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    0. Introduction 2Normalizing well, ~(P,) and strategy comparison, solidity and universalitytheorem, conjectures + proof outline.

    1. Normalizing iteration trees 9Definition of W (T ,U), coarse case, full normalization, diagrams

    1.1. Normalizing trees of length 2 101.2. Normalizing T aF 161.3. Normalizing T aU 281.4. Normalization commutes with copying 391.5. The extender tree Vext 451.6. The branches of W (T ,U) 472. Strategies that normalize well 56

    Background constructions, conversion systems, lift(T ,M,k,C). Backgroundinduced strategies normalize well. (Theorem 2.8)

    2.1. The definition 562.2. A strategy for V that normalizes well 612.3. Strategies induced by full background extender constructions 723. Strong hull condensation 90

    Pseudo-hull embeddings. Background induced strategies have strong hull con-densation.

    4. Comparing iteration strategies 985. Fine structure for the least-branch hierarchy 132

    Basic fine structural definitions and results for hod mice in the least-branchhierarchy

    5.1. Least branch premice 1325.2. Complete strategies 1355.3. Copying 1375.4. Least branch hod pairs 139

    Date: January 2016.1


    5.5. The Dodd-Jensen Lemma 1425.6. Background constructions 1445.7. Comparison and the hod-pair order 1495.8. The existence of cores 1535.9. Proofs of theorems 0.4 and 0.5 1716. Phalanx iteration into a backgrounded construction 1746.1. The Bicephalus Lemma 1746.2. Proof of Lemma 5.9 1816.3. UBH holds in hod mice 1947. HOD in the derived model of hod mouse 201References 215

    0. Introduction

    In this paper, we shall prove a general comparison lemma for iteration strategies.We then use it to develop the basic theory of hod mice in the least branch hierarchy.12

    Our comparison lemma relies heavily on an analysis of the normalization of a finitestack of iteration trees. Recall that an iteration tree W on a premouse M is normaliff the extenders EW used in W have lengths increasing with , and each EW isapplied to the longest possible initial segment of the earliest possible model in W .Suppose now ~T is a finite stack of iteration trees, with T0 being a normal tree onM,and Ti+1 being a normal tree on the last model of Ti. Let N be the last model ofthe last tree. There is a natural attempt to construct a minimal normal iterationtree W on M having last model N . This attempt may break down by reaching anillfounded model. If it does not break down, it will in the end produce a model Pand : N P such that i~T = iW . We call W the embedding normalization of~T .

    If ~T is played according to a reasonable iteration strategy , thenW is also by ,so the W-construction does not break down.

    Although it is embedding normalization that is important to us here, one can alsoask whether there is a normal tree on M whose last model is equal to N . We shall1This research was partially done while the author was a Simons Foundation fellow at the IsaacNewton Institute for Mathematical Sciences in the programme Mathematical, Foundational andComputational Aspects of the Higher Infinite (HIF) funded by EPSRC grant EP/K032208/1. Theauthor extends special thanks to the INI and its staff, for having provided an environment that isideal for mathematical research.2The author thanks Xianghui Shi and Nam Trang for their invaluable help in the preparation ofthis paper.


    show that this is true if M is an iterable premouse, and ~T is a finite stack of finitetrees. The proof gives that there is a full normalization of ~T in other cases as well.

    Some of our work on normalization was done earlier (but never written up) withItay Neeman, and then later with Grigor Sargsyan. Fuchs, Neeman and Schindler([2]) and Mitchell ([4]), and probably others, have considered the question. Muchof what seems to be new in this part of the paper was done independently, and atroughly the same time, by Farmer Schlutzenberg. (See [18].) Schlutzenberg andthe author have carried this work further, and in particular analyzed embeddingnormalization and full normalization for infinite stacks of normal trees. See [19].

    The reasonableness of iteration strategies with respect to embedding normalizationis isolated in

    Definition 0.1. Let be an iteration strategy for a (hod) premouse M. We saythat normalizes well iff whenever ~T is a finite stack of normal trees by , and Wis an embedding normalization of ~T , then W is by .

    The concept is defined more fully in section 2.1, and 2.3 should be considered theofficial definition of normalizing well.

    Embedding normalization actually makes sense for coarse-structural stacks ~T oncoarse-structural M. Granted the appropriate form of UBH in V , the iterationstrategy for V normalizes well. In particular, if we assume AD+, and then let Vbe a coarse -Woodin model Nx as in Theorem 10.1 of [20] (due to Woodin), thenthe iteration strategy we get for Nx normalizes well.

    We shall show that the property of normalizing well passes from (for V ) tothe iteration strategy ofM, wheneverM is a level of (hod or pure-extender) mouseconstruction in V . The proof of this is like Sargsyans proof that hull condensationpasses to induced strategies (Lemma 2.9 of [11]). It is important here that we definednormalizing well in terms of embedding normalizations. We do show in [22] that if

    is induced by for N as above, and ~T is by , and U is its full normalization,then U is by . However, the proof does not proceed by some direct, combinatorialroute. It involves a comparison argument, and so cannot be used until a comparisontheorem for iteration strategies has already been proved.

    We shall define a slight strengthening of hull condensation, and show that it passesfrom for V to forM, where is the induced iteration strategy for a levelM ofa full background construction. We shall call this property strong hull condensation.The details are in 4.

    With these properties in hand, we can state our strategy comparison theorem. Westate first a version that has AD+ as its hypothesis.

    Notation. Let be a strategy forM, ~T a stack onM with last model P , and Q aninitial segment of P . Then ~T ,Q is the ~T -tail of restricted to stacks on Q. So for


    ~U on Q:~T ,Q(

    ~U) = (~T a ~U).

    Theorem 0.1. Assume AD+, and forM and N be countable (hod or pure-extender)premice, with Suslin-co-Suslin (, , 1)-iteration strategies and respectively.Suppose and normalize well and have strong hull condensation. Then there arecountable normal trees T on M and U on N by and , with last models P andQ respectively, such that either

    (1) P Q, and T ,P agrees with U ,P on finite stacks of normal trees, or(2) Q P, and U ,Q agrees with T ,Q on finite stacks of normal trees.

    This seems to be new even in the case of pure extender premice. Of course, if wedrop the strategy-agreement condition, it becomes in that case the usual ComparisonLemma.

    We prove 0.1 by puttingM and N into a common -Woodin universe N, where and are in . We then iterate (M,) and (N ,) into the full backgroundconstruction (of the appropriate type) of N. Here is a definition encapsulating themethod:

    Definition 0.2 (AD+). Let P be a countable (hod or pure extender) premouse, and an (, , 1 + 1)-strategy for P ; then ~(P ,) is the assertion:

    Whenever N is a coarse -Woodin model with iteration strategy as in 10.1 of[20] (so is inductive-like and has the scale property), and P HCN , and is Suslin captured by (N,), and R is a level of the full background construction ofN of the appropriate type, with strategy induced by , then in the comparisonof (P ,) and (R,):

    (a) no strategy disagreements on finite stacks of iteration trees show up, and(b) only the P-side moves.

    The meaning of (a) is: suppose the comparison of P with R has produced anormal tree T on P with last model Q, with T by . Suppose Q| = R|, thenT ,Q| = R|, on finite stacks of normal trees. Thus the least disagreement betweenQ and R is an extender disagreement. Part (b) says that if E on Q and F on R arethe extenders involved in it, then F = .

    We shall show (cf. Theorem 4.1 below)

    Theorem 0.2. Assume AD+. Let P be a (hod or pure extender) premouse with(, , 1 + 1)-iteration strategy such that is Suslin-co-Suslin, and normalizeswell and has strong hull condensation; then ~(P ,).

    We note

    Proposition 0.3. Theorem 0.2 implies Theorem 0.1.


    Proof. Let (M,) and (N ,) be as in the hypotheses of 0.1. Let (N,) witness~(M,) and ~(N ,) simultaneously. Let C be the full background extender con-struction of N of the appropriate type.

    Claim. There is a level R of C such that R is a -iterate of M.

    Proof. Suppose first C breaks down, in that it has a least level Q such that Q is not-solid. Since M is -solid, and this is preserved by iteration, Q is not an initialsegment of an iterate ofM. By ~(M,), M iterates to a proper initial segment ofQ, with no strategy disagreement. This implies that some R properly before Q in Cis a iterate of M.

    If C never breaks down, then letQ = (N)C, where is the Woodin of N. ThenMcannot -iterate past Q by the usual universality argument. (Note here .)SoM iterates to a proper initial segment of Q, and thus some R properly before Qin C is a -iterate of M. a

    Claim. There is a level S of C such that S is an -iterate of N .

    Proof. Symmetric. aNotice that the iterations provided by our two claims do not drop. Letting R andS be the last models, we may assume without lost of generality that R is before Sin C, or R = S. Let T be the normal tree onM by with last model R. Let U bethe normal tree on N by that comes from com