{q}^{∗}(N, τ) then |p_{N′(μ)}| = |p_{N^{∗}(μ,q)}| whenever N^{∗}(μ, q) is either passive or active with 24 λ_{N^{∗}(μ,q)} > μ, and E_{top}^{N} | μ ∉ N′(μ) whenever N^{∗}(μ, q) is active with λ_{N^{∗}(μ,q)} = μ. 25 The same proof goes through even if we allow μ = τ, which we excluded in the 26 definition of a divisor. Since obviously N′(τ) = N, τ cannot be a limit point of 27 𝒟_{q}^{∗}(N, τ) if q ≠ ∅, which guarantees that 𝒟_{q}^{∗}(N, τ) is bounded in τ in this case. 28 If q = ∅ then 𝒟_{q}(N, τ) is bounded in τ, as N is a singularizing structure for τ 29 which is exact for τ. It follows that 𝒟_{q}^{∗}(N, τ) ⊆ 𝒟_{q}(N, τ) is bounded in τ as well. 30 ⊓ ⊔(Lemma 3.20)

31 Lemma 3.21. Let N be the singularizing L[E]-level for τ where τ is inaccessible 32 in N and let (μ, q) be a strong divisor for N. If q′ is a proper bottom part of q then 33 there is no μ′ ≤ μ such that (μ′, q′) is a divisor for N.

34 The canonical divisor (μ(N, τ), q(N, τ)) is chosen as follows.

q(N, τ) ≃ the unique q such that 𝒟_{q}^{∗}(N, τ) ≠ ∅
μ(N, τ) ≃ max(𝒟_{q}^{∗}(N, τ)).

35 The uniqueness of q is a direct consequence of Lemma 3.21. 36 If N is pluripotent and does not admit any strong divisors, we let

q(N, τ) = d_{N}
μ(N, τ) = cr(E_{top}^{N}).

37 In this particular case we will consider (μ(N, τ), q(N, τ)) a strong divisor for (N, τ) 38 and let N(μ(N, τ), q(N, τ)) = N. Even though (μ(N, τ), q(N, τ)) is not a divisor 1 in the usual sense, the pair (μ(N, τ), q(N, τ)) satisfies the criterion formulated in 2 Lemma 3.19, namely that ϑ_{N}(μ(N, τ), q(N, τ)) = ϑ_{N}, being the cardinal successor 3 of μ(N, τ) in N, is closed in N relative to q(N, τ), and in fact relative to any finite 4 q ⊂ λ(E_{top}^{N}).

5 4. Global square sequence

6 In this section we give a proof of Theorem 1.2. Recall that 𝒮 is the class of all 7 singular cardinals of the extender model L[E]. Given a singular cardinal τ, let N_{τ} 8 denote the level of L[E] which singularizes τ. That is, N_{τ} is of the form L[E] || α 9 for the unique α such that τ is regular in L[E] || α but singular in L[E] ||(α+1). We 10 also allow the situation where τ ∉ N_{τ}, in which case τ = α. Since τ is a cardinal, 11 the ultimate projectum of N_{τ} is τ. We say that N_{τ} is pluripotent just in case the 12 pair (τ, N_{τ}) is pluripotent. We fix the following notation. 13 • n_{τ} = n(τ, N_{τ}) and n_{τ}^{∗} = n^{∗}(τ, N_{τ}). 14 • p_{τ} = p_{N_{τ}}. 15 • If N_{τ} is pluripotent then d_{τ} = d_{N_{τ}} and e_{τ} = e_{N_{τ}} (see Lemma 3.6). 16 • ϱ_{τ} = ϱ_{N_{τ}}^{n_{τ}} and ℋ_{τ} is the domain of J_{ρ_{τ}}^{E_{τ}^{N}}, that is ℋ_{τ} = S_{ωϱ_{τ}}^{E_{τ}^{N}}. 17 • h̃_{τ} = h̃_{N_{τ}}^{n_{τ} +1}. 18 • If N_{τ} is pluripotent then h_{τ}^{∗} = h_{N_{τ}}^{∗}. 19 Given two acceptable J-structures M̄, M for the same language, parameters p̄, p 20 in these structures and an ordinal τ̄ ∈ M̄ such that p̄∩τ̄ = ∅ and h̃n+1 ; M̄(τ̄∪{p̄}) = M̄, 21 there is at most one Σ_{0}^{(n)}-preserving map σ : M̄ → M satisfying σ ↾ τ̄ = id and 22 σ(p̄) = p. If such a map exists, it is unique and σ : h̃n+1 ; M̄(i, ⟨x, p̄⟩) ↦ h̃_{M}^{n+1}(i, ⟨x, p⟩) 23 for all i ∈ ω and x ∈ [τ̄]^{<ω}. Obviously, if M̄ = M and p̄ = p then σ = id; it will be 24 notationally convenient to consider also this trivial case. 25 • σ_{M̄,M,τ̄}^{n,p̄,p} : M̄ → M is the unique map σ just described above, if exists. 26 • σ_{τ̄,τ} = σ_{Nτ̄,N_{τ} ,τ̄}^{nτ̄,pτ̄,p_{τ}} whenever τ̄ ≤ τ are elements of 𝒮 and σ_{Nτ̄,N_{τ} ,τ̄}^{nτ̄,pτ̄,p_{τ}} exists. 27 When we write “σ_{τ̄,τ} ”, we implicitly assume that τ̄ ≤ τ. 28 Similarly as it is done in the construction of a □_{κ}-sequence in [14], we split 𝒮 29 into two disjoint classes 𝒮^{0} and 𝒮^{1}.

𝒮^{1} = {τ ∈ 𝒮 | N_{τ} is exact for τ and (q(N_{τ}), μ(N_{τ})) is defined.}
𝒮^{0} = 𝒮 − 𝒮^{1}.

30 We would like to stress at this point that for each τ ∈ 𝒮 there is a uniquely 31 determined singularizing structure which enables us to define a set C_{τ} that belongs 32 to the canonical global □^{𝒮}-sequence. This is one of the main differences between the 33 construction of the canonical □^{𝒮}-sequence and that of the canonical □_{κ}-sequence, 34 where we had to restrict ourselves on a club of ordinals that do not index extenders. 35 Hence □^{𝒮} always holds in extender models, although it should be mentioned that 36 □^{𝒮} might be quite a weak statement, as the class of all singular ordinals of the 37 model might be very “thin” in comparison with all singular cardinals of V. 38 For τ ∈ 𝒮^{1} we define: 39 • q_{τ} = q(N_{τ} , τ), m_{τ} = |q_{τ} | and μ_{τ} = μ(N_{τ} , τ). 40 • M_{τ} = N_{τ} (μ_{τ} , q_{τ}). 41 • N_{τ}^{∗} = N_{τ}^{∗}(μ_{τ} , q_{τ}). 42 • r_{τ} = p_{τ} − q_{τ}. If N_{τ} is pluripotent then r_{τ}^{∗} = d_{τ} − q_{τ}.

1 • λ_{τ} = λ_{N_{τ}} (μ_{τ} , q_{τ}), ϑ_{τ} = ϑ_{N_{τ}} (μ_{τ} , q_{τ}), ν_{τ} = ν_{N_{τ}} (μ_{τ} , q_{τ}) and F_{τ} = F_{N_{τ}} (μ_{τ} , q_{τ}). 2 • If M_{τ} is a protomouse, h_{τ} = h_{M_{τ}}. If M_{τ} is a premouse, h_{τ} = h_{τ}^{∗}. 3 • If (μ, q) is a divisor for N_{τ}, we let h_{τ}^{(μ,q)} = h_{N_{τ} (μ,q)}. If N_{τ} is pluripotent 4 and (μ, q) = (cr(E_{top}^{N_{τ}}, d_{τ}), we let h_{τ}^{(μ,q)} = h_{N_{τ}}^{∗} = h_{τ}. 5 We first construct a global square sequence ⟨C_{τ} τ ∈ 𝒮⟩ and then we produce the 6 set A′ ⊆ A as required by Theorem 1.2. Following the strategy in [14], we now 7 define approximations

ℬ = ⟨B_{τ} | τ ∈ 𝒮⟩  and  ℬ^{∗} = ⟨B_{τ}^{∗} | τ ∈ 𝒮⟩

8 to our global square sequence. As before, we work on 𝒮^{0} and 𝒮^{1} separately. There 9 are, however some differences between our current construction and the construction 10 of a □_{κ}-sequence. In the construction of a □_{κ}-sequence ⟨C_{τ} ⟩_{τ}, the direct limit of 11 the collapsing L[E]-levels for ordinals from C_{τ} ∩ τ̄ always results in the collapsing 12 L[E]-level for τ̄. This is a consequence of the fact that τ^{∗} is the cardinal successor 13 of κ in N_{τ^{∗}} for each τ^{∗} ∈ 𝒮^{0}, and is crucial in the verification that the set C_{τ} 14 arising in the construction is closed. In the case of a global square sequence, it 15 is not automatically true that a direct limit of this kind results in a singularizing 16 structure for the respective ordinal, and we need to introduce an extra tool to 17 enforce this. The idea again comes from Jensen’s construction in L in [4]. Even 18 with this extra tool it is not possible to prove that a tail-end of B_{τ} is closed if τ ∈ 𝒮^{0}. 19 One approach here would be to amend the definition of B_{τ} so that B_{τ} would be 20 closed, and indeed this can be done, as explained in Remark 4.13. However, we 21 favor the strategy of working with the sets B_{τ} defined in the standard way, then 22 define B_{τ}^{∗} the same way as in [14] and finally thin out B_{τ}^{∗} to obtain some C_{τ}^{∗} that is 23 closed on a tail-end and has small order-type. We believe this construction gives us 24 more information about the square sequence than that discussed in Remark 4.13. 25 Unlike in the construction of a □_{κ}-sequence, B_{τ}^{∗} here will not in general be closed 26 on a tail-end. The thinning out procedure is the same as in [14], but we apply it to 27 a slightly different situation that also requires a new element, namely a proof that 28 the result C_{τ}^{∗} of thinning is closed. We start with the extra tool mentioned above.

29 Definition 4.1. Let τ ∈ 𝒮. We let

α_{τ} = the largest α < τ satisfying τ ∩ h̃_{τ} (α ∪ {p_{τ} }) = α.

30 Notice that α_{τ} exists for every τ ∈ 𝒮. This is true, as the set of all ordinals α 31 satisfying τ ∩ h̃_{τ} (α ∪ {p_{τ} }) = α is closed and bounded in τ. That this set is closed 32 is obvious; its boundedness in τ follows from the fact that N_{τ} is a singularizing 33 structure for τ, that is, h̃_{τ} (δ ∪ {p_{τ} }) is cofinal in τ for some δ < τ. Also, it may 34 happen that α_{τ} = 0. Looking at the definition of α_{τ}, one might get an impression 35 that (α_{τ} , p_{τ}) is a divisor for N_{τ} if this level is exact for τ. Although this might be 36 true in some cases, it is false in many typical cases that arise in the construction 37 of the global square sequence for a simple reason that the hull h̃_{τ} (α_{τ} ∪ {p_{τ} }) may 38 be bounded in ωϱ_{τ}. Finally note that any α with τ ∩ h̃_{τ} (α ∪ {p_{τ} } is closed under 39 Gödel pairing, as it is the critical point of the inverse to the Mostowski collapsing 40 isomorphism associated with h̃_{τ} (α ∪ {p_{τ} }). So h̃_{τ} (α ∪ {p_{τ} }) is the set of all values 41 h̃_{τ} (i, ⟨ξ, p_{τ} ⟩) for ξ < α and i < ω. In particular, this is true of α = α_{τ}. 42 The following lemma establishes basic facts about maps σ_{τ̄,τ} that will be relevant 43 in the construction of our global square sequence. This lemma is an analogue to 44 Lemma 3.2 in [14], but the proof needs a new argument, as the proof in [14] relied 1 on the fact that the critical points of σ_{τ̄,τ} were local cardinal successors, which is 2 false in the present case. We formulate the lemma for the broader class of maps of 3 the form σ_{(N̄,τ̄),N_{τ}}^{n} defined at the beginning of this section, as this generality will 4 be needed in several applications.

5 Lemma 4.2. Let ατ^{∗} < τ̄ < τ be ordinals, M^{∗}, M̄, M be acceptable J-structures 6 for the same language, p^{∗}, p̄, p be parameters in these structures and n ∈ ω. Assume 7 further that the following requirements are met.

8 (i) α is the largest ordinal α′ < τ such that τ ∩ h̃_{M}^{n+1}(α′ ∪ {p}) = α′. 9 (ii) α is the largest ordinal α′ < τ̄ such that τ̄ ∩ h̃n+1 ; M̄(α′ ∪ {p̄}) = α′.

10 (iii) h̃_{M^{∗}}^{n+1}(τ^{∗} ∪ {p^{∗}}) = M^{∗} and h̃n+1 ; M̄(τ̄ ∪ {p̄}) = M̄.

11 (iv) Both maps σ^{∗} = σ_{M^{∗},M,τ^{∗}}^{n,p^{∗},p} and σ̄ = σ_{M̄,M,τ̄}^{n,p̄,p} exist and σ^{∗}(τ^{∗}) = τ = σ̄(τ̄).

12 Then the following holds.

13 (a) σ^{∗}, σ̄ are not Σ_{1}^{(n)}-preserving hence not cofinal at the n-th level. 14 (b) sup(ωϱ_{τ} ∩ rng(σ^{∗})) < sup(ωϱ_{τ} ∩ rng(σ̄)). 15 (c) rng(σ^{∗}) ⊆ rng(σ̄).

16 Proof. Regarding (a), assume σ^{∗} is Σ_{1}^{(n)}-preserving. Then h̃_{M}^{n+1}(τ^{∗} ∪ {p}) = 17 rng(σ^{∗}), hence τ ∩ h̃_{M}^{n+1}(τ^{∗} ∪ {p}) = τ^{∗} which contradicts the maximality of α_{τ}. 18 Regarding (b), assume for a contradiction that ≥ is the case. Let x ⊆ τ^{∗} 19 be finite and i ∈ ω. Assume h̃n+1 ; M̄(i, ⟨x, p̄⟩) is defined and is an element of the 20 n-th reduct of M̄; denote the value by ȳ. Then M̄ ⊧ ψ[z, ȳ, i, ⟨x, p̄⟩] where 21 ψ(v^{n}, u_{0}^{0}, u_{1}^{0}, u_{2}^{0}) is a Σ_{0}^{(n)}-formula and (∃v^{n})ψ is a functionally absolute defini-22 tion of the Σ_{1}^{(n)}-Skolem function h̃n+1 ; M̄. It follows that M ⊧ ψ[σ̄(z), y, i, ⟨x, p⟩] 23 where y = σ̄(ȳ) and both σ̄(z), y are elements of the n-th reduct of M. By our 24 assumption ≥ holds in (b) instead of <, so there is an ordinal ζ < ωϱ_{M^{∗}}^{n} such that 25 σ̄(z), y ∈ σ^{∗}(S_{ζ}^{E}) where we recall that E is the extender sequence of L[E]. It follows 26 that M ⊧ (∃v^{n} ∈ σ^{∗}(S_{ζ}^{E}))(∃u^{n} ∈ σ^{∗}(S_{ζ}^{E}))ψ[v^{n}, u^{n}, i, ⟨x, p⟩] where ψ(v^{n}, u^{n}, u_{1}^{0}, u_{2}^{0}) 27 is a specialization of ψ(v^{n}, u_{0}^{0}, u_{1}^{0}, u_{2}^{0}) obtained by substituting u^{n} for u_{0}^{0}; see [18], 28 Section 1.6. Since this statement is Σ_{0}^{(n)}, it is preserved under σ^{∗} and we conclude 29 that y^{∗} = h̃_{M^{∗}}^{n+1}(i, ⟨x, p^{∗}⟩) is defined and σ^{∗}(y^{∗}) = y. If ȳ ∈ τ̄ then y ∈ τ by as-30 sumption (iv) above and the Σ_{0}-elementarity of στ̄ and similarly we obtain that 31 y^{∗} ∈ τ^{∗}. This means that ȳ = y = y^{∗} < τ^{∗}. It follows that τ̄ ∩ h̃n+1 ; M̄(τ^{∗} ∪ {p̄}) = τ^{∗}, 32 which contradicts the maximality of α_{τ}. 33 Regarding (c), we run the argument from (b) with the roles of τ^{∗} and τ̄ switched 34 and in higher generality, as now we have to consider values of Skolem functions 35 that are not necessarily elements of the n-th reduct. If y ∈ rng(σ^{∗}) then y = 36 σ^{∗}(y^{∗}) where y^{∗} = h̃_{M^{∗}}^{n+1}(i, ⟨x, p^{∗}⟩) for some finite x ⊆ τ^{∗}. It follows that M^{∗} ⊧ 37 (∃w^{0})(w^{0} = h̃_{M^{∗}}^{n+1}(i, ⟨x, p^{∗}⟩). This statement is a Σ_{1}^{(n)} statement, as it is obtained 38 by substituting the good Σ_{1}^{(n)}-function h̃_{M^{∗}}^{n+1} into the Σ_{1}^{(0)}-formula (∃w^{0})(w^{0} = u_{0}^{0}); 39 see [18], Section 1.8. Hence the above statement can be expressed in the form 40 M^{∗} ⊧ (∃v^{n})ψ′[v^{n}, i, ⟨x, p^{∗}⟩] where ψ′ is a Σ_{0}^{(n)}-formula. Now similarly as in the 41 previous paragraph we show that M̄ ⊧ (∃v^{n})ψ′[v^{n}, i, ⟨x, p̄⟩], this time using the 42 inequality ≤ in (b). As the translation between this statement and the statement 43 M̄ ⊧ (∃w^{0})(w^{0} = h̃n+1 ; M̄(i, ⟨x, p̄⟩)) is uniform, latter statement is true. Hence ȳ = 1 h̃n+1 ; M̄(i, ⟨x, p̄⟩) is defined. It is now obvious that σ̄(ȳ) = h̃_{M}^{n+1}(i, ⟨x, p⟩) = σ^{∗}(y^{∗}). 2 This proves (c). ⊓ ⊔(Lemma 4.2)

3 We are now ready to define B_{τ} for τ ∈ 𝒮^{0}.

4 Definition 4.3. Let τ ∈ 𝒮^{0}. The set B_{τ} consists of all τ̄ ∈ τ ∩ 𝒮^{0} that meet the 5 following requirements. 6 (1) Nτ̄ is a premouse of the same type as N_{τ}. 7 (2) nτ̄ = n_{τ}. 8 (3) ατ̄ = α_{τ}. 9 (4) There is a map σ : Nτ̄ → N_{τ} that is Σ_{0}^{(n_{τ})}-preserving with respect to the 10 language of premice and such that: 11 (a) σ ↾ τ̄ = id ↾ τ̄ and if τ̄ ∈ Nτ̄ then σ(τ̄) = τ. 12 (b) σ(pτ̄) = p_{τ}. 13 (c) If β ∈ p_{τ} then there is a generalized witness Q_{τ}^{∗}(β) = ⟨Q_{τ} (β), t_{τ} (β)⟩ 14 for β with respect to N_{τ} and p_{τ} satisfying Q_{τ}^{∗}(β) ∈ rng(σ).

15 Since n_{τ}^{∗} ≤ n_{τ}, the map σ from (4) is identical with σ_{τ̄,τ}. Three basic facts about 16 maps σ_{τ̄,τ} for τ̄, τ ∈ B_{τ} are summarized in the following lemma.

17 Lemma 4.4. Let τ^{∗} < τ̄ be two elements of B_{τ} and n = n_{τ}. 18 (a) σ_{τ̄,τ} is not Σ_{1}^{(n)}-preserving, and therefore is not cofinal at the n-th level. 19 (b) rng(σ_{τ^{∗},τ}) ⊆ rng(σ_{τ̄,τ}) and sup(ωϱ_{τ} ∩ rng(σ_{τ^{∗},τ})) < sup(ωϱ_{τ} ∩ rng(σ_{τ̄,τ})). 20 (c) (σ_{τ̄,τ})^{−1} ∘ σ_{τ^{∗},τ} = σ_{τ^{∗},τ̄}, so σ_{τ^{∗},τ̄} exists. 21 (d) σ_{τ^{∗},τ̄} : N_{τ^{∗}} → Nτ̄ is not cofinal at the n-th level. 22 (e) σ_{τ^{∗},τ̄} witnesses (4) in Definition 4.3 with (τ^{∗}, τ̄) in place of (τ̄, τ). 23 (f) B_{τ} ∩ τ̄ = Bτ̄ − min(B_{τ}).

24 Proof. Clause (a) follows from (a) in Lemma 4.2, the first part of (b) follows 25 from (c) in Lemma 4.2 and the second part of (b) follows from (b) in Lemma 4.2. 26 Clauses (c) and (d) follow directly from (b). Clause (e) follows from (b) and (c) 27 via a straightforward verification; this verification uses the fact that generalized 28 witnesses for β ∈ p_{τ} with respect to N_{τ} and p_{τ} (and similarly in the case of pτ̄ and 29 Nτ̄) are characterized by Π_{1}^{(n)}-statements; see also the proof of Lemma 3.3 in [14]. 30 Finally ⊆ in (f) follows directly from (c) and (e); ⊇ in (f) follows by (c) and an 31 argument similar to the proof of (e). ⊓ ⊔(Lemma 4.4)

32 Given τ ∈ 𝒮^{0}, we define the following objects by recursion.

τ(0) = B_{τ}
τ(i + 1) ≃ min(B_{τ(i)})
ℓ_{τ} = the largest i such that B_{τ(i)} is defined
B_{τ}^{∗} = B_{τ(0)} ∪ B_{τ(1)} ∪ ⋯ ∪ B_{τ(ℓ_{τ})}.

33 By construction, the set B_{τ} is a tail-end of B_{τ}^{∗}. The following lemma summarizes 34 basic properties of sets B_{τ}^{∗} and B̄_{τ}^{∗}.

35 Lemma 4.5. Let τ̄ ∈ B_{τ}^{∗} ∪ {τ}, τ′ < τ^{∗} be elements of B_{τ}^{∗} ∩ τ̄ and n = n_{τ}. Then:

36 (a) B∗ ; τ̄ = B_{τ}^{∗} ∩ τ̄.

37 (b) Clauses (1) – (3) in Definition 4.3 hold. 38 (c) σ_{τ^{∗},τ̄} exists, is Σ_{0}^{(n)}-preserving and if τ^{∗} ∈ N_{τ^{∗}} then σ_{τ^{∗},τ̄}(τ^{∗}) = τ̄. 39 (d) σ_{τ^{∗},τ̄} is not Σ_{1}^{(n)}-preserving, and therefore is not cofinal at the n-th level.

1 (e) rng(σ_{τ′,τ̄}) ⊆ rng(σ_{τ^{∗},τ̄}) and sup(ωϱτ̄ ∩ rng(σ_{τ′,τ̄})) < sup(ωϱτ̄ ∩ rng(σ_{τ^{∗},τ̄})). 2 (f) σ_{τ^{∗},τ̄} ∘ σ_{τ′,τ^{∗}} = σ_{τ′,τ̄}.

3 Proof. Clause (a) follows from the definition of B_{τ}^{∗} and Lemma 4.4(b) by a straight-4 forward verification. The rest follows immediately from Lemma 4.2, Lemma 4.4, 5 Definition 4.3 and the definition of σ_{τ^{∗},τ̄}. ⊓ ⊔(Lemma 4.5)

6 It is not clear that B_{τ}^{∗} is closed on a tail-end. It will be convenient to work with 7 a closed set, so we add all missing limit points to B_{τ}^{∗}. For τ ∈ 𝒮^{0} let

B̄_{τ}^{∗} = B_{τ}^{∗} ∪ lim(B_{τ}^{∗}) ∪ {τ}
N̄τ̄ = \{ \begin{matrix}Nτ̄ & if τ̄ ∈ B_{τ}^{∗} \\ lim(⟨N_{τ′,τ^{∗}}, σ_{τ′,τ^{∗}} | τ′, τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄⟩) & otherwise \end{matrix}

h̄τ̄ = \{ \begin{matrix}h̃τ̄ & if τ̄ ∈ B∗ ; τ̄ \\ h̃n_{τ} +1 ; N̄τ̄ & otherwise. \end{matrix}
ϱ̄τ̄ = ϱn_{τ} ; N̄τ̄
p̄τ̄ = pN̄τ̄.

8 The direct limit of the diagram ⟨N_{τ′,τ^{∗}}, σ_{τ′,τ^{∗}} | τ′, τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄⟩ in the above defi-9 nition is of course considered transitive. This possible due to its well-foundedness, 10 which in turn is a consequence of the fact that the map σ̄_{τ̄,τ} : N̄τ̄ → N_{τ} defined by 11 σ̄_{τ̄,τ} : σ̄_{τ^{∗}τ̄}, (x) ↦ σ_{τ^{∗},τ} (x) is Σ_{0}-preserving. Here σ̄_{τ^{∗}τ̄} : N_{τ^{∗}} → N̄τ̄ are the direct 12 limit maps for τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄. It follows that N̄τ̄ is an acceptable J-structure, so 13 it does make sense to talk about fine structure of N̄τ̄. Of course, if τ̄ ∈ B_{τ}^{∗} then 14 N̄τ̄ = Nτ̄. In general, it is not clear whether N̄τ̄ is a singularizing structure for τ̄. 15 The following lemma summarizes basic facts about N̄τ̄ for τ̄ ∈ B̄_{τ}^{∗}.

16 Lemma 4.6. Let τ̄ ∈ lim(B_{τ}^{∗}), τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄ and n = n_{τ}. Then the following holds.

17 (a) ωϱn+1 ; N̄τ̄ = τ̄.

18 (b) The direct limit maps σ̄_{τ^{∗},τ̄} as well as the map σ̄_{τ̄,τ} are Σ_{0}^{(n)}-preserving. 19 (c) σ̄_{τ^{∗},τ̄} ↾ τ^{∗} = id ↾ τ^{∗} and if τ^{∗} ∈ N_{τ^{∗}} then σ̄_{τ^{∗},τ̄}(τ^{∗}) = τ̄. Similarly, 20 σ̄_{τ̄,τ} ↾ τ̄ = id ↾ τ̄ and if τ̄ ∈ N̄τ̄ then σ̄_{τ̄,τ} (τ̄) = τ. 21 (d) σ̄_{τ^{∗},τ̄}(p_{τ^{∗}}) = p̄τ̄ and σ̄_{τ̄,τ} (p̄τ̄) = p_{τ}. 22 (e) σ̄_{τ^{∗},τ̄} = σ_{(N̄_{τ^{∗}} ,τ^{∗}),N̄τ̄}^{n,p_{τ^{∗}}} and σ̄_{τ̄,τ} = σ_{(N̄τ̄,τ̄),N̄_{τ}}^{n,p̄τ̄}. 23 (f) N̄τ̄ is an L[E]-level. 24 (g) α_{τ} is the largest α < τ̄ such that τ̄ ∩ h̄τ̄(α ∪ {p̄τ̄}) = α.

25 Proof. We may without loss of generality assume that τ̄ ∉ B_{τ}^{∗}. Clauses (b) and (c) 26 follow from properties of direct limits by a straightforward verification. From (b),(c) 27 and the soundness of each N_{τ^{∗}} we obtain h̄τ̄(τ̄ ∪ {p̄}) = N̄τ̄ where p̄ is the common 28 value of σ̄_{τ^{∗},τ̄}(p_{τ^{∗}}) for τ^{∗} ∈ B_{τ}^{∗}∩τ̄, from which we conclude that ωϱn+1 ; N̄τ̄ ≤ τ̄. Since τ̄ 29 is a cardinal in L[E] and N̄τ̄ agrees with L[E] up to τ̄, we conclude that τ̄ = ωϱn+1 ; N̄τ̄. 30 This gives (a). Let i be such that τ(i + 1) < τ̄ < τ(i) and τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄ be such 31 that τ(i + 1) ≤ τ^{∗}. As each β̄ ∈ p̄ is of the form σ̄_{τ^{∗},τ̄}(β^{∗}) for some β^{∗} ∈ p_{τ^{∗}}, 32 the map σ_{τ^{∗},τ(i)} has some generalized witness for β(i) = σ_{τ^{∗},τ(i)}(β^{∗}) with respect 33 to N_{τ(i)} and p_{τ(i)} in its range. Since rng(σ_{τ^{∗},τ(i)}) ⊆ rng((σ_{τ(i),τ})^{−1} ∘ σ̄_{τ̄,τ}) and 34 β(i) = (σ_{τ(i),τ})^{−1} ∘ σ̄_{τ̄,τ} (β̄), the same applies to the composition (σ_{τ(i),τ})^{−1} ∘ σ̄_{τ̄,τ}. 35 As this composition is Σ_{0}^{(n)}-preserving and being a generalized witness as above is 1 a Π_{1}^{(n)}-property, it is preserved downwards under σ̄_{τ̄,τ(i)}. It follows that for each 2 element of p̄ there is a generalized witness with respect to N̄τ̄ and p̄ in N̄τ̄, so p̄ = p̄τ̄ 3 by Corollary 1.12.4 in [18] and N̄τ̄ is sound. Now apply the Condensation Lemma 4 to the triple σ̄_{τ̄,τ} : N̄τ̄ → N_{τ}. We conclude that N̄τ̄ is solid and obtain four options. 5 Option (a) in the Condensation Lemma fails, as N_{τ} and N̄τ̄ have different (n+1)-st 6 projecta and options (c) and (d) in the Condensation Lemma fail, as τ̄ is a limit 7 cardinal in N̄τ̄. It follows that N̄τ̄ is a proper initial segment of N_{τ}, and hence 8 a level of L[E], which proves (f) in the current lemma. Clause (e) then follows 9 immediately. Regarding (g), assume α_{τ} < α < τ̄ and pick some τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄ such 10 that α < τ^{∗}. As α_{τ^{∗}} = α_{τ}, we have some finite x ⊆ α and some i < ω such that 11 α ≤ h̄_{τ^{∗}}(i, ⟨x, p_{τ^{∗}}⟩) < τ^{∗}. Since σ̄_{τ^{∗},τ̄} is Σ_{0}^{(n)}-preserving, σ̄_{τ^{∗},τ̄} ↾ τ^{∗} = id and h̄_{τ^{∗}} has 12 a functionally absolute Σ_{1}^{(n)}-definition, α < h̄τ̄(i, ⟨x, p̄τ̄⟩) < τ^{∗} < τ̄. This verifies 13 (g). Finally (h) follows by the same argument as Lemma 4.4(a). ⊓ ⊔(Lemma 4.6)

14 We next summarize the basic facts about B̄_{τ}^{∗} relevant for our construction. To 15 unify the notation, for τ^{∗} ≤ τ̄ in B̄_{τ}^{∗} we let

(3)  σ̄_{τ^{∗},τ̄} = σ_{(N̄_{τ^{∗}} ,τ^{∗}),N̄τ̄}^{n_{τ} ,p̄_{τ^{∗}}}.

16 Of course, if τ^{∗}, τ̄ ∈ B_{τ}^{∗} ∪ {τ} then σ̄_{τ^{∗},τ̄} = σ_{τ^{∗},τ̄} and if τ̄ ∈ B̄_{τ}^{∗} − B_{τ}^{∗} then σ̄_{τ^{∗},τ̄} is 17 the same as in Lemma 4.6, so this notation is consistent with that used before.

18 Lemma 4.7. Let τ′, τ^{∗}, τ̄ ∈ B̄_{τ}^{∗} such that τ′ < τ^{∗} and let n = n_{τ}. Then the 19 following holds. 20 (a) B̄∗ ; τ̄ = B̄_{τ}^{∗} ∩ τ̄ whenever τ̄ ∈ B_{τ}^{∗}. 21 (b) σ̄_{τ^{∗},τ̄} exists and σ̄_{τ^{∗},τ̄}(τ^{∗}) = τ̄ whenever τ^{∗} ∈ N̄_{τ^{∗}}. 22 (c) σ̄_{τ′,τ^{∗}} : N̄_{τ′} → N̄_{τ^{∗}} is not Σ_{1}^{(n)}-preserving hence not cofinal at the n-th level. 23 (d) rng(σ̄_{τ′,τ̄}) ⊆ rng(σ̄_{τ^{∗},τ̄}) and sup(ωϱ̄τ̄ ∩ rng(σ̄_{τ′,τ̄})) < sup(ωϱ̄τ̄ ∩ rng(σ̄_{τ^{∗},τ̄})). 24 (e) σ̄_{τ^{∗},τ̄} ∘ σ̄_{τ′,τ^{∗}} = σ̄_{τ′,τ̄}. 25 (f) α_{τ} is the largest α < τ̄ such that τ̄ ∩ h̄τ̄(α ∪ {p̄τ̄}) = α.

26 Proof. Clause (a) follows from the coherency of the sets B_{τ}^{∗} (see Lemma 4.5(a)) 27 and from the construction of B̄_{τ}^{∗}. The remaining clauses follow from Lemma 4.5 28 and Lemma 4.6 in the straightforward way. ⊓ ⊔(Lemma 4.5)

29 In the following we prove that B_{τ}^{∗} is unbounded in τ whenever τ has uncountable 30 cofinality. We split the proof into two lemmata. It will be convenient to introduce 31 the following approximation to B_{τ}.

(4) B̂_{τ} = {τ̄ ∈ τ ∩ 𝒮 | τ̄ meets the requirements (1) – (4) from Definition 4.3}

32 Thus, the only difference between B_{τ} and B̂_{τ} is that B̂_{τ} is allowed to contain 33 elements that are outside 𝒮^{0}. However, we still require the cardinal τ to be an 34 element of 𝒮^{0}.

35 Lemma 4.8. B̂_{τ} is unbounded in τ whenever τ ∈ 𝒮^{0} has uncountable cofinality.

36 Proof. This is an interpolation argument similar to that in the proof of Lemma 3.7 37 in [14]. Given an ordinal τ′ < τ, we want to find some τ̃ ∈ B̂_{τ} such that τ′ ≤ τ̃ < τ. 38 We form a fully elementary countable hull X ≺ N_{τ} such that τ, τ′, p_{τ} , α_{τ} , ∈ X, and 39 for each β ∈ p_{τ}, some generalized witness Q_{τ}^{∗}(β) for β with respect to N_{τ} and p_{τ} 40 is in X as well. (Of course τ ∈ X only if τ ∈ N_{τ}.) We then collapse X to some 41 transitive N̄ and obtain a fully elementary embedding σ : N̄ → N_{τ} whose range is 1 X. Obviously, N̄ is a sound premouse of the same type as N_{τ} and σ can be viewed 2 as a fully elementary, and so Σ^{∗}-preserving embedding with respect to the language 3 for premice. Also n(τ̄, N̄) = n_{τ}; we denote this common value by n. Let τ̄, p̄, ᾱ be 4 the σ-preimages of τ, p_{τ} , α_{τ}, respectively. 5 We let τ̃ = sup(σ[τ̄]). Obviously τ̃ > τ′. Since τ̄ is countable and τ has 6 uncountable cofinality, we have τ̃ < τ. Notice also that τ̃ is a limit cardinal in the 7 sense of L[E]. Using the interpolation lemma (see [14] and [18], Lemma 3.6.10) we 8 obtain an acceptable J-structure Ñ and maps σ̃ : N̄ → Ñ and σ′ : Ñ → N_{τ} such 9 that σ̃ is Σ_{0}^{(n)}-preserving, σ′ is Σ_{0}^{(n−1)}-preserving^{2}, σ′ ↾ τ̃ = id ↾ τ̃, and if τ̃ ∈ Nτ̃ 10 then σ′(τ̃) = τ. This is true even if ht(N̄) = τ̄, in which case σ̃ is essentially the 11 same object as σ, up to its codomain. Let us stress that Ñ is constructed as the 12 fine pseudoultrapower (in the sense of [18]) of N̄ by σ ↾ J Ē ; τ̄ where Ē = EN̄ and 13 the functions used to construct this pseudoultrapower are precisely all total good 14 Σ_{1}^{(i)}(N̄)-functions with domains in J Ē ; τ̄ where i < n. Our goal is to show that

τ̃ ∈ 𝒮, Ñ = Nτ̃ and nτ̃ = n_{τ} (5)
σ̃ = σ_{τ̃,τ} (6)
ατ̃ = α_{τ} (7)

15 We let p̃ = σ̃(p̄). Notice that σ̃(ᾱ) = α_{τ}. The following sketch summarizes the 16 proof of (5). The details, which are left to the reader, follow from the general fine 17 structure theory described in [18], Chapters 1–3. 18 Since τ ∈ 𝒮^{0}, the pair (N_{τ} , τ) is not pluripotent, hence (N̄, τ̄) is not pluripotent 19 either. This means that if N̄ is active and cr(EN̄ ; top) < τ̄ then n = n(τ̄, N̄) > 0, so 20 the map σ̃, being a pseudoultrapower map in the sense of [18], is Σ_{2}-preserving. It 21 follows that Ñ is a potential premouse in the sense of [18], that is, the top extender 22 of Ñ is total on Ñ. To see that Ñ is a premouse one has to verify the initial segment 23 condition, and this is immediate if N̄ is a type A or type B premouse. If N̄ is a 24 type C premouse, the initial segment condition follows from the fact that σ̃ maps 25 λ(N̄) cofinally into λ(Ñ) if n = 1 and that σ̃ is Σ_{2}^{(1)}-preserving if n > 1. 26 By the elementarity of σ we have p̄ ∈ Rn+1 ; N̄, which has two immediate con-27 sequences. First, σ̃ is cofinal at the n-th level hence Σ_{1}^{(n)}-preserving and σ′ is 28 Σ_{0}^{(n)}-preserving. Second, h̃n+1 ; Ñ(τ̃ ∪ {p̃}) = Ñ, so ωϱn+1 ; Ñ ≤ τ̃. Thus Ñ is solid, as 29 follows from the first part of the Condensation Lemma. Since τ̃ is a cardinal and Ñ 30 agrees with L[E] up to τ̃, we conclude that ωϱn+1 ; Ñ = τ̃ and p̃ ∈ Rn+1 ; Ñ. Recall that

31 the property of being a generalized witness for β is a Π_{1}^{(n)}-property whenever β is 32 larger than or equal to the (n + 1)-st projectum of the structure in question. Since 33 the generalized solidity witness Q_{τ}^{∗}(β) is an element of rng(σ) for each β ∈ p_{τ} and 34 rng(σ′) ⊇ rng(σ), for each β̃ ∈ p̃ there is a generalized solidity witness with respect 35 to Ñ and β̃ in Ñ, namely (σ′)^{−1}(Q_{τ}^{∗}(σ′(β̃))). Then p̃ = pÑ by Corollary 1.12.4 in 36 [18] and Ñ is sound. We now show that Ñ is an initial segment of N_{τ}, and therefore 37 an initial segment of L[E]. This is clear if τ̃ = ht(Ñ), so assume that τ̃ ∈ Ñ. We 38 have τ̃ = cr(σ′) and the condensation lemma applied to σ′ gives us four possibili-39 ties. Options (c) and (d) can easily be ruled out since either of them implies that 1 τ̃ must be a successor cardinal in Ñ. Hence (a) or (b) is the case, and so Ñ is an 2 initial segment of N_{τ}. Notice that in fact (b) holds, as ωϱn+1 ; Ñ = τ̃ ≠ τ = ωϱ_{N_{τ}}^{n+1}. 3 We now complete the proof of (5). Since N̄ is a singularizing structure for τ̄ 4 with n(τ̄, N̄) = n by the elementarity of σ, we have some ordinal γ̄ < τ̄ such 5 that h̃n+1 ; N̄(γ̄ ∪ {p̄}) is cofinal in τ̄. Let γ = σ̃(γ̄) < τ̃. As σ̃ is Σ_{1}^{(n)}-preserving, 6 h̃n+1 ; Ñ(γ ∪ {pÑ}) ⊇ σ̃[h̃n+1 ; N̄(γ̄ ∪ {p̄})] and the latter is cofinal in τ̃ since σ̃ maps 7 τ̄ cofinally into τ̃. It follows that Ñ is the singularizing level of L[E] for τ̃ and 8 nτ̃ = n(τ̃, Ñ) ≤ n. To see that nτ̃ = n, we show that τ̃ ∩ h̃k+1 ; Ñ(δ ∪ {p̃}) is bounded 9 in τ̃ for every δ < τ̃ and k < n. Given such a δ, let δ̄ = σ̃^{−1}[δ]. Since σ̃ maps τ̄ 10 cofinally into τ̃, the ordinal δ̄ is strictly smaller than τ̄. Since n(τ̄, N̄) = n, the hull 11 h̃k+1 ; N̄(δ̄ ∪ {p̄}) is bounded below τ̄, say by some γ̄. This fact can be expressed in a 12 Π_{1}^{(k)} manner over N̄ as follows.

2Here we let n − 1 = 0 if n = 0.

(∀x^{k} ∈ [δ̄]^{<ω})(∀ζ^{k} < τ̄)(∀i^{k} ∈ ω)(ζ^{k} = h̃k+1 ; N̄(i^{k}, ⟨x^{k}, p̄⟩) ⟶ ζ^{k} < γ̄).

13 The Σ_{1}^{(k)}-elementarity of σ̃ then guarantees that the above statement is transferred 14 by σ̃ to Ñ, so τ̃ ∩ h̃k+1 ; Ñ(σ̃(δ̄) ∪ {pÑ}) is bounded by σ̃(γ̄) < τ̃. As σ̃(δ̄) ≥ δ, we 15 obtain the desired conclusion, and thereby the equality nτ̃ = n. This completes the 16 proof of (5). Clause (6) then follows immediately. 17 It remains to prove (7). To see that ατ̃ ≤ α_{τ}, pick any α′ such that α_{τ} < α′ < τ̃ 18 and let ᾱ′ = σ̃^{−1}[α′]. Then ᾱ < ᾱ′ < τ̄ where the second inequality follows 19 again from the fact that σ̃ maps τ̄ cofinally into τ̃; recall also that σ̃(ᾱ) = α. By 20 the full elementarity of σ, there is some finite x ⊆ ᾱ′ and some i ∈ ω such that 21 ᾱ′ ≤ h̃n+1 ; N̄(i, ⟨x, pN̄⟩) < τ̄. If we apply σ̃ to these inequalities, we obtain σ̃(ᾱ′) ≤ 22 h̃Ñ(i, ⟨σ̃(x), pτ̃⟩) < τ̃. By the definition of ᾱ′ each element of x is mapped into α′, so 23 σ̃(x) is a finite subset of α′. Since σ̃(ᾱ′) ≥ α′, we see that τ̃∩h̃Ñ(α′∪{pτ̃}) ≠ α′. As 24 this is true of any α′ satisfying α_{τ} < α′ < τ̃, we conclude that ατ̃ ≤ α_{τ}. To see the 25 converse, we verify the inclusion τ̃∩h̃Ñ(α_{τ} ∪{pτ̃}) ⊆ α_{τ}. Let x ⊆ α_{τ} be finite, i ∈ ω 26 and ζ = h̃Ñ(i, ⟨x, pτ̃⟩) < τ̃. Since σ′ is Σ_{0}^{(n)}-preserving, σ′(ζ) = h̃_{τ} (i, ⟨x, p_{τ} ⟩) < τ. 27 It follows that ζ < α_{τ}, which completes the proof of (7) and thereby the proof of 28 the entire lemma. ⊓ ⊔(Lemma 4.8)

29 Lemma 4.9. Assume τ ∈ 𝒮^{0} and B̂_{τ} is unbounded in τ. Then B̂_{τ} is almost 30 contained in 𝒮^{0}. It follows that there is some τ̂ < τ such that B̂_{τ} − τ̂ = B_{τ} − τ̂.

31 Proof. Let n = n_{τ}. The premouse Nτ̄ is Σ_{0}-embeddable into N_{τ} whenever τ̄ ∈ B̂_{τ}, 32 so Nτ̄ is active if and only if N_{τ} is active and cr(E_{top}^{Nτ̄}) < τ̄ if and only if cr(E_{top}^{N_{τ}}) < τ. 33 It follows that none of Nτ̄ is pluripotent, as N_{τ}, being an element of 𝒮^{0}, is not 34 pluripotent. Assume for a contradiction that unboundedly many τ ∈ B̂_{τ} fail to 35 be in 𝒮^{0}. For each such τ̄ the structure Nτ̄ is exact for τ̄, which means that 36 ωϱτ̄ > τ̄. As σ_{τ̄,τ} is Σ_{0}^{(n)}-preserving, ωϱ_{τ} > τ hence N_{τ} is exact for τ. Find a 37 strictly increasing sequence ⟨τ_{ι} | ι < γ⟩ cofinal in τ such that each N_{τ_{ι}} is exact 38 for τ_{ι}, the sequence ⟨μ_{τ_{ι}}⟩_{ι} is monotonic, which includes the possibility that it is 39 constant, and the parameters q_{τ_{ι}} have the same number of elements for all ι < γ; 40 see the beginning of this section for the definition of μ_{τ_{ι}} and q_{τ_{ι}}. The monotonicity 41 of μ_{τ_{ι}} is obtained by choosing τ_{ι+1} with minimal possible value of μ_{τ_{ι+1}} at each step; 42 more details can be found in [14], proof of Lemma 3.9. The number of elements in 1 q_{τ_{ι}} can be fixed by the pigeonhole argument. Let μ be the supremum of all μ_{τ_{ι}} and 2 q = σ_{τ_{ι},τ} (q_{τ_{ι}}); this value obviously does not depend on ι. 3 For μ < τ the proof is similar to the proof of Lemma 3.9 in [14]. We first verify 4 that (μ, q) is a divisor for N_{τ}. Given η < μ and i ∈ ω such that ζ = h̃_{τ} (i, ⟨η, p_{τ} ⟩) ≤ 5 max(q_{τ}), let ι < γ be such that η < μ_{τ_{ι}} and ζ_{ι} = h̃_{τ_{ι}}(i, ⟨η, p_{τ_{ι}}⟩) is defined. This 6 is possible, as the ordinals τ_{ι} are unbounded in τ. As (μ_{τ_{ι}}, q_{τ_{ι}}) is a divisor for 7 N_{τ_{ι}}, ζ_{ι} < μ_{ι} and so ζ = σ_{τ_{ι},τ} (ζ_{ι}) < μ_{τ_{ι}} ≤ μ. This verifies (d) in Definition 3.5 8 of a divisor. Since h̃_{τ_{ι}}(μ_{τ_{ι}} ∪ {q_{τ_{ι}}}) is unbounded in ωϱ_{τ_{ι}} for all ι < γ and ωϱ_{τ} 9 is the union of all σ_{τ_{ι},τ} [ωϱ_{τ_{ι}}] for ι < γ (this is again a consequence of the fact 10 that τ_{ι}’s are unbounded in τ), the hull h̃_{τ} (μ ∪ {q}) is unbounded in ωϱ_{τ}, which 11 verifies (c) in the definition of a divisor, and thereby proves that (μ, q) is a divisor 12 for N_{τ}. Because this divisor cannot be strong, Lemmata 3.16 and 3.17 give us 13 an ordinal β′ = max(p_{N′_{τ} (μ)}) − π′_{τ}^{−1}(q); here we follow the notation from those 14 lemmata, so N′_{τ} (μ) is the transitive collapse of h̃_{τ} (μ ∪ {p_{τ} }) and π′_{τ} is the inverse 15 to the collapsing map. Then β = π′_{τ} (β′) ≥ μ and, letting Q^{∗} = π′_{τ} (⟨W_{N′_{τ} (μ)}^{β′,p′}, t⟩)

16 where p′ = π′_{τ}^{−1}(q) and t is the inverse image of p′ under the associated witness 17 map, the pair Q^{∗} is a generalized n-witness for β with respect to N_{τ} and q; see 18 the text immediately preceding Lemma 3.11. Let η, ξ < μ and i, j ∈ ω be such 19 that Q^{∗} = h̃_{τ} (i, ⟨η, p_{τ} ⟩) and β = h̃_{τ} (j, ⟨ξ, p_{τ} ⟩). Pick ι < γ large enough that 20 η, ξ < μ_{τ_{ι}} and both Q_{τ_{ι}}^{∗} = h̃_{τ_{ι}}(i, ⟨η, p_{τ_{ι}}⟩) and β_{τ_{ι}} = h̃_{τ_{ι}}(j, ⟨ξ, p_{τ_{ι}}⟩ are defined; the 21 existence of such an ι is again guaranteed by the fact that τ_{ι}’s are unbounded in 22 τ. As σ_{τ_{ι},τ} is Σ_{0}^{(n)}-preserving and the property of being a generalized n-witness 23 is a Π_{1}^{(n)}-property, Q_{τ_{ι}}^{∗} is a generalized n-witness for β_{τ_{ι}} = σ_{τ_{ι},τ}^{−1} (β) with respect 24 to N_{τ_{ι}} and q_{τ_{ι}}. Moreover, Q_{τ_{ι}}^{∗}, β_{τ_{ι}} ∈ h̃_{τ_{ι}}(μ_{τ_{ι}} ∪ {p_{τ_{ι}}}) = rng(π′_{τ_{ι}}) by our choice 25 of ι and β_{τ_{ι}} ≥ μ ≥ μ_{τ_{ι}}. It follows that π′_{τ_{ι}}^{−1}(Q_{τ_{ι}}^{∗}) is a generalized witness for 26 β′_{τ_{ι}} = π′_{τ_{ι}}^{−1}(β_{τ_{ι}}) with respect to N′_{τ_{ι}}(μ_{τ_{ι}}) and π′_{τ_{ι}}^{−1}(q_{τ_{ι}}). Moreover β′_{τ_{ι}} ≥ μ_{τ_{ι}}, 27 as β_{τ_{ι}} ≥ μ_{τ_{ι}}, hence π′_{τ_{ι}}^{−1}(q_{τ_{ι}}) is not the standard parameter of N′_{τ_{ι}}, but merely a 28 proper top segment of it. By Lemma 3.16, the divisor (μ_{τ_{ι}}, q_{τ_{ι}}) is not strong, which 29 contradicts our assumption. 30 Now consider the situation where μ = τ. We claim that q ≠ ∅. Notice that if 31 q = ∅ then q_{τ_{ι}} = ∅ for all ι < γ. It follows directly from definition of α_{τ_{ι}} that 32 no pair (μ′, p_{τ_{ι}}) can be a divisor for N_{τ_{ι}} if μ′ > α_{τ_{ι}}, so necessarily μ_{τ_{ι}} ≤ α_{τ_{ι}} = α_{τ} 33 for all ι < γ. Then μ ≤ α_{τ}, a contradiction. Once we know that q ≠ ∅, we can 34 proceed similarly as in the previous paragraph. Letting β = max(q), the standard 35 solidity witness ⟨W_{N_{τ}}^{β,p_{τ}} , t⟩ is of the form h̃_{τ} (η, p_{τ}) for some η < τ, this time by the 36 soundness of N_{τ}. Choose ι large enough such that η < μ_{ι} and h̃_{τ_{ι}}(η, p_{τ}) is defined. 37 Similarly as above we then show that (μ_{τ_{ι}}, q_{τ_{ι}}) fails to be strong. Contradiction. 38 ⊓ ⊔(Lemma 4.9)

39 Corollary 4.10. Assume τ ∈ 𝒮^{0} has uncountable cofinality. Then B_{τ} is unbounded 40 in τ. Consequently, B_{τ}^{∗} is unbounded in τ.

41 We are now ready to define the sets C_{τ}^{∗} which constitute a very close approxima-42 tion to our global square sequence. Each C_{τ}^{∗} will be a closed subset of B̄_{τ}^{∗} cofinal 43 in τ for uncountably cofinal τ, the successor points of C_{τ}^{∗} will be in B_{τ}^{∗} and C_{τ}^{∗} 44 will have small order type. It will soon turn out that C_{τ}^{∗} is almost contained in 45 B_{τ}^{∗} whenever τ ∈ 𝒮^{0} has uncountable cofinality. The definition of C_{τ}^{∗} proceeds by 1 recursion; along with C_{τ}^{∗} we also define ordinals τ_{ι} and ξ_{ι}^{τ} by recursion on ι. In what 2 follows we will be making use of the properties of B̄_{τ}^{∗} summarized in Lemma 4.7. 3 Recall also the definitions of parameters N̄τ̄, h̄τ̄, ϱ̄τ̄ and p̄τ̄ that are stated immedi-4 ately above Lemma 4.6. Recall also that if τ̄ ∈ B_{τ}^{∗} then h̄τ̄ = h̃τ̄ and the versions 5 of the remaining parameters with “bars” agree with those without “bars”, so for 6 instance N̄τ̄ = Nτ̄ and ϱ̄τ̄ = ϱτ̄.

τ_{0} = min(B_{τ}^{∗} ∪ {τ})
ξ_{ι}^{τ} = the least ξ < τ such that h_{τ} ({ξ} ∪ {p_{τ} }) ⊈ rng(σ̄_{τ_{ι},τ})
τ_{ι+1} = the least τ̄ ∈ B_{τ}^{∗} ∪ {τ} such that h_{τ} ({ξ_{ι}^{τ} } ∪ {p_{τ} }) ⊆ rng(σ_{τ̄,τ})
τ_{ι} = sup({τῑ| ῑ < ι}) for limit ι
ι_{τ} = the least ι such that τ_{ι} = τ

7 We let

C_{τ}^{∗} = {τ_{ι} | ι < ι_{τ} }.

8 C_{τ}^{∗} is obviously a closed subset of B̄_{τ}^{∗}. An easy induction on ι using Lemma 4.7(d) 9 yields that both ⟨τ_{ι} | ι < ι_{τ} ⟩ and ⟨ξ_{ι}^{τ} | ι < ι_{τ} ⟩ are strictly increasing. By Corol-10 lary 4.10 and the fact that each h_{τ} ({ξ} ∪ {p_{τ} }) is countable, C_{τ}^{∗} is unbounded in τ 11 whenever τ has uncountable cofinality. We stress that successor points in C_{τ}^{∗} are 12 elements of B_{τ}^{∗}, whereas the membership of limit points of C_{τ}^{∗} to B_{τ}^{∗} is not clear at 13 this point. The following lemma summarizes some less obvious facts about C_{τ}^{∗}.

14 Lemma 4.11. The following is true of the sets C_{τ}^{∗}. 15 (a) If τ̄ > α_{τ} is a limit point of C_{τ}^{∗} then τ̄ ∈ 𝒮, N̄τ̄ = Nτ̄, nτ̄ = n_{τ} and 16 ατ̄ = α_{τ}.

17 (b) If τ̄ ∈ C_{τ}^{∗} ∩ B_{τ}^{∗} then C∗ ; τ̄ = C_{τ}^{∗} ∩ τ̄.

18 (c) otp(C_{τ}^{∗}) < τ for all τ ∈ 𝒮^{0}.

19 Proof. Regarding (a), from Lemma 4.6 (f) and (c) we know that N̄τ̄ is an L[E]-level 20 in which τ̄ is inaccessible where we allow the option τ̄ = ht(N̄τ̄). Once we have 21 proved that nτ̄ = n_{τ}, it follows from Lemma 4.6(g) that ατ̄ = α_{τ}. Let n = n_{τ}. The 22 following simple argument shows that τ̄ ∩ h̃k+1 ; N̄τ̄(δ ∪ {p̄τ̄}) is bounded in τ̄ whenever 23 δ < τ̄ and k < n. If τ^{∗} ∈ B_{τ}^{∗} ∩ τ̄ is larger than δ then τ^{∗} ∩ h̃_{N_{τ^{∗}}}^{k+1}(δ ∪ {p_{τ^{∗}}}) is

24 bounded in τ^{∗} by some β, so for each i ∈ ω the Π_{1}^{(k)}-statement

(∀ζ^{k})(∀x^{k})[(ζ^{k} = h̃_{N_{τ^{∗}}}^{k+1}(i, ⟨x^{k}, p_{τ^{∗}}⟩) & x^{k} ∈ [δ]^{<ω} & ζ^{k} < τ^{∗}) ⟶ ζ^{k} < β]

25 holds in N_{τ^{∗}}. As σ̄_{τ^{∗},τ̄} is Σ_{1}^{(k)}-preserving, it transfers this statement to N̄τ̄, hence 26 τ̄ ∩ h̃k+1 ; N̄τ̄(δ ∪ {p̄τ̄}) is bounded by β. In order to complete the proof of (a) it is 27 therefore sufficient to show that τ̄ ∩ h̄τ̄(δ ∪ {p̄τ̄}) is cofinal in τ̄ for some δ < τ̄. 28 We first observe that ξ_{ι}^{τ} < τ_{ι} for every ι < ι_{τ}. Otherwise we would have 29 h̃_{τ} (i, ⟨ξ, p_{τ} ⟩) ∈ rng(σ̄_{τ_{ι},τ}) for all ξ < τ_{ι} and i ∈ ω hence h̃_{τ} (τ_{ι} ∪ {p_{τ} }) ⊆ rng(σ̄_{τ_{ι},τ}) 30 as every finite x ⊆ τ_{ι} can be coded as a single ordinal ξ < τ_{ι} using the Gödel pairing 31 function due to the fact that τ_{ι} is primitive recursively closed. This would mean 32 that τ ∩ h̃_{τ} (τ_{ι} ∪ {p_{τ} }) = τ_{ι} which contradicts the maximality of α_{τ}. 33 We next show that h̄τ̄(ξτ ; ῑ ∪ {α_{τ} , τ̄} ∪ {p̄τ̄}) is cofinal in τ̄ where ῑ is such that 34 τ̄ = τῑ. An important technical tool here is the observation that if τ̄ ∈ B̄_{τ}^{∗}, X ⊆ N̄τ̄ 35 and h̃_{τ} (σ̄_{τ̄,τ} [X] ∪ {p_{τ} }) ⊆ rng(σ̄_{τ̄,τ}) then

(8)  h̃_{τ} (σ̄_{τ̄,τ} [X] ∪ {p_{τ} }) = σ̄_{τ̄,τ} [h̄τ̄(X ∪ {p̄τ̄})].

1 This holds because h̃_{τ} (σ̄_{τ̄,τ} [X] ∪ {p_{τ} }) is a Σ_{1}^{(n)}-elementary substructure of N_{τ}, so 2 there are witnesses to existential quantifiers of the form (∃v^{n}) inside the substruc-3 ture. More precisely, if y ∈ h̃_{τ} (σ̄_{τ̄,τ} [X] ∪ {p_{τ} }) then y = h̃_{τ} (i, ⟨x, p_{τ} ⟩) for some 4 finite x ⊆ σ̄_{τ̄,τ} [X] and i ∈ ω hence N_{τ} ⊧ (∃v^{n})ψ[v^{n}, y, i, ⟨x, p_{τ} ⟩] where the formula 5 (∃v^{n})ψ(v^{n}, u_{0}^{0}, u_{1}^{0}, u_{2}^{0}) constitutes a functionally absolute definition of h̃_{τ} and ψ is a 6 Σ_{0}^{(n)}-formula. By the Σ_{1}^{(n)}-uniformization, see [18], Lemma 3.1.4, some witness z to 7 the existential quantifier (∃v^{n}) has the form h̃_{τ} (j, ⟨x, p_{τ} ⟩) where j ∈ ω and therefore 8 z ∈ rng(σ̄_{τ̄,τ}). So N_{τ} ⊧ ψ[z, y, i, ⟨x, p_{τ} ⟩] and, letting z̄, ȳ, x̄ be the inverse image of 9 z, y, x under σ̄_{τ̄,τ}, we conclude that N̄τ̄ ⊧ ψ[z̄, ȳ, i, ⟨x̄, p̄τ̄⟩], that is, h̄τ̄(i, ⟨x̄, p̄τ̄⟩) is 10 defined, its value is ȳ and is mapped to y by σ̄_{τ̄,τ}. 11 As a next step we prove that for each ι < ι_{τ} such that τ_{ι} ∈ B_{τ}^{∗},

(9)  τ_{ι+1} ∩ h̃_{τ_{ι+1}}({ξ_{ι}^{τ} , α_{τ} , τ_{ι+1}} ∪ {p_{τ_{ι+1}}}) ⊈ τ^{∗} for any τ^{∗} < τ_{ι}.

12 Assume for a contradiction that this fails. Let N̄ be the transitive collapse of 13 h̃_{τ_{ι+1}}({ξ_{ι}^{τ} , α_{τ} , τ_{ι+1}} ∪ {p_{τ_{ι+1}}}) and σ̄ : N̄ → N_{τ_{ι+1}} be the inverse to the collapsing 14 isomorphism. Let and (τ̄, p̄) = σ̄^{−1}(p_{τ_{ι+1}}, τ_{ι+1}). It follows by standard fine struc-15 tural considerations that N̄ is an acceptable J-structure for the same language as 16 N̄_{τ_{ι+1}}, the map σ̄ is Σ_{1}^{(n)}-preserving and h̃n+1 ; N̄(τ̄ ∪ {p̄}) = N̄. The ordinal τ̄ is 17 obviously regular in N̄. In fact, h̃k+1 ; N̄(δ ∪ {p̄}) is bounded in τ̄ whenever δ < τ̄ 18 and k < n, hence there is no good Σ_{1}^{(k)}(N̄)-function singularizing τ̄. To see this 19 notice that for δ < τ̄ the boundedness of τ_{ι+1} ∩ h̃_{τ_{ι+1}}(σ̄(δ) ∪ {p_{τ_{ι+1}}}) in τ_{ι+1} can be 20 expressed as the requirement that for each i ∈ ω the Σ_{1}^{(n)}-statement

(∃β^{n})(∀ζ^{k})(∀x^{k})(β^{n} < τ_{ι+1} & φ(β^{n}, ζ^{k}, i, x^{k}, p_{τ_{ι+1}}, σ̄(δ))

21 holds in N_{τ_{ι+1}} where φ(β^{n}, ζ^{k}, i, x^{k}, p_{τ_{ι+1}}, σ̄(δ)) is the Σ_{1}^{(k)}-statement

(ζ^{k} = h̃_{N_{τ_{ι+1}}}^{k+1} (i, ⟨x^{k}, p_{τ_{ι+1}}) & x ∈ [σ̄(δ)]^{<ω} & ζ^{k} < τ_{ι+1}) ⟶ ζ^{k} < β^{n}

22 and that σ̄ is sufficiently preserving to transfer these statements down to N̄. Let 23 τ̃ = sup(σ̄[τ̄]). Notice that α_{τ} < τ̃ < τ_{ι} where the first inequality follows from the 24 fact that α_{τ} ∈ rng(σ̄) and the second inequality is a consequence of our assumption 25 that (9) fails. Applying the interpolation lemma to σ̄ : N̄ → N_{τ_{ι+1}} we obtain an 26 acceptable J-structure Ñ for the same language as N̄ and N_{τ_{ι+1}}, a Σ_{0}^{(n)}-preserving 27 embedding σ̃ : N̄ → Ñ cofinal at the n-th level that extends σ̄ ↾ τ̄ and a Σ_{0}^{(n)}- 28 preserving embedding σ′ : Ñ → N_{τ_{ι+1}} such that σ′ ↾ τ̃ = id and σ′ ∘ σ̃ = σ̄. These 29 preservation properties of σ̃ and σ′ follow from the fact that h̃n+1 ; N̄(τ̄ ∪ {p̄}) = N̄; 30 this fact moreover yields that h̃n+1 ; Ñ(τ̃ ∪ {p̃}) = Ñ where p̃ = σ̃(p̄). Since there

31 is no good Σ_{1}^{(k)}(N̄)-function singularizing τ̄ for k < n we have σ̃(τ̄) = τ̃ hence 32 σ′(τ̃) = σ̄(τ̄) = τ_{ι}; see the proof of Theorem 4.8 above for more details concerning 33 the application of the interpolation lemma. Since τ̃ < τ_{ι}, we can apply Lemma 4.2 34 to Ñ, N_{τ_{ι}}, σ′, σ_{τ_{ι},τ_{ι+1}} and N_{τ_{ι+1}} in place of N^{∗}, N̄, σ^{∗}, σ̄ and N_{τ} to conclude that 35 rng(σ′) ⊆ rng(σ_{τ_{ι},τ_{ι+1}}). It follows that h̃_{τ_{ι+1}}({ξ_{ι}^{τ} } ∪ {p_{τ_{ι+1}}}) ⊆ rng(σ_{τ_{ι},τ_{ι+1}}) hence 36 h̃_{τ} ({ξ_{ι}^{τ} }∪{p_{τ} }) = σ_{τ_{ι+1},τ} [h̃_{τ_{ι+1}}({ξ_{ι}^{τ} }∪{p_{τ_{ι+1}}})] ⊆ rng(σ_{τ_{ι+1},τ} ∘σ_{τ_{ι},τ_{ι+1}}) = rng(σ_{τ_{ι},τ}) 37 where the equality on the left follows from (8) and the definition of τ_{ι+1}. The 38 resulting inclusion contradicts the requirement imposed on ξ_{ι}^{τ} by definition and 39 thereby completes the proof of (9).

1 We are now ready to complete the proof of (a). Let τ̄ = τῑ be as above. If β < τ̄, 2 pick ι^{∗} < ῑ such that τ_{ι^{∗}} ∈ B_{τ}^{∗} ∩ τ̄ and β < τ_{ι^{∗}}. Then

h̄τ̄(ξτ ; ῑ ∪ {α_{τ} , τ̄} ∪ {p̄τ̄}) ⊇ h̄τ̄({ξ_{ι^{∗}}^{τ}, α_{τ} , τ̄} ∪ {p̄τ̄})
⊇ σ̄_{τ_{ι+1},τ̄}[h̃_{τ_{ι^{∗}+1}}({ξ_{ι^{∗}}^{τ}, α_{τ} , τ_{ι^{∗}+1}} ∪ {p_{τ_{ι^{∗}+1}}})]

3 and the last set above has some elements in the interval [β, τ̄) by (9). The second 4 inclusion above follows by the fact that σ̄_{τ_{ι^{∗}+1},τ̄} is Σ_{0}^{(n)}-preserving, whereas the first 5 one is trivial. This completes the proof of (a). 6 Now turn to the proof of (b). Let τ̄ ∈ C_{τ}^{∗} ∩ B_{τ}^{∗} and ῑ < ι_{τ} be such that τ̄ = τῑ. 7 Then B_{τ}^{∗} ∩ τ̄ = B_{τ}^{∗} by Lemma 4.5(a). To prove (b), we show by induction on ι < ῑ 8 that τ̄_{ι} = τ_{ι} for each ι ≤ ῑ. Since the induction at limit steps ι is trivial and τ̄_{0} = τ_{0} 9 by coherency, the proof can be obviously reduced to showing that ξτ̄ ; ι = ξ_{ι}^{τ} and 10 τ̄_{ι+1} = τ_{ι+1} whenever ι < ῑ. Without loss of generality we can assume that τ̄_{ι′} = τ_{ι′} 11 for all ι′ ≤ ι and ξτ̄ ; ι′ = ξ_{ι′}^{τ} for all ι′ < ι. Since ι + 1 ≤ ῑ, from (8) we obtain

h̃_{τ} ({ξ} ∪ {p_{τ} }) = σ_{τ̄,τ} [h̃τ̄[({ξ} ∪ {p_{τ} })]

12 whenever ξ < ξτ ; ῑ. It follows that h̃τ̄({ξ} ∪ {pτ̄})] ⊆ rng(σ̄_{τ_{ι},τ̄}) just in case that 13 h̃_{τ} ({ξ} ∪ {p_{τ} }) ⊆ rng(σ̄_{τ_{ι},τ}) for each such ξ; here we apply σ_{τ̄,τ} to both sides of the 14 first inclusion. Hence ξ_{ι}^{τ} is the least ξ with the property h̃τ̄({ξ}∪{pτ̄})] ⊆ rng(σ̄_{τ_{ι},τ̄}), 15 that is, ξ_{ι}^{τ} = ξτ̄ ; ι. The same kind of argument yields that τ_{ι+1} is the least element

16 τ′ of B∗ ; τ̄ = B_{τ}^{∗} ∩ τ̄ such that h̃τ̄({ξτ̄ ; ι} ∪ {pτ̄}) ⊆ rng(σ_{τ′,τ̄}), so τ̄_{ι+1} = τ_{ι+1}. This

17 completes the proof of (b). 18 Regarding (c), it suffices to show that the set Z_{τ} = {ξ_{ι}^{τ} | ι < ι_{τ} } is bounded in τ. 19 As we have seen that the assignment ι ↦ ξ_{ι}^{τ} is strictly monotonic, we conclude that 20 otp(C_{τ}) = otp(Z_{τ}) < τ. Assume for the contradiction that Z_{τ} is unbounded in 21 τ. As τ has uncountable cofinality, the intersection C_{τ} ∩ lim(Z_{τ}) is nonempty, and 22 in fact is closed unbounded in τ. Pick ι < ι_{τ} with τ_{ι} is in this intersection. Then 23 Z_{τ} ∩ τ_{ι} = {τῑ | ῑ < ι} is unbounded in τ_{ι} by the choice of ι and h̃_{τ} ({ξ} ∪ {p_{τ} }) ⊆ 24 rng(σ_{τ_{ι},τ}) for all ξ < τ_{ι}, as follows directly from the definition of τ_{ι}. Since τ_{ι} is 25 primitive recursively closed, we conclude that h̃_{τ} (τ_{ι} ∪ {p_{τ} }) ⊆ rng(σ_{τ_{ι},τ}), which 26 implies that τ ∩ h̃_{τ} (τ_{ι} ∪ {p_{τ} }) = τ_{ι} > α_{τ}. This contradicts the properties of α_{τ}. 27 ⊓ ⊔(Lemma 4.11)

28 Remark 4.12. With some extra effort, we could obtain a stronger conclusion 29 [τ_{ι}, τ_{ι+1}) ∩ h̃_{τ_{ι+1}}({ξ_{ι}^{τ} , α_{τ} , τ_{ι+1}} ∪ {p_{τ_{ι+1}}}) ≠ ∅ instead of (9).

30 We we are now ready to give the definition of the global sequence ⟨C_{τ} | τ ∈ 𝒮^{0}⟩. 31 By Lemma 4.6 and Lemma 4.11, if τ has uncountable cofinality then a tail-end of 32 C_{τ}^{∗} is contained in B̂_{τ}; see (4). By Lemma 4.9, B̂_{τ} agrees with B_{τ} on a tail-end, 33 so a tail-end of C_{τ}^{∗} is actually contained in B_{τ}^{∗} ⊇ B_{τ}. For each τ ∈ 𝒮^{0} let γ_{τ} be the 34 least element γ of C_{τ}^{∗} ∪ {τ} such that C_{τ}^{∗} − γ ⊆ B_{τ}^{∗}. We let

(10)  C_{τ} = C_{τ}^{∗} − γ_{τ} .

35 C_{τ} is obviously a closed subset of τ and is unbounded in τ whenever τ has un-36 countable cofinality, as follows immediately from the definition of C_{τ}. It is also 37 obvious that otp(C_{τ}) ≤ otp(C_{τ}^{∗}) < τ, as follows from Lemma 4.11(c). Regarding 38 the coherency, if τ̄ ∈ C_{τ} then τ̄ ∈ C_{τ}^{∗} ∩ B_{τ}^{∗} hence C∗ ; τ̄ = C_{τ}^{∗} ∩ τ̄ by Lemma 4.11(b),

39 so γτ̄ = γ_{τ}. It follows that Cτ̄ = C∗ ; τ̄ − γτ̄ = (C_{τ}^{∗} ∩ τ̄) − γ_{τ} = (C_{τ}^{∗} − γ_{τ}) ∩ τ̄ = C_{τ} ∩ τ̄.

40 This completes the construction of the global square sequence on 𝒮^{0}.

1 Remark 4.13. As mentioned immediately above Definition 4.1, there is a way of 2 defining the sets B_{τ} so that these sets will be closed on a tail-end. Together with 3 α_{τ} in Definition 4.1 we consider the monotonic sequence of ordinals ⟨α_{τ}^{i} | i < ω⟩ 4 defined inductively by equalities α_{τ}^{0} = α_{τ} + 1 and α_{τ}^{i+1} = sup(τ ∩ h̃_{τ} (α_{τ}^{i} ∪ {p_{τ} }). 5 It is not hard to see that there is some k_{τ} ∈ ω such that ⟨α_{τ}^{i} | i < k_{τ} + 1⟩ is 6 strictly increasing and α_{τ}^{i} = τ for all i > k_{τ}. The amendment to the definition of 7 B_{τ} consists in adding requirements to Definition 4.3 that kτ̄ = k_{τ} and αk_{τ} ; τ̄ = α_{τ}^{k_{τ}}. 8 This approach to the construction makes certain parts of the proof simpler, but 9 other parts more complicated. In general, the two approaches yield constructions of 10 roughly the same level of complexity.

11 We now proceed with the construction of C_{τ} for τ ∈ 𝒮^{1}. Recall the basic notation 12 for objects associated with such τ introduced at the beginning of this section. Since 13 N_{τ} is a singularizing structure for τ that is exact for τ, Lemma 3.7 guarantees that 14 M_{τ} is a singularizing structure for τ. Also, q_{τ} = p_{M_{τ}} if M_{τ} is a protomouse, as 15 follows from Lemma 3.11, and q_{τ} = d_{τ} if M_{τ} is a premouse. Of course, if M_{τ} is a 16 premouse then M_{τ} is pluripotent. In either case, M_{τ} is sound and solid with respect 17 to the language for coherent structures; if M_{τ} is a premouse, this is equivalent to 18 saying that M_{τ} is Dodd sound and Dodd solid. 19 Certain aspects of the construction become simpler for τ ∈ 𝒮^{1}. One such aspect 20 involves ordinals α_{τ} which we used to show that the embeddings between singular-21 izing structures are not cofinal at the level n_{τ} and which played an important role 22 in the proof that C_{τ} is closed and has small order type. Recall the definition of 23 Skolem functions h_{τ}^{(μ_{τ} ,q_{τ})} from the beginning of this section.

24 Lemma 4.14. Let τ ∈ 𝒮^{1} and α < τ be such that τ ∩ h_{τ} (α ∪ {q_{τ} }) = α. Then 25 α = 0.

26 Proof. We first observe that if α satisfies the assumptions of the lemma and 0 < α 27 then τ ∩h̃_{τ} (α∪{p_{τ} }) = α. This is clear if M_{τ} = N_{τ}, so let us focus on the case where 28 M_{τ} is a protomouse. Notice that if 0 < α then α is a limit cardinal and μ_{τ} < α so 29 actually ϑ_{τ} < α. Given x ∈ [α]^{<ω} and i ∈ ω, assume ζ = h̃_{τ} (i, ⟨x, p_{τ} ⟩) < τ. Then 30 {ζ} is a singleton that is Σ_{1}^{(n_{τ})}-definable^{3} over N_{τ} in the parameters x, i and p_{τ}, so 31 by Lemma 3.7(a) {ζ} is Σ_{1}-definable over M_{τ} in the parameters x, i, ϑ_{τ} and q_{τ}. It 32 follows that ζ = h_{τ} (j, ⟨x ∪ {ϑ_{τ} }, q⟩) for some j ∈ ω. Then ζ < α, as ϑ_{τ} < α. 33 We next show that α_{τ} ≤ μ_{τ}. If ωϱ_{τ} ∩h̃_{τ} (α_{τ} ∪{p_{τ} }) is cofinal in ωϱ_{τ} then (α_{τ} , ∅) 34 is a divisor for τ. Because the second component in this divisor is ∅, this divisor is 35 strong, so by maximality of α_{τ} we have that (α_{τ} , ∅) = (μ_{τ} , q_{τ}) hence α_{τ} = μ_{τ}. If 36 ωϱ_{τ} ∩h̃_{τ} (α_{τ} ∪{p_{τ} }) is bounded in ωϱ_{τ} then μ_{τ} > α_{τ} because of (c) in Definition 3.5. 37 Now assume that α > 0 satisfies the assumptions of the lemma. By the above 38 results α_{τ} ≤ μ_{τ} < α and τ ∩ h̃_{τ} (α ∪ {p_{τ} }) = α, which is impossible. It follows that 39 α = 0. ⊓ ⊔(Lemma 4.14)

40 Following the standard strategy, we next define the sets B_{τ} for τ ∈ 𝒮^{1}.

41 Definition 4.15. Let τ ∈ 𝒮^{1}. The set B_{τ} consists of all τ̄ ∈ τ ∩ 𝒮^{1} for which 42 there is a map σ : Mτ̄ → M_{τ} that is Σ_{0}-preserving with respect to the language of 43 coherent structures and such that: 44 (a) σ ↾ τ̄ = id ↾ τ̄ and σ(τ̄) = τ.

3Recall that N_{τ} is exact for τ

1 (b) σ(qτ̄) = q_{τ}. 2 (c) If β ∈ q_{τ} then there is a generalized witness Q_{τ}^{∗}(β) = ⟨Q_{τ} (β), t_{τ} (β)⟩ for β 3 with respect to M_{τ} and q_{τ} satisfying Q_{τ}^{∗}(β) ∈ rng(σ).

4 This definition obviously corresponds to (4) in Definition 4.3. Clauses (1) and 5 (2) do not apply here for obvious reasons and (3) can be omitted because of 6 Lemma 4.14. Since τ ∈ M_{τ} whenever τ ∈ 𝒮^{1}, the second part in (a) always 7 makes sense. The map σ in the above definition is unique and we denote it by σ_{τ̄,τ}^{∗}, 8 as “σ_{τ̄,τ} ” was already reserved. The following lemma summarizes basic properties 9 of B_{τ} and is an obvious analogue to Lemma 4.4.

10 Lemma 4.16. Let τ^{∗} < τ̄ be two elements of B_{τ}. 11 (a) σ_{τ̄,τ}^{∗} is not Σ_{1}-preserving, and therefore is not cofinal. 12 (b) rng(σ_{τ^{∗},τ}^{∗}) ⊆ rng(σ_{τ̄,τ}^{∗}) and sup(On ∩ rng(σ_{τ^{∗},τ}^{∗}) < sup(On ∩ rng(σ_{τ̄,τ}^{∗})). 13 (c) (σ_{τ̄,τ}^{∗})^{−1} ∘ σ_{τ^{∗},τ}^{∗} = σ_{τ^{∗},τ̄}^{∗}, so σ_{τ^{∗},τ̄}^{∗} exists. 14 (d) σ_{τ^{∗},τ̄}^{∗} : M_{τ^{∗}} → Mτ̄ is not cofinal. 15 (e) σ_{τ^{∗},τ̄}^{∗} witnesses Definition 4.15 with (τ^{∗}, τ̄) in place of (τ̄, τ). 16 (f) B_{τ} ∩ τ̄ = Bτ̄ − min(B_{τ}).

17 Proof. The proof is virtually identical with the proof of Lemma 4.2. One again 18 uses Lemma 4.2 and Lemma 4.14 to prove (a) and (b). ⊓ ⊔(Lemma 4.16)

19 We define the set B_{τ}^{∗} and maps σ_{τ^{∗},τ̄}^{∗} : M_{τ^{∗}} → Mτ̄ for τ^{∗} ≤ τ̄ from B∗ ; τ̄ ∪ {τ} as 20 before.

21 Lemma 4.17. Let τ̄ ∈ B_{τ}^{∗} ∪ {τ}, τ′ < τ^{∗} be elements of B_{τ}^{∗} ∩ τ̄ and n = n_{τ}. Then:

22 (a) B∗ ; τ̄ = B_{τ}^{∗} ∩ τ̄.

23 (b) σ_{τ^{∗},τ̄}^{∗} exists, is Σ_{0}-preserving, σ_{τ^{∗},τ̄}^{∗} ↾ τ^{∗} = id and σ_{τ^{∗},τ̄}^{∗}(τ^{∗}, q_{τ^{∗}}) = (τ̄, qτ̄). 24 (c) σ_{τ^{∗},τ̄}^{∗} is not Σ_{1}-preserving, and therefore is not cofinal. 25 (d) rng(σ_{τ′,τ̄}^{∗}) ⊆ rng(σ_{τ^{∗},τ̄}^{∗}) and sup(On ∩ rng(σ_{τ′,τ̄}^{∗})) < sup(On ∩ rng(σ_{τ^{∗},τ̄}^{∗})). 26 (e) σ_{τ^{∗},τ̄}^{∗} ∘ σ_{τ′,τ^{∗}}^{∗} = σ_{τ′,τ̄}^{∗}.

27 Proof. Virtually identical to the proof of Lemma 4.5. Note also that (c) can be 28 viewed as a consequence of Lemma 4.14. ⊓ ⊔(Lemma 4.17)

29 Notice that μτ̄ = μ_{τ} and | qτ̄ | = | q_{τ} | for all τ̄ ∈ B_{τ}^{∗}. Clause (d) in Lemma 4.17 30 yields the following monotonicity property of the sequence ⟨ϑτ̄ | τ̄ ∈ B_{τ}^{∗} ∪ {τ}⟩:

(11)  τ^{∗} < τ̄ ⟶ ϑ_{τ^{∗}} < ϑτ̄.

31 This immediately implies that otp(B_{τ}^{∗}) ≤ ϑ_{τ} ≤ μ_{τ}^{+} < τ. Unlike the situation in 32 the case where τ ∈ 𝒮^{0} we will be further able to prove that B_{τ}^{∗} is closed on a 33 tail-end, which will make it possible to define C_{τ} to be the canonical tail-end of B_{τ}^{∗} 34 and thereby simplify the construction. There are, of course, new complications in 35 the present situation related to issues with canonical divisors. Similarly as in the 36 previous case we introduce the sets B̂τ.

37  B̂_{τ} is the set of all ordinals τ̄ ∈ τ ∩ 𝒮^{1} satisfying the following.

38 (a) Nτ̄ has a strong divisor (μ_{τ} , q_{τ̄,τ}) such that, letting M_{τ̄,τ} = Nτ̄(μ_{τ} , q_{τ̄,τ}), 39 there is a Σ_{0}-preserving map σ̂_{τ̄,τ} : N_{τ̄,τ} → M_{τ} satisfying the require-40 ments from Definition 4.15 with q_{τ̄,τ} in place of qτ̄. 41 (b) 0 is the only ordinal α < τ̄ is such that h_{M_{τ̄,τ}} (α ∪ {q_{τ̄,τ} }) = α.

1 Obviously B_{τ} ⊆ B̂_{τ}. If τ̄ ∈ B_{τ} then q_{τ̄,τ} = qτ̄. But B̂_{τ} is also allowed to contain 2 elements τ̄ for which some protomouse associated with Nτ̄, that is not the canon-3 ically chosen one, embeds into M_{τ}. The definition of B̂_{τ} makes use of the fact 4 that these protomice are uniquely determined by τ and τ̄. Our aim is to show that 5 on a tail-end of B̂_{τ}, these protomice agree with the canonical ones. Notice that 6 | q_{τ̄,τ} | = | q_{τ} | for all τ̄ ∈ B̂_{τ}.

7 Lemma 4.18. If τ ∈ 𝒮^{1} has uncountable cofinality then B̂_{τ} is unbounded in τ.

8 Proof. We combine arguments from the proof of Lemma 3.10 in [14] and that of 9 Lemma 4.11 above. Given some τ′ < τ, we are again looking for some τ̃ ∈ B̂_{τ} 10 such that τ′ ≤ τ̃. By taking elementary hulls and collapsing, we find a countable 11 coherent structure M̄, an elementary map σ : M̄ → M_{τ} that has τ′, τ, q_{τ} and some 12 generalized solidity witnesses Q_{τ}^{∗}(β) for each β ∈ q_{τ} in its range. We then let τ̄ be 13 the σ-preimage of τ and τ̃ = sup(τ ∩ σ′′τ̄); as before τ̃ < τ since τ is not ω-cofinal. 14 Also, τ̃ is a limit cardinal in the sense of L[E]. Our aim is to show that τ̃ ∈ B̂_{τ}. 15 The interpolation lemma gives us an acceptable structure M̃ together with Σ_{0}- 16 preserving maps σ̃ : M̄ → M̃ and σ′ : M̃ → M_{τ} such that σ′ ∘ σ̃ = σ. The map σ̃, 17 being a fine pseudoultrapower map, is cofinal, σ′ ↾ τ̃ = id ↾ τ̃ and σ̃(τ̃) = τ. Since 18 q_{τ} ∈ R_{M_{τ}} and τ = ωϱ_{M_{τ}}^{1} , (all fine structure here is done with respect to the language 19 of coherent structures), the elementarity of σ guarantees that q̄ ∈ RM̄ and ωϱ1 ; M̄ = τ̄.

20 The fact that σ̃ is a pseudoultrapower map then yields q̃ =^{def} σ̃(q̄) ∈ RM̃ and ωϱ1 ; M̃ = 21 τ̃. Finally, since rng(σ) ⊆ rng(σ′) and σ′(q̃) = q_{τ}, the generalized witness Q_{τ}^{∗}(σ′(β)) 22 is in the range of σ′ for each β ∈ q̃, and since σ′ is Σ_{0}-preserving, its σ′-preimage 23 is a generalized witness for β with respect to M̃ and q̃. By Corollary 1.12.4 in [18], 24 q̃ = pM̃, and M̃ is sound and solid. The fact that σ̃ is cofinal also yields that M̃ is a 25 coherent structure with ϑM̃ = sup(ϑ_{τ} ∩σ̃[ϑτ̄]). The map σ′ is Σ_{0}-preserving but not 26 cofinal, as otherwise τ̃ = τ ∩h_{τ} (τ̃∪{p_{τ} }) which would contradict Lemma 4.14. This 27 verifies the assumptions of the condensation lemma for protomice, Lemma 3.14. We 28 conclude that Ñ = N(M̃) is a proper level of M hence a level of L[E] and (μ, q̃) is 29 a divisor for Ñ such that M̃ = Ñ(μ, q̃). Since M_{τ} is a singularizing structure for τ 30 there is some δ < τ such that τ ∩ h_{τ} (δ ∪ {q_{τ} }) is cofinal in τ. Then τ̄ ∩ hM̄(δ̄∪ {q̄}) 31 is cofinal in τ̄ where q̄ = σ^{−1}(q_{τ}) and δ̄ = σ^{−1}[δ]. As σ̃ maps τ̄ cofinally into τ̃, 32 we conclude that δ̃ ∩ hM̃(δ ∪ {q̃}) is cofinal in τ̃, so M̃ is a singularizing structure 33 for τ̃ with n(τ̃, M̃) = 0. By the second conclusion in Lemma 3.14 and the above, 34 Ñ = Nτ̃ and is exact for τ̃. 35 We next verify that (μ_{τ} , q̃) is a strong divisor for Nτ̃. This can be done using an 36 argument similar to that in the proof of Lemma 3.10 in [14]; however, we present a 37 different argument that avoids the use of Fodor’s pressing down lemma. Recall the 38 notion of a closed ordinal from Definition 3.18. The goal is to show:

(12) There are cofinally many ϑ^{∗} < ϑ_{τ} that are closed in M_{τ} relative to q_{τ}.

39 From (12) we conclude that there are cofinally many ordinals ϑ^{∗} < ϑM̄ that are 40 closed in M̄ relative to q̄. Since σ̃ maps ϑM̄ cofinally into ϑM̃, there are cofinally 41 many ordinals ϑ^{∗} < ϑM̃ that are closed in M̃ relative to q̃. But then ϑM̃ itself is 42 closed in M̃ relative to q̃, as in any coherent structure M, any limit of ordinals that 43 are closed in M relative to a fixed parameter is itself closed in M relative to the 44 same parameter. By Lemma 3.19, (μ_{τ} , q̃) is strong as desired.

1 To see (12), fix arbitrary ϑ_{0} < ϑ_{τ} larger than μ_{τ}. Since μ_{τ} is the largest cardinal 2 in J_{ϑ_{τ}}^{E}, there is a function g ∈ J_{ϑ_{τ}}^{E} that maps μ_{τ} surjectively onto ^{μ_{τ}} 𝒫(μ_{τ}) ∩ J_{ϑ_{0}}^{E}. 3 This function can be viewed as a function g^{∗} : μ_{τ} → 𝒫(μ_{τ}) where

g^{∗}(ζ) = {≺η, η′, η′′≻ | η′′ ∈ g(η)(≺ζ, η′≻)}.

4 Now if ξ, ξ′ < μ_{τ} then

F_{τ} (g(ξ))(q_{τ} , ξ′) ∩ μ_{τ} = π_{M_{τ}} (g(ξ))(q_{τ} , ξ′) ∩ μ_{τ}
= π_{M_{τ}} (g)(ξ)(q_{τ} , ξ′) ∩ μ_{τ}
= {ξ′′ < μ_{τ} | ≺ξ, ξ′, ξ′′≻ ∈ π_{M_{τ}} (g^{∗})(q_{τ})}
= {η′′ < μ_{τ} | ≺ξ, ξ′, η′′≻ ∈ F_{τ} (g^{∗})(q_{τ})},

5 and it is clear that F (g^{∗})(q_{τ}) can be replaced by a^{∗} = F (g^{∗})(q_{τ}) ∩ μ_{τ} on the 6 bottom line above. Since ϑ_{τ} is closed in M_{τ} relative to q_{τ} the set a^{∗} is an element 7 of J_{ϑ_{τ}}^{E}, so we can pick some ϑ_{1} < ϑ_{τ} larger than ϑ_{0} such that a^{∗} ∈ J_{ϑ_{1}}^{E}. Then 8 F (f)(q_{τ} , ξ′) ∩ μ_{τ} ∈ J_{ϑ_{1}}^{E} whenever f : μ_{τ} → 𝒫(μ_{τ}) is in J_{ϑ_{0}}^{E} and ξ′ < μ_{τ}, as 9 F (f)(q_{τ} , ξ′) ∩ μ_{τ} = {ξ′′ < μ_{τ} | ≺ξ, ξ′, ξ′′≻ ∈ a^{∗}} where ξ is such that f = g(ξ). 10 Repeating this argument, we inductively construct a strictly increasing sequence 11 ⟨ϑ_{i} | i ∈ ω⟩ of ordinals below ϑ_{τ} such that for every f : μ_{τ} → 𝒫(μ_{τ}) in J_{ϑ_{i}}^{E} and 12 ξ′ < μ_{τ} the set F (f)(q_{τ} , ξ′) ∩ μ_{τ} is in J_{ϑ_{i+1}}^{E}. Let ϑ^{∗} = sup({ϑ_{i} | i ∈ ω}). Then 13 ϑ^{∗} < ϑ_{τ} since ϑ_{τ} has uncountable cofinality. By construction, ϑ^{∗} is closed in M_{τ} 14 relative to q_{τ}. 15 It remains to prove that τ̃ satisfies (b) in the definition of B̂_{τ}. Assume 0 < α < τ̃. 16 Let ᾱ = σ^{−1}[α]. By Lemma 4.14, there is some finite x ⊆ α and some i ∈ ω such 17 that α ≤ h_{τ} (i, ⟨x, q_{τ} ⟩) < τ and since σ is fully elementary, there is some finite 18 x̄ ⊆ ᾱ satisfying ᾱ ≤ hM̄(i, ⟨x̄, q̄⟩) < τ̄. Since σ̃ ↾ τ̄ = σ ↾ τ̄, we conclude that 19 α ≤ σ̃(ᾱ) ≤ hM̃(i, ⟨σ̃(x̄), q̃⟩) < τ̃. Obviously σ̃(x̄) = σ(x̄) ⊆ α. ⊓ ⊔(Lemma 4.18)

20 Lemma 4.19. Assume τ ∈ 𝒮^{1}. Then B̂_{τ} is closed.

21 Proof. We assume without loss of generality that B̂_{τ} is unbounded in τ. Given a 22 limit point τ̃ < τ of B̂, we prove that τ̃ ∈ B̂_{τ}. Following the strategy from the case 23 where τ ∈ 𝒮^{0}, we form the direct limit of the diagram

⟨M_{τ̄,τ} , σ_{τ̄,τ̄′}^{∗} | τ̄ < τ̄′ & τ̄, τ̄′ ∈ B̂_{τ} ∩ τ̃⟩.

24 We obtain the transitivized direct limit structure M̃ together with the direct limit 25 maps σ̃τ̄ : M_{τ̄,τ} → M̃ and the canonical embedding σ̃ : M̃ → M_{τ} such that 26 σ̃ ∘ σ̃τ̄ = σ_{τ̄,τ}^{∗} for all τ̄ ∈ B̂_{τ} ∩ τ̃. All these maps are Σ_{0}-preserving, so M̃ is a 27 coherent structure, as this is a Π_{2}-property that is true about all structures in the 28 direct limit diagram. Also σ̃τ̄ ↾ τ̄ = id ↾ τ̄, σ̃τ̄(τ̄) = τ̃, σ̃ ↾ τ̃ = id ↾ τ̃ and σ̃(τ̃) = τ. 29 We let q̃ = σ̃τ̄(q_{τ̄,τ}); this object is again independent of the choice of τ̄ ∈ B̂_{τ} ∩ τ̃. 30 Our intention is to show that τ̃ is a singular cardinal in L[E], M̃ = Nτ̃(μ_{τ} , q̃) and 31 σ̃ witnesses that τ̃ ∈ B̂_{τ}. 32 Since M_{τ̄,τ} is sound whenever τ̄ ∈ B̂_{τ} ∩ τ̃, the Σ_{0}-elementarity of the maps 33 σ̃_{τ} guarantees that hM̃(τ̃ ∪ {q̃}) = M̃, and so ωϱ1 ; M̃ = τ̃ and q̃ ∈ RM̃. Since 34 rng(σ̃) ⊇ rng(σ̂_{τ̄,τ}) and rng(σ̂_{τ̄,τ}) contains a generalized witness Q^{∗}(β) for β with 35 respect to M_{τ} and q_{τ} whenever β ∈ q_{τ}, any such witness is in the range σ̃. By 36 the Σ_{0}-elementarity of σ̃, the object σ̃^{−1}(Q^{∗}(σ̃(β̃))) is a generalized witness for 37 β̃ whenever β̃ ∈ q̃. By Corollary 1.12.4 in [18], q̃ = pM̃ and the structure M̃ is 1 solid and sound. Finally, the map σ̃ is not Σ_{1}-preserving, and therefore not cofinal. 2 Otherwise h_{τ} (τ̃ ∪ {q_{τ} }) = σ̃[hM̃(τ̃ ∪ {q̃})], so τ ∩ h_{τ} (τ̃ ∪ {q_{τ} }) = τ̃, contradicting 3 Lemma 4.14. This verifies the assumptions of Lemma 3.14. It follows that Ñ = 4 N(M̃) is a level of N_{τ} hence a level of L[E] and M̃ = Ñ(μ_{τ} , q̃). 5 As a next step we prove that M̃ is a singularizing structure for τ̃ with n(τ̃, M̃) = 6 0. Then again by Lemma 3.14, Ñ = Nτ̃ and is exact for τ̃. Let τ̄ ∈ B̂_{τ} ∩ τ̃, 7 ζτ̄ = sup(σ̃τ̄[On ∩ M_{τ̄,τ}]) and M̃τ̄ = ⟨S_{ζτ̄}^{E}, F̃ ∩ S_{ζτ̄}^{E}⟩ where F̃ is the top extender of 8 M̃. Then σ̃τ̄ when viewed as a map from M_{τ̄,τ} to M̃τ̄ is cofinal hence Σ_{1}-preserving. 9 By Lemma 4.14, τ̄ is the least ordinal τ′ smaller than τ̃ satisfying the condition 10 τ̃ ∩ hM̃τ̄(τ′ ∪ {q̃}) = τ′. This observation enables us to show that the map ϑτ̄ ↦ τ̄ 11 is a singularizing function for τ̃ that is Σ_{1}-definable over M̃. Notice that ζτ̄ is fully 12 determined by ϑτ̄ as the least ordinal ζ such that F̃(x) ∈ S_{ζ}^{E} for all x ∈ 𝒫(μ_{τ})∩J_{ϑτ̄}^{E}. 13 Thus, letting ψ(w, u, v) be the Σ_{1}-formula that constitutes a functionally absolute 14 definition of a Σ_{1}-Skolem function, the map ϑτ̄ ↦ τ̄ can be defined as follows: ξ = τ̄ 15 just in case that M̃ satisfies the Σ_{1}-statement

(∃ζ, X, G, M, h)
• X = S_{ζ}^{E} & G = F ∩ X & M = ⟨X, G⟩ & ξ < τ̃
• (∀ζ̄ < ζ)(∀X̄ ∈ X)(∃x ∈ X)(∃y ∈ X)(⟨x, y⟩ ∈ G & y ∉ X̄)
• h = {⟨y, i, x⟩ ∈ X × ω × X | M ⊧ ψ[y, i, x]}
• (∀x ∈ [ξ]^{<ω})(∀i ∈ ω)(∀η < τ̃)η = h(i, ⟨x, q̃⟩) → η < ξ)
• (∀ξ′ < ξ)(∃x′ ∈ [ξ′]^{<ω})(∃i′ ∈ ω)(∃η′ < τ̃)[η′ = h(i′, ⟨x′, q̃⟩) & η ≥ ξ′)].

16 We now finish the proof that τ̃ satisfies (a) in the definition of B̂_{τ} and σ̃ = σ_{τ̃,τ} 17 it remains to verify that (μ_{τ} , q̃) is a strong divisor for Nτ̃. By Lemma 3.19, this is 18 equivalent to showing that ϑM̃ is closed in M̃ relative to q̃. We first observe that 19 ϑ_{N_{τ̄,τ}} is closed in M̃ relative to q̃ whenever τ̄ ∈ B̂_{τ} ∩ τ̃. Pick a map f : μ_{τ} → 𝒫(μ_{τ}) 20 from J_{ϑ_{Nτ̄,τ}}^{E} and an ordinal ξ < μ_{τ}. Letting F̃ and F_{τ̃,τ} be the top extender of M̃ 21 and M_{τ̃,τ} respectively, the facts that σ̃τ̄ is Σ_{0}-preserving and μ_{τ} < cr(σ̃τ̄) guarantee 22 that F̃(f)(q̃, ξ)∩μ_{τ} = F_{τ̃,τ} (f)(q_{τ̃,τ} , ξ)∩μ_{τ}. The set on the right side of this equality 23 is an element of J_{ϑ_{Nτ̄,τ}}^{E}, as (μ_{τ} , q_{τ̃,τ}) is a strong divisor for N_{τ̃,τ}. Finally since the 24 property of being strong relative to a fixed parameter is closed under limits and 25 ϑM̃ = sup{ϑ_{N_{τ̄,τ}} | τ̄ ∈ B̂_{τ} ∩ τ̃}, we conclude that ϑM̃ is closed relative to q̃ in M̃. 26 To complete the proof that τ̃ ∈ B̂_{τ} we have to verify that M̃ and q̃ satisfy (b) 27 in the definition of B̂_{τ}. Assume there is an ordinal α such that 0 < α < τ̃ and 28 τ̃ ∩ hM̃(α ∪ {q̃}) = α. Let τ̄ ∈ B̂_{τ} ∩ τ̃ be such that α < τ̄. Pick a finite x ⊆ α and 29 an i ∈ ω. If ζ = h_{M_{τ̄,τ}} (i, ⟨x, q_{τ̄,τ} ⟩) is defined and ζ < τ̄ then using the fact that 30 σ̃τ̄ is Σ_{0}-preserving we obtain hM̃(i, ⟨x, q̃⟩) = ζ < τ̃. By our assumption on α then 31 ζ < α. This means that τ̄ ∩ h_{M_{τ̄,τ}} (α ∪ {q_{τ̄,τ} }) = α, which contradicts the fact that 32 τ̄ ∈ B̂_{τ}. ⊓ ⊔(Lemma 4.19)

33 The last essential lemma in the treatment of the case τ ∈ 𝒮^{1} is an analogue to 34 Lemma 4.9.

35 Lemma 4.20. Assume τ ∈ 𝒮^{1} and B̂_{τ} is unbounded in τ. Then B̂_{τ} agrees with B_{τ} 36 on a tail-end. That is, there is some τ̂ < τ such that B̂_{τ} − τ̂ = B_{τ} − τ̂.

37 Proof. We have to show that (μ_{τ} , q_{τ̄,τ}) = (μτ̄, qτ̄) for all τ̄ in some tail-end of 38 B̂_{τ}. So suppose for a contradiction that this fails. Then for cofinally many τ̄ ∈ B̂_{τ}, 1 the strong divisor (μ_{τ} , q_{τ̄,τ}) fails to be the canonical one. Letting B̂′_{τ} be the set of 2 all such τ̄, it follows from Lemma 3.21 that μ_{τ} < μτ̄ and qτ̄ is a (not necessarily 3 proper) bottom segment of q_{τ̄,τ} whenever τ̄ ∈ B̂′_{τ}. As in the proof of Lemma 4.9, 4 we find a strictly monotonic sequence ⟨τ_{ι} | ι < γ⟩ cofinal in τ such that the ordinals 5 τ_{ι} are all in B̂′_{τ}, the sequence ⟨μ_{τ_{ι}} | ι < γ⟩ is (not necessarily strictly) monotonic and 6 q_{τ_{ι}} are of the same size for all ι < γ; say this size is m. Let μ = sup{μ_{ι} | ι < γ} and 7 q be the bottom segment of q_{τ} of size m. Obviously μ_{τ} < μ ≤ τ and σ̂_{τ_{ι},τ} (q_{τ_{ι}}) = q 8 for all ι < γ. 9 We first verify that (μ, q) is a divisor for N_{τ} whenever μ < τ. This is a straight-10 forward application of Lemma 3.10; an argument of this kind can also be found 11 in the proof of Lemma 3.12 in [14]. To make this text self-contained, we give the 12 argument. Let β = max(q) where max(∅) = τ. If ξ < μ and f : μ_{τ} → μ_{τ} are 13 such that ζ =^{def} F_{τ} (f)(q_{τ} − q, ξ) ≤ β, pick ι < γ large enough that ξ < μ_{τ_{ι}} and 14 f ∈ J_{ϑ_{τι,τ}}^{E} where ϑ_{τ_{ι},τ} = ϑ(M_{τ_{ι},τ}). Letting F_{τ_{ι},τ} be the extender F_{N_{τι,τ}} (μ_{τ} , q_{τ_{ι},τ}),

15 the fact that σ̂_{τ_{ι},τ} is Σ_{0}-preserving implies ζ_{τ_{ι}} =^{def} F_{τ_{ι},τ} (f)(q_{τ_{ι},τ} − q_{τ_{ι}}, ξ) ≤ β_{τ_{ι}} where 16 β_{τ_{ι}} = max(q_{τ_{ι}}) and max(∅) = τ_{ι}. Since (μ_{τ_{ι}}, q_{τ_{ι}}) is a divisor for N_{τ_{ι}}, we have 17 ζ_{τ_{ι}} < μ_{τ_{ι}}, so ζ = σ̂_{τ_{ι},τ} (ζ_{τ_{ι}}) = ζ_{τ_{ι}} < μ. 18 We next observe that q ≠ ∅. Assuming the contrary, we have q_{τ_{ι}} = ∅ for 19 some/any ι < γ. By the definition of a divisor, τ_{ι} ∩ h̃_{τ_{ι}}(μ_{τ_{ι}} ∪ {q_{τ_{ι}}}) = μ_{ι}. Since 20 μ_{τ_{ι}} > ϑ_{τ_{ι},τ}, Lemma 3.7(b) yields τ ∩ h_{M_{τι,τ}} (μ_{τ_{ι}} ∪ {q_{τ_{ι},τ} }) = μ_{τ_{ι}}, which contradicts 21 clause (b) in the definition of B̂_{τ}. 22 We are now heading towards the final contradiction. Notice that if μ < τ then 23 (μ, q) is a divisor for N_{τ} that is not strong. Using this for μ < τ and the fact 24 that h̃_{τ} (μ ∪ {p_{τ} }) = N_{τ} if μ = τ we show that (μ_{τ_{ι}}, q_{τ_{ι}}) cannot be strong after all, 25 which yields the desired contradiction. The proof makes use in a crucial way of the 26 following claim that describes how to transfer witnesses between N_{τ} and N_{τ_{ι},τ}. A 27 version of this claim was also used in [14], proof of Lemma 3.12.

28 Claim. Let μ ≤ β < min(q_{τ} − q) where min(∅) = λ_{τ}. Let further s_{τ} = p_{τ} − q if 29 N_{τ} is either passive or active with λ_{N_{τ}^{∗}} > μ_{τ} and s_{τ} = d_{τ} − q if N_{τ} is active with 30 λ_{N_{τ}^{∗}} = μ_{τ}. Similarly, for ι < γ let s_{τ_{ι}} = p_{τ_{ι}} − q_{τ_{ι}} if N_{τ_{ι}} is either passive or active 31 with λ_{N_{τι,τ}^{∗}} > μ_{τ} and s_{τ_{ι}} = d_{τ_{ι}} − q_{τ_{ι}} if N_{τ_{ι}} is active with λ_{N_{τι,τ}^{∗}} = μ_{τ}; here N_{τ_{ι},τ}^{∗} 32 denotes the structure N_{τ_{ι}}^{∗}(μ_{τ} , q_{τ_{ι},τ}). Assume that Q_{τ}^{∗}(β) ∈ J_{λ_{τ}}^{E} and x, x′ ∈ [μ]^{<ω} 33 and i, i′ ∈ ω are objects satisfying one of the following hypotheses.

34 (a) N_{τ} is either passive or active with λ_{N_{τ}^{∗}} > μ_{τ}, Q_{τ}^{∗}(β) = h̃_{τ} (i, ⟨x, p_{τ} ⟩) is a gen-35 eralized n_{τ} -witness for β with respect to N_{τ} and s_{τ} and β = h̃_{τ} (i′, ⟨x′, p_{τ} ⟩). 36 (b) N_{τ} is active with λ_{N_{τ}^{∗}} = μ_{τ} or N_{τ} = M_{τ}, Q_{τ}^{∗}(β) = h_{τ}^{∗}(i, ⟨x, d_{τ} ⟩) is a gener-37 alized Dodd witness for β with respect to N_{τ} and s_{τ} and β = h_{τ}^{∗}(i′, ⟨x′, d_{τ} ⟩). 38 Granting these assumptions there is an ι^{∗} such that if ι^{∗} ≤ ι < γ then β, Q_{τ}^{∗}(β) ∈ 39 rng(σ̂_{τ_{ι},τ}) and there are y, y′ ∈ [μ_{τ_{ι}}]^{<ω} and i, i′ ∈ ω such that the following is true. 40 Let β_{τ_{ι}} and Q_{τ_{ι}}^{∗} be the inverse image of β and Q_{τ}^{∗}(β) under σ̂_{τ_{ι},τ}, respectively.

41 (c) If N_{τ_{ι}} is either passive or active with λ_{N_{τι,τ}^{∗}} > μ_{τ} then Q_{τ_{ι}}^{∗} = h̃_{τ_{ι}}(i, ⟨y, p_{τ_{ι}}⟩) 42 is a generalized n_{τ_{ι}}-witness for β_{τ_{ι}} with respect to N_{τ_{ι}} and s_{τ_{ι}} and β_{τ_{ι}} = 43 h̃_{τ_{ι}}(i′, ⟨y′, p_{τ_{ι}}⟩). 44 (d) If N_{τ_{ι}} is active with λ_{N_{τι,τ}^{∗}} = μ_{τ} then Q_{τ_{ι}}^{∗} = h_{τ_{ι}}^{∗}(i, ⟨y, d_{τ_{ι}}⟩) a generalized 45 Dodd witness for β_{τ_{ι}} with respect to N_{τ_{ι}} and s_{τ_{ι}} and β_{τ_{ι}} = h_{τ_{ι}}^{∗}(i′, ⟨y′, d_{τ_{ι}}⟩).

1 Proof. For N_{τ} we have to consider three options: (i) N_{τ} is either passive or active 2 with λ_{N_{τ}^{∗}} > μ_{τ}, (ii) N_{τ} is active with λ_{N_{τ}^{∗}} = μ_{τ} and N_{τ} = M_{τ}. For N_{τ_{ι}} we only 3 have two options: (iv) N_{τ_{ι}} is either passive or active with λ_{N_{τι,τ}^{∗}} > μ_{τ} and (v) N_{τ_{ι}} 4 is active and λ_{N_{τι,τ}^{∗}} = μ_{τ}. The option N_{τ_{ι}} = M_{τ_{ι}} does not apply as N_{τ_{ι}} cannot 5 be a pluripotent premouse; this is a consequence of the fact that σ̂_{τ_{ι},τ} fails to be 6 cofinal. Thus, we have altogether six cases. As an example we treat the case where 7 N_{τ} is as in (i) and N_{τ_{ι}} is as in (v). The proofs in the remaining cases are similar. 8 By Lemma 3.12, the standard witness W_{M_{τ}}^{0,β,q_{τ} −q} is Σ_{0}-definable from Q_{τ}^{∗}(β), β 9 and ϑ_{τ} inside J_{λ_{τ}}^{E}. Moreover, this witness can be encoded into a set w ⊆ β in 10 a canonical way so that w is Σ_{0}-definable from w inside any transitive admissi-11 ble structure containing the respective objects. It follows that w is of the form 12 h̃_{τ} (k, ⟨z, p_{τ} ⟩) for some k ∈ ω where z = x ∪ x′ ∪ {ϑ_{τ} }. By Lemma 3.8, there 13 are functions f, f′ ∈ J_{ϑ_{τ}}^{E} such that w = F_{τ} (f)(q_{τ} , z) and β = F (f′)(q_{τ} , x′). Fix 14 some ι^{∗} such that z ⊂ μ_{τ_{ι^{∗}}} and f, f′ ∈ J_{ϑ_{τ_{ι^{∗}}}}^{E}. Such an ι^{∗} exists, as the ordi-15 nals μ_{τ_{ι}} and ϑ_{τ_{ι},τ} monotonically converge to μ and ϑ_{τ}, respectively. Now con-16 sider any ι ≥ ι^{∗}. Letting w_{ι} = F_{τ_{ι},τ} (f)(q_{τ_{ι},τ} , z) and β_{τ_{ι}} = F_{τ_{ι},τ} (f′)(q_{τ_{ι},τ} , x′) 17 where F_{τ_{ι},τ} denotes the extender F_{N_{τι,τ}} (μ_{τ} , q_{τ_{ι},τ}), obviously σ̂_{τ_{ι},τ} (β_{τ_{ι}}) = β and 18 σ̂_{τ_{ι},τ} (w_{ι}) = w, so σ̂_{τ_{ι},τ} (Q′_{τ_{ι}}) = W_{M_{τ}}^{β,q_{τ} −q}. The last equality here follows from the 19 uniformity of the coding used to produce w. Since σ̂_{τ_{ι},τ} is Σ_{0}-preserving, Q′_{τ_{ι}} is a 20 generalized witness for β_{τ_{ι}} with respect to M_{τ_{ι},τ} and q_{τ_{ι},τ} − q_{τ_{ι}}. By Lemma 3.8, 21 w_{ι} = h_{τ_{ι}}^{∗}(i_{2}, ⟨z ∪ {ξ}, q_{τ_{ι},τ} ⟩) and β_{τ_{ι}} = h_{τ_{ι}}^{∗}(i′, ⟨x′ ∪ {ξ′}, q_{τ_{ι},τ} ⟩) for some ξ, ξ′ < μ_{τ} 22 and i_{2}, i′ ∈ ω (r = ∅ here where r is the parameter as in Lemma 3.8). Due to 23 the uniformity of the coding, Q′_{τ_{ι}} is lightface definable from w_{ι} inside N_{τ_{ι}}, so there 24 is an i_{1} ∈ ω such that Q′_{τ_{ι}} = h_{τ_{ι}}^{∗}(i_{1}, ⟨z ∪ {ξ}, q_{τ_{ι},τ} ⟩). Applying Lemma 3.13, the 25 standard witness W_{M_{τι,τ}}^{β_{τι} ,q_{τι,τ} −q_{τι}} is Σ_{0}-definable in N_{τ_{ι}} from Q′_{τ_{ι}}, β_{τ_{ι}} and ϑ_{τ_{ι},τ}, so

26 by Lemma 3.11(e), the standard Dodd witness ^{∗}W _{N_{τι}}^{β_{τι} ,q_{τι,τ} −q_{τι}} is Σ_{0}-definable in N_{τ} 27 from the same parameters. Since q_{τ_{ι},τ} = d_{τ_{ι}} in this case, we have d_{τ_{ι},τ} − q_{τ_{ι}} = s_{τ_{ι}}. 28 Letting y = z ∪ {ξ, ξ′, ϑ_{τ_{ι},τ} } and y′ = x′ ∪ {ξ′}, obviously y, y′ ∈ [μ_{τ_{ι}}]^{<ω} and there 29 is an i ∈ ω such that

∗W _{N_{τι}}^{β_{τι} ,s_{τι}} = h_{τ_{ι}}^{∗}(i, ⟨y, d_{τ_{ι}}⟩)
β_{τ_{ι}} = h_{τ_{ι}}^{∗}(i′, ⟨y′, d_{τ_{ι}}⟩),

30 which is in fact a little bit more than we intended to prove. ⊓ ⊔(Claim)

31 Returning to the proof of the lemma, we are now ready to reach the final contra-32 diction by showing that for sufficiently large ι < γ, the canonical divisor (μ_{τ_{ι}}, q_{τ_{ι}}) 33 cannot be strong after all. Here we apply the above Claim. 34 For τ we have to consider five possibilities: (i) N_{τ} is either passive or active with 35 λ_{N_{τ}^{∗}} > μ_{τ}, (ii) N_{τ} is active with λ_{N_{τ}^{∗}} = μ_{τ} and (iii) N_{τ} = M_{τ}. If (i) applies and N_{τ} 36 is active then λ_{N_{τ}^{∗}(μ,q)} > μ, as follows directly by inspecting the associated hulls. 37 If (ii) applies than N_{τ} is active and either of the two options λ_{N_{τ}^{∗}(μ,q)} = μ and 38 λ_{N_{τ}^{∗}(μ,q)} > μ must be considered. The same is true if (iii) applies, that is, we again 39 have to consider two options. Hence five options have to be considered for τ. For 40 τ_{ι} we only have to consider two cases: either λ_{N_{τι}^{∗}} > μ_{τ_{ι}} or else λ_{N_{τι}^{∗}} = μ_{τ_{ι}}, as M_{τ_{ι}} 41 cannot be a premouse. This makes altogether ten cases; all these cases are treated 42 in a similar way. As an example we treat the case where N_{τ} is active, λ_{N_{τ}^{∗}} = μ_{τ}, 43 λ_{N_{τ}^{∗}(μ,q)} > μ, N_{τ_{ι}} is active with λ_{N_{τι}^{∗}} = μ_{τ_{ι}}, hence λ_{N_{τι,τ}^{∗}} = μ_{τ}.

1 Let s_{τ} = d_{τ} − q. If μ = τ, let β = max(q) and Q_{τ}^{∗}(β) = ^{∗}W _{N_{τ}}^{β,s_{τ}}. We have 2 seen above that q ≠ ∅, so β exists. By the Dodd solidity of N_{τ}, the witness 3 Q_{τ}^{∗}(β) is an element of N_{τ} and by acceptability, Q_{τ}^{∗}(β) ∈ J_{λ_{τ}}^{E}. Since N_{τ} is Dodd 4 sound, we can find x, x′ ∈ [μ]^{<ω} and i, i′ ∈ ω such that Q_{τ}^{∗}(β) = h_{τ}^{∗}(i, ⟨x, d_{τ} ⟩) and 5 β = h_{τ}^{∗}(i′, ⟨x′, d_{τ} ⟩). Now assume that μ < τ. Let r_{τ} = p_{τ} − q. Since (μ, q) is a 6 divisor for N_{τ} and μ > μ_{τ}, this divisor cannot be strong. Letting r′ = π′^{−1}(r_{τ}) 7 where π′ : N′_{τ} (μ) → N_{τ} is the inverse to the Mostowski collapsing isomorphism 8 associated with the hull h̃_{τ} (μ ∪ {p_{τ} }) (see Definition 3.15), the parameter r′ is 9 a proper top segment of p_{N′_{τ} (μ)}. Notice that N′_{τ} (μ) is a premouse; this follows 10 from the fact that π′ is Σ_{1}-preserving^{4} with respect to the language of premice. 11 Since ωϱ_{N′_{τ} (μ)}^{1} = μ and π′ ↾ N′_{τ} (μ) = id, the first part of the condensation lemma 12 (Lemma 3.2) yields that N′_{τ} (μ) is solid. From all of this we conclude that there is 13 some β′ such that μ ≤ β < π′^{−1}(λ_{N_{τ}} (μ, q)) and W_{N′_{τ} (μ)}^{0,β′,r′} ∈ N′_{τ} (μ). Let β = π′(β′).

14 Since π′ is Σ_{1}-preserving, π′(W_{N′_{τ} (μ)}^{0,β′,r′}) is a generalized witness for β with respect 15 to N_{τ} and r_{τ} and this generalized witness is essentially a subset of β < λ_{N_{τ}} (μ, q), 16 modulo the natural uniform coding. Hence this witness is an element of J_{λ_{τ}}^{E}. Notice 17 that rng(π′) = h̃_{τ} (μ ∪ {p_{τ} }) = h_{τ}^{∗}(μ ∪ {d_{τ} }), as follows by Lemma 3.6. Hence 18 π′(W_{N′_{τ} (μ)}^{0,β′,r′}) = h_{τ}^{∗}(i_{1}, ⟨x_{1}, d_{τ} ⟩) and β = h_{τ}^{∗}(i′, ⟨x′, d_{τ} ⟩) for suitably chosen x_{1}, x′ ∈

19 [μ]^{<ω} and i_{1}, i′ ∈ ω. By Lemma 3.12, W_{N_{τ}}^{0,β,r} is Σ_{0}-definable from π′(W_{N′_{τ} (μ)}^{0,β′,r′}), β

20 and ϑ_{τ} inside J_{λ_{τ}}^{E}, so there are x ∈ [μ]^{<ω} and i ∈ ω satisfying W_{N_{τ}}^{0,β,r} = h_{τ}^{∗}(i, ⟨x, p_{τ} ⟩). 21 By (2), W_{N_{τ}}^{0,β,r} = ^{∗}W _{N_{τ}}^{β,s}, so we can let Q^{∗}(β) = ^{∗}W _{N_{τ}}^{β,s}. To summarize, in either 22 case μ = τ or μ < τ we proved that the hypothesis (b) in the above Claim is met. 23 Applying the Claim, we obtain some ι < γ and Q_{τ_{ι}}^{∗} ∈ N_{τ_{ι}} such that conclusion 24 in the Claim is satisfied. Since we are treating the case where N_{τ_{ι}} is active with 25 λ_{τ_{ι},τ} = μ_{τ}, conclusion (d) applies. It follows that Q_{τ_{ι}}^{∗} is a generalized witness for 26 β_{τ_{ι}} = σ̂_{τ_{ι},τ}^{−1} (β) with respect to N_{τ_{ι}} and s_{τ_{ι}} = d_{τ_{ι}} − q_{τ_{ι}}. Moreover, both Q_{τ_{ι}}^{∗} and β_{τ_{ι}} 27 are in the hull h_{τ}^{∗}(μ_{τ_{ι}} ∪ {d_{τ_{ι}}}). Arguing as in the previous paragraph, we conclude 28 that the standard Dodd witness ^{∗}W _{N_{τι}}^{β_{τι} ,s_{τι}} is in the same hull. Let N′_{τ_{ι}}(μ_{τ_{ι}}) be the 29 transitive collapse of h_{τ}^{∗}(μ_{τ_{ι}} ∪ {d_{τ_{ι}}}), π′_{τ_{ι}} : N′_{τ_{ι}}(μ_{τ_{ι}}) → N_{τ_{ι}} be the inverse to the 30 collapsing isomorphism and β′_{τ_{ι}}, Q′_{τ_{ι}} be the inverse image of β_{τ_{ι}},^{∗}W _{N_{τι}}^{β_{τι} ,s_{τι}} under 31 π′, respectively. Then Q′_{τ_{ι}} is a generalized Dodd witness for β′_{τ_{ι}} with respect to 32 N′_{τ_{ι}} and s′_{τ_{ι}} where β′_{τ_{ι}} and s′_{τ_{ι}} are the π′_{τ_{ι}}-preimages of β_{τ_{ι}} and s_{τ_{ι}}, respectively. 33 Since this generalized Dodd witness is an element of N′_{τ_{ι}}(μ_{τ_{ι}}), also the standard

34 Dodd witness ^{∗}W _{N′_{τι} (μ_{τι})}^{β′_{τι} ,s′_{τι}} is in N′_{τ_{ι}}(μ_{τ_{ι}}); denote this standard Dodd witness by ^{∗}W.

35 Since μ_{τ_{ι}} ≤ β′_{τ_{ι}}, we conclude that E_{top}^{N′_{τι} (μ_{τι})} | μ_{τ_{ι}} = E_{top}^{^{∗}W} | μ_{τ_{ι}} ∈ N′_{τ_{ι}}(μ_{τ_{ι}}), so by 36 Lemma 3.16(e,f) the divisor (μ_{τ_{ι}}, q_{τ_{ι}}) cannot be strong. ⊓ ⊔(Lemma 4.20)

37 The discussion of various other cases is simpler than that treated in the above 38 proof, as it is not always necessary to extract standard witnesses from generalized 39 ones. Indeed, the Claim in the previous lemma is formulated for generalized wit-40 nesses. The reason we had to look at standard witnesses in the above proof was 41 that we needed to convert a solidity witness into a Dodd solidity witness in order to 42 meet the assumptions of the Claim. One might suggest that this conversion should 1 also be included in the Claim; however, it is not clear how to do this in an elegant 2 way at the level of generality the Claim is formulated, as the conversion between 3 solidity witnesses and Dodd solidity witnesses was granted by Lemma 3.6 and (2).

4Recall that n_{τ} = 0 in our case

4 We are now ready to define C_{τ} for τ ∈ 𝒮^{1}:

γ_{τ} = the least γ ∈ B_{τ} ∪ {τ} such that B_{τ} − γ is closed.
C_{τ} = B_{τ} − γ_{τ} .

5 Then each C_{τ} ⊆ 𝒮^{1}, the same argument as in the proof of (10) shows that the 6 sets C_{τ} are coherent, Lemma 4.18, Lemma 4.19 and Lemma 4.20 imply that C_{τ} is 7 unbounded whenever τ has uncountable cofinality and otp(C_{τ}) ≤ otp(B_{τ}) ≤ ϑ_{τ}, 8 as was pointed out immediately below (11). This completes the construction of C_{τ} 9 for τ ∈ 𝒮^{1}.

10 So far we have constructed a □𝒮-sequence ⟨C_{τ} | τ ∈ 𝒮⟩. To complete of the 11 proof of Theorem 1.2, we present a procedure that produces, given a class A ⊆ 𝒮, 12 a class A′ ⊆ A such that A′ ∩ κ is stationary in κ whenever A ∩ κ is, and a □𝒮-13 sequence ⟨C_{τ}^{A′} | τ ∈ 𝒮⟩ such that lim(C_{τ}^{A′}) ∩ A′ = ∅ for all τ ∈ 𝒮. We begin with 14 the construction of the class A′. We will follow Jensen’s idea from [4]; see also [3]. 15 Let A ⊆ 𝒮 be given. The class A′ ⊆ A comprises all τ ∈ A for which there is an 16 L[E]-level P = J_{β}^{E} and a parameter a ∈ P such that

(13)

(a) P ⊧ ZFC^{−}, τ is a largest cardinal in P and is regular in P.
(b) X ∩ τ ∉ A whenever X ≺ P is such that a ∈ X and X ∩ τ ∈ τ.

17 Here X ≺ Z means that X is a fully elementary substructure of P. A′ is clearly a 18 well-defined class; we verify that A′ is as required.

19 Lemma 4.21. Assume κ is an inaccessible cardinal and A ∩ κ is stationary in κ. 20 Then A′ is stationary in κ.

21 Proof. Let C ⊆ κ be closed unbounded in κ. By acceptability, C ∈ J_{κ^{+}}^{E}. Since 22 A∩κ is stationary in κ there is some X ≺ J_{κ^{+}}^{E} with C ∈ X and X∩κ ∈ A. From now 23 on assume that τ = X ∩κ ∈ A is the least possible. Let J Ē ; β be the transitive collapse 24 of X and σ : J Ē ; β → J_{κ^{+}}^{E} be the inverse to the Mostowski collapsing isomorphism. 25 Then τ is the largest cardinal in J Ē ; β, the map σ is fully elementary, τ = cr(σ), 26 σ(τ) = κ and σ(C ∩ τ) = C. Clearly τ ∈ C, as τ is a limit point of C and C is 27 closed. We prove that τ ∈ A′. 28 Obviously J Ē ; β ⊧ ZFC^{−}. Since J Ē ; τ has no largest cardinal, a straightforward 29 application of the Condensation Lemma (that is, Lemma 3.2) to σ ↾ Q where Q 30 is an arbitrarily large level of J Ē ; β projecting to τ yields that all such levels Q are 31 L[E]-levels, hence Ē = E ↾ β. It follows that J_{β}^{E} is an L[E]-level in which τ is 32 regular. Now if Y ≺ J_{β}^{E} is such that C ∩ τ ∈ Y and Y ∩ τ ∈ τ then necessarily 33 τ̄ = Y ∩ τ ∉ A, as σ[Y] ≺ J_{κ^{+}}^{E} is such that C ∈ σ[Y] and κ ∩ σ[Y] = τ̄ < τ. 34 Thus, it suffices to set P = J_{β}^{E} and a = C ∩ τ in (13) for the current value of τ. 35 ⊓ ⊔(Lemma 4.21)

1 We next have to modify our global square sequence slightly. Recall the definition 2 of λ_{τ} and ϑ_{τ} from the beginning of this section. We let

𝒮_{∗}^{1} = {τ ∈ 𝒮^{1} | J_{ϑ_{τ}}^{E} ⊧ ZFC^{−}}
𝒮_{∗}^{0} = 𝒮 − 𝒮_{∗}^{1}.

3 This way 𝒮_{∗}^{0} becomes larger than 𝒮^{0}, as it may contain cardinals τ for which (μ_{τ} , q_{τ}) 4 is defined, but for such τ the premouse N_{τ} is passive and λ_{τ} is the largest cardinal 5 in N_{τ}. To see this notice that if λ_{τ} is not the largest cardinal in N_{τ} then ϑ_{τ} ∈ N_{τ}^{∗} 6 hence J_{ϑ_{τ}}^{E} ⊧ ZFC^{−}. If λ_{τ} is the largest cardinal in N_{τ} and N_{τ} is active then 7 J_{ν_{τ}}^{E} ⊧ ZFC^{−} and the ultrapower map π : J_{ϑ_{τ}}^{E} → J_{ν_{τ}}^{E} associated with E_{top}^{M_{τ}} is 8 Σ_{0}-preserving and cofinal, so the standard argument then yields that π is fully 9 elementary and J_{ϑ_{τ}}^{E} ⊧ ZFC^{−}. 10 For τ ∈ 𝒮^{0} we let C_{τ} be defined as before. For τ ∈ 𝒮_{∗}^{0} − 𝒮^{0} the canonical divisor 11 (μ_{τ} , q_{τ}) exists; we define the set C_{τ} similarly as for elements of 𝒮^{0}. We first let 12 B_{τ} be the set of all τ̄ ∈ τ ∩ 𝒮_{∗}^{0} − 𝒮^{0} such that: 13 (i) τ̄ satisfies all requirements in Definition 4.3. 14 (ii) μτ̄ = μ_{τ} and | qτ̄ | = | q_{τ} |. 15 We then define B_{τ}^{∗} exactly as in the case where τ ∈ 𝒮^{0}; we also define B̂_{τ} analogously 16 to the definition for τ ∈ 𝒮^{1}; here of course we disregard clause (b) in that definition. 17 An argument using (12) in a way similar to the proof of Lemma 4.18 shows that 18 B̂_{τ} is unbounded in τ whenever τ has uncountable cofinality. As in the proof of 19 Lemma 4.19 we show that B̂_{τ} is closed in τ; here of course we work with structures 20 N_{τ} instead of M_{τ}. Finally, as in Lemma 4.9 we show that B̂_{τ} agrees with B_{τ} 21 on a tail-end. We can let C_{τ} = B_{τ}^{∗} in this case; it follows exactly as in the case 22 τ ∈ 𝒮^{1} above that otp(C_{τ}) ≤ ϑ_{τ}. Notice that the construction yields C_{τ} ⊆ 𝒮_{∗}^{0} − 𝒮^{0} 23 whenever τ ∈ 𝒮_{∗}^{0} − 𝒮^{0}.

24 Lemma 4.22. Let τ ∈ 𝒮_{∗}^{0} and τ̄ ∈ lim(C_{τ}) ∩ A′. Then there is some τ^{∗} ∈ C_{τ} ∩ τ̄ 25 such that τ′ ∉ A whenever τ′ ∈ C_{τ} ∩ (τ^{∗}, τ̄).

26 Proof. Let (P, a) be a pair witnessing that τ̄ ∈ A′. Since Nτ̄ is a singularizing 27 structure for τ̄, τ is regular in P and P is an L[E]-level, we conclude that P is an 28 initial segment of J_{β}^{E} where Nτ̄ = ⟨J_{β}^{E}, E_{ωβ}⟩. 29 First assume that P ∈ Nτ̄. Let τ^{∗} ∈ C_{τ} ∩ τ̄ be such that P, a ∈ rng(σ_{τ^{∗},τ̄}). For 30 any τ′ ∈ C_{τ} ∩ (τ^{∗}, τ̄) let P′ be the preimage of P under σ_{τ′,τ̄} and X = σ_{τ′,τ̄}[P′]. 31 Then X ≺ P and a ∈ X, so τ′ = X ∩ τ̄ ∉ A. 32 Now consider the case where P ∉ Nτ̄, that is, P = J_{β}^{E}. Since P ⊧ ZFC^{=} 33 and Nτ̄ is a singularizing structure for τ̄, necessarily τ ∈ 𝒮^{0} and E_{top}^{Nτ̄} ≠ ∅. As 34 τ̄ is the largest cardinal in P, we have λ(E_{top}^{Nτ̄}) = τ̄, so μ = cr(E_{top}^{Nτ̄}) < τ̄. Since 35 τ ∈ 𝒮^{0} and the construction of C_{τ} guarantees that C_{τ} ⊆ 𝒮^{0} we conclude that N_{τ′} 36 is a premouse with cr(E_{top}^{N_{τ′}}) = μ < τ′ and actually μ^{+} < τ′ whenever τ′ ∈ C_{τ}; 37 notice that n_{τ} > 0 in this case as N_{τ} is not pluripotent, so all maps σ_{τ′,τ′′} for 38 τ′ ≤ τ′′ in C_{τ} are Σ_{1}-preserving. As a consequence of all of the above we have 39 that dom(E_{top}^{N_{τ′}}) = 𝒫(μ) = dom(E_{top}^{Nτ̄}) and E_{top}^{Nτ̄}(x) = σ_{τ′,τ̄}(E_{top}^{N_{τ′}} (x)) whenever 40 τ′ ∈ C_{τ} ∩ τ̄. Letting β′ = ht(N_{τ′}) and ϑ = μ^{+}, we have J_{β′}^{E} = Ult(J_{ϑ}^{E}, E_{top}^{N_{τ′}}) and 41 J_{β}^{E} = Ult(J_{ϑ}^{E}, E_{top}^{Nτ̄}). It follows that σ_{τ′,τ̄} : π′(f)(α) ↦ π(f)(α) where π′, π are the 42 corresponding ultrapower maps, so σ_{τ′,τ̄} is fully elementary when viewed as a map 1 from J_{β′}^{E} to J_{β}^{E} and X = rng(σ_{τ′,τ̄}) ≺ J_{β}^{E}. We can now proceed as above to show 2 that τ′ ∉ A whenever a ∈ rng(σ_{τ′,τ̄}). ⊓ ⊔(Lemma 4.20)

3 For each τ ∈ 𝒮^{0} ∩ A′ let

δ_{τ} = the least ordinal δ ∈ C_{τ} such that (δ, τ) ∩ A = ∅.

4 and

(14)  C_{τ}^{A′} = C_{τ} − ⋃{(δτ̄, τ̄) | τ̄ ∈ lim(C_{τ}) ∩ A′}

5 It follows immediately from the definition that each C_{τ}^{A′} is a closed subset of τ and 6 otp(C_{τ}^{A′}) ≤ otp(C_{τ}) < τ. Because the sequence ⟨C_{τ} ⟩_{τ} is coherent, it follows easily 7 that ⟨C_{τ}^{A′}⟩_{τ} is also coherent; this is similar to the verification that the sets defined 8 in (10) are coherent. To complete the verification that ⟨C_{τ}^{A′}⟩_{τ} is a □^{𝒮{0 \atop ∗}}-sequence 9 we show that C_{τ}^{A′} is unbounded in τ whenever C_{τ} is. This is clear if lim(C_{τ}) ∩ A′ is 10 bounded in τ, so assume without loss of generality that lim(C_{τ}) ∩ A′ is unbounded 11 in τ. We will rely on the following immediate consequence of Lemma 4.20.

(15)  τ^{∗}, τ̄ ∈ lim(C_{τ}) ∩ A′ & τ^{∗} < τ̄ ⟹ τ^{∗} < δτ̄

12 It follows that C_{τ}^{A′} is unbounded in τ, as it contains lim(C_{τ}) ∩ A′. 13 Finally notice that any limit point τ̄ of C_{τ}^{A′} is a limit point of C_{τ}, but τ̄ cannot 14 be an element of A′ since all ordinals from lim(C_{τ})∩A′ are successor points of C_{τ}^{A′}. 15 This shows that A′ ∩lim(C_{τ}^{A′}) = ∅ and thereby completes the proof of Theorem 1.2 16 in the case where τ ∈ 𝒮_{∗}^{0}.

17 We next turn to the case where τ ∈ 𝒮_{∗}^{1}. We need the analogue to Lemma 4.20, 18 but in order to prove it we have to refine our sets C_{τ}. We let

C′_{τ} = {τ̄ ∈ C_{τ} | ϑτ̄ ≺ ϑ_{τ} }

19 We first verify that ⟨C′_{τ} ⟩_{τ} is a □^{𝒮{1 \atop ∗}}-sequence. Using the coherency of the sequence 20 ⟨C_{τ} ⟩_{τ} and elementary properties of ≺ it is easy to see that the sequence ⟨C′_{τ} ⟩_{τ} is 21 coherent; again, this is similar to the situation in (10). Since ϑ_{τ^{∗}} converges to ϑτ̄ 22 as τ^{∗} converges to τ, we see that J_{ϑτ̄}^{E} ≺ J_{ϑ_{τ}}^{E} whenever τ̄ is a limit point of C′_{τ}, so 23 τ̄ ∈ C′_{τ}. This shows that C′_{τ} is closed. It remains to show that C′_{τ} is unbounded in 24 τ whenever τ has uncountable cofinality.

25 Lemma 4.23. Let τ ∈ 𝒮_{∗}^{1} be of uncountable cofinality. Then the set

B̂′_{τ} = {τ̄ ∈ B̂_{τ} | J_{ϑ_{τ̄,τ}}^{E} ≺ J_{ϑ_{τ}}^{E} }

26 is unbounded in τ.

27 Proof. It is sufficient to show that in the proof of Lemma 4.18, the interpolant M̃ 28 emerging from the construction satisfies J E ; ϑ̃ ≺ J_{ϑ_{τ}}^{E} where ϑ̃ = ϑM̃. We follow the 29 notation from that proof. Recall that σ : M̄ → M_{τ} is fully elementary where M̄ 30 is countable. We have μ̄, ϑ̄ ∈ M̄ such that σ(μ̄, ϑ̄) = (μ_{τ} , ϑ_{τ}). We have seen that 31 ϑ̃ = sup(σ[ϑ̄]). Let Ē = EM̄. 32 Assume first that M_{τ} = N_{τ}; in this case ϑ_{τ} = μ_{τ}^{+} < ht(M_{τ}). Then ϑ̄ = 33 μ̄^{+M̄} < ht(M̄), so we can apply the interpolation lemma to obtain an interpolant 34 M̃ extending J E ; ϑ̃ and Σ_{0}-preserving maps σ_{0} : M̄ → M̃ and σ_{1} : M̃ → M_{τ} such 35 that σ_{1} ↾ ϑ̃ = id and σ_{1}(ϑ̃) = ϑ_{τ}. Here M̃ is the ultrapower of M̄ by σ ↾ J E ; ϑ̄ and σ_{0} 1 is the associated ultrapower map. It follows from the above properties of σ_{1} that

2 J E ; ϑ̃ ≺ J_{ϑ_{τ}}^{E}.

3 From now on assume that M_{τ} is a protomouse. If λ_{N_{τ}^{∗}(μ_{τ} ,q_{τ})} > μ_{τ} then N_{τ}^{∗} 4 is a proper extension of J_{ϑ_{τ}}^{E} and ϑ_{τ} = μ_{τ}^{+N_{τ}^{∗}} where N̄^{∗} be the inverse image of 5 N_{τ}^{∗} under σ. Then N̄^{∗} is the collapsing level of N̄ for ϑ̄ and J Ē ; ϑ̄ is a proper 6 initial segment of N̄^{∗}. Hence we can use the interpolation lemma to obtain an 7 interpolant N̄′ end-extending J E ; ϑ̃ and Σ_{0}-preserving embeddings σ_{0} : N̄^{∗} → N̄′ 8 and σ_{1} : N̄′ → N_{τ}^{∗} such ϑ̃ = cr(σ_{1}) and σ_{1}(ϑ̃) = ϑ_{τ}. (For the present purpose 9 we do not even need to use a fine structural ultrapower in the construction of the 10 interpolant; it is sufficient to construct σ_{0} as the ultrapower map associated with 11 the coarse ultrapower Ult(N̄^{∗}, σ ↾ J Ē ; ϑ̄).) Then obviously σ_{1} ↾ J E ; ϑ̃ : J E ; ϑ̃ → J_{ϑ_{τ}}^{E} is 12 fully elementary, so J E ; ϑ̃ ≺ J_{ϑ_{τ}}^{E}. 13 Finally consider the case where λ_{N_{τ}^{∗}}(μ_{τ} , q_{τ}) = μ_{τ} and M_{τ} is a protomouse. 14 Then ϑ_{τ} < μ_{τ}^{+}, so N_{τ}^{∗} = ⟨J_{ϑ_{τ}}^{E} , E_{ϑ_{τ}} ⟩. Let μ_{1} = cr(E_{ϑ_{τ}}) and ϑ_{1} = μ_{1}^{+}; then 15 μ_{1} < μ_{τ} = λ(E_{ϑ_{τ}}). Letting μ̄_{1}, ϑ̄_{1} be the inverse images of μ_{1}, ϑ_{1} under σ and 16 ϑ̃_{1} = sup(σ[ϑ̄_{1}]), we see that ϑ̄_{1} = μ̄_{1}^{+M̄} and J Ē ; ϑ̄_{1} is a proper initial segment of 17 M̄, so we can apply the interpolation lemma to obtain an interpolant M′ and 18 Σ_{0}-preserving maps σ_{0} : M̄ → M′ and σ_{1} : M′ → M_{τ} such that σ_{0} is cofinal, 19 σ_{0}(ϑ̄_{1}) = ϑ̃_{1}, cr(σ_{1}) = ϑ̃_{1} and σ_{1}(ϑ̃_{1}) = ϑ_{1}. As above we conclude that J E ; ϑ̃_{1} ≺ J_{ϑ_{1}}^{E}.

20 Since σ_{0} maps ϑ̄_{1} cofinally into ϑ̃_{1} and σ_{0} ↾ ϑ̄_{1} = id, the restriction σ_{0} ↾ M̄ || ϑ̄ 21 maps the coherent structure M̄ || ϑ̄ cofinally into the coherent structure M′ || ϑ′ 22 where ϑ′ = σ_{0}(ϑ̄), so sup(σ_{1}[ϑ′]) = ϑ̃. Now notice that σ_{1} : M′ || ϑ′ → M^{∗} is cofinal 23 where M^{∗} = ⟨J E ; ϑ̃, E_{ϑ_{τ}} ∩ J E ; ϑ̃⟩, so M^{∗} is a coherent structure and ϑ_{M^{∗}} = ϑ̃_{1}. Since 24 J E ; ϑ̃_{1} ≺ J_{ϑ_{τ}}^{E}, it follows by a straightforward verification that the map σ^{∗} : J E ; ϑ̃ → J_{ϑ_{τ}}^{E} 25 defined by σ^{∗} : π̃(f)(α) ↦ π(f)(α) for f ∈ J E ; ϑ̃ and α < μ_{τ} is fully elementary. 26 ⊓ ⊔(Lemma 4.23)

27 We can now prove the analogue to Lemma 4.22.

28 Lemma 4.24. Let τ ∈ 𝒮_{∗}^{1} and τ̄ ∈ lim(C′_{τ})∩A′. Let (P, a) be a pair witnessing that 29 τ̄ ∈ A′. Then there is some τ^{∗} ∈ C′_{τ} ∩τ̄ such that τ′ ∉ A whenever τ′ ∈ C′_{τ} ∩(τ^{∗}, τ̄).

30 Proof. If P ∈ Mτ̄ we argue exactly as in the proof of Lemma 4.22. Otherwise 31 P = J_{ντ̄}^{E}. For every τ′ ∈ C′_{τ} ∩ τ̄ we have J_{ϑ_{τ′}}^{E} ≺ J_{ϑτ̄}^{E}, so σ_{τ′,τ̄} : J_{ν_{τ′}}^{E} → J_{ντ̄}^{E} = P is 32 fully elementary as σ_{τ′,τ̄} : F_{τ′}(f)(α) ↦ Fτ̄(f)(σ_{τ′,τ̄}(α)) for f ∈ J_{ϑ_{τ′}}^{E} and α < λ_{τ′}. 33 So we can let τ^{∗} ∈ C′_{τ} ∩ τ̄ be such that a ∈ rng(σ_{τ^{∗},τ̄}) and again proceed as in the 34 proof of Lemma 4.22. ⊓ ⊔(Lemma 4.23)

35 With Lemma 4.24 in hand we can define C_{τ}^{A′} for τ ∈ 𝒮_{∗}^{1} similarly as in the 36 previous case. We let

δ_{τ} = the least ordinal δ ∈ C′_{τ} such that (δ, τ) ∩ A = ∅

37 and

C_{τ}^{A′} = C′_{τ} − ⋃{(δτ̄, τ̄) | τ̄ ∈ lim(C′_{τ}) ∩ A′}.

38 The verification that ⟨C_{τ}^{A′} | τ ∈ 𝒮_{∗}^{1}⟩ is a □^{𝒮{1 \atop ∗}}(A′)-sequence is then the same as 39 before.

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35 Department of Mathematics, University of California at Irvine, Irvine, CA 92697

36  E-mail address: mzeman@math.uci.edu