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\fancyhead[LO,RE]{\small Under measurables, $\HOD_{\Ord^{\omega}}$ is locally arbitrary}
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\title[Under measurables, $\HOD_{\Ord^{\omega}}$ is locally arbitrary]{Under measurables, $\mathrm{HOD}_{\mathrm{Ord}^{\omega}}$ is locally arbitrary}
\author{}
\date{October 12, 2026}
\subjclass[2020]{03E25, 03E35, 03E45, 03E55}
\keywords{ordinal definability, dependent choice, symmetric extensions, Prikry forcing, measurable cardinals, Woodin cardinals, inner models}


\begin{document}
\begin{abstract}
For the inner model $\HOD_{\Ord^{\omega}}$, we derive an analogue
for Roguski's classical result that without nontrivial assumptions
on $V$, $\HOD$ is an arbitrary model of $\ZFC$.
For any sentence $\sigma$, letting $\CM$ denote ``class of measurables,''
the following theories are equiconsistent:
\begin{enumerate}
\item $\ZFC+\CM+
  [\HOD_{\Ord^{\omega}}\models
    (\exists\kappa\,V_{\kappa}\models\sigma)]$;
\item $\ZFC+\GCH+\CM+
  [\HOD_{\Ord^{\omega}}\models
    (\SVC\wedge\CM\wedge\exists\kappa\,V_{\kappa}\models\sigma)]$;
\item $\ZF+\DC+\CM+
  \exists\kappa\,V_{\kappa}\models\sigma$.
\end{enumerate}
Thus, under measurables the only local constraint on $\HOD_{\Ord^{\omega}}$ is the requisite choice principle.

We also derive an analogue for Woodins. Letting $\CW$ denote ``class of Woodins,'' the following are equiconsistent:
\begin{enumerate}
	\item $\ZFC+\CW+
	[\HOD_{\Ord^{\omega}}\models
	(\exists\kappa\,V_{\kappa}\models\sigma)]$;
	\item $\ZFC+\GCH+\CW+
	[\HOD_{\Ord^{\omega}}\models
	(\exists\kappa\,V_{\kappa}\models\sigma)]+$ $\HOD_{\Ord^{\omega}}$ is a ground of $V$;
	\item $\ZFC+\CW+$ some symmetric extension $V(G) \models \DC+\exists\kappa\,V_{\kappa}\models\sigma$.
\end{enumerate}
\end{abstract}
\maketitle

\setcounter{tocdepth}{1}
\tableofcontents


\section{External HOD and local arbitrariness}\label{sec:intro}

Roguski's work on the theory of $\HOD$ is the starting point for this
paper \cite{Roguski}. In the countable transitive formulation of his
realization theorem, every countable transitive model $M$ of $\ZFC$
has a larger model $N$ of $\ZFC$ such that
\[
 M=\HOD^N.
\]
Ordinary $\HOD$ can thus realize a prescribed $\ZFC$ universe.
We seek an analogue when definitions may use countable sequences
of ordinals as parameters. Our first result concerns the theories of rank
segments of the resulting inner model under a class of measurable
cardinals. A second result relates its rank theory under a proper
class of Woodins to that of symmetric extensions of Woodin grounds.

For $U\models\ZFC$, write
\[
 \Hseq^U=\HOD_{\Ord^\omega}^U.
\]
An allowed parameter is one function from $\omega$ to the ordinals,
together with finitely many ordinal parameters; the hereditary
requirement is retained. This inner model contains every ordinal
$\omega$-sequence of $U$ and satisfies $\ZF+\DC$
(Lemma~\ref{lem:HDC}). The local analogue of Roguski's theorem will
therefore compare its rank theory with that of choiceless models
satisfying dependent choice.

Write $\CM$ for the assertion that measurable cardinals are unbounded
in the ordinals, abbreviated as ``class of measurables.'' A measurable
cardinal is uncountable and carries a nonprincipal, cardinal-complete
ultrafilter. We require normality only in choice models. Put
\[
 T_0=\ZF+\DC+\CM.
\]
For each first-order sentence $\sigma$, consider the theories
\begin{align*}
 E_1(\sigma)&=\ZFC+\CM+
  [\Hseq\models\exists\kappa\,(V_\kappa\models\sigma)],\\
 E_2(\sigma)&=\ZFC+\GCH+\CM+
  [\Hseq\models\SVC\wedge\CM\wedge\exists\kappa\,(V_\kappa\models\sigma)],\\
 E_3(\sigma)&=T_0+\exists\kappa\,(V_\kappa\models\sigma).
\end{align*}
In the first two theories, the witnessing rank is computed inside
$\Hseq$. Theory $E_2$ requires GCH in the ambient universe and
SVC together with a class of measurables inside $\Hseq$.
Our first main result is the following.

\begin{theorem}[External rank equiconsistency]\label{thm:external-main}
For every $\sigma$,
\[
 \Con(E_1(\sigma))\ \longleftrightarrow\
 \Con(E_2(\sigma))\ \longleftrightarrow\
 \Con(E_3(\sigma)).
\]
\end{theorem}

This is the sense in which $\HOD_{\Ord^\omega}$ is locally arbitrary
under measurables. Every rank-sentence assertion consistent with
$\ZF+\DC+\CM$ can hold inside the sequence-parameter HOD of a
$\ZFC+\GCH+\CM$ universe, even when that HOD is itself required to
satisfy $\SVC+\CM$. GCH is imposed on the ambient universe, so it
does not restrict the rank theory being realized inside $\Hseq$.
The comparison concerns consistency of rank-sentence theories.
Exact realization of an entire starting model is available under
the prepared-ground hypotheses stated below.

\subsection{The Woodin equiconsistency}

Write $\PCW$ for the assertion that Woodin cardinals are unbounded in
the ordinals. For a fixed first-order sentence $\sigma$, put
\[
 R_\sigma\equiv\exists\alpha\,(V_\alpha\models\sigma).
\]
Consider the following three theories:
\begin{align*}
 W_1(\sigma)&=\ZFC+\PCW+[\Hseq\models R_\sigma],\\
 W_2(\sigma)&=\ZFC+\PCW+
 [\text{some set-symmetric extension satisfies }\DC+R_\sigma],\\
 W_3(\sigma)&=\ZFC+\GCH+\PCW+[\Hseq\models R_\sigma]\\
 &\qquad+[\Hseq\text{ is a symmetric ground of }V].
\end{align*}
The symmetric extension in $W_2$ is a model of $\ZF$. The assertion
$\PCW$ is imposed on its $\ZFC$ ground, and need not hold in the
symmetric extension. To read $W_2$ syntactically, assert the existence
of a set symmetric system and a condition forcing $\DC+R_\sigma$
under the hereditary-symmetric interpretation. The symmetric forcing
theorem supplies $\ZF$ in that interpretation.
In $W_3$, both GCH and $\PCW$ are assertions about the ambient
universe. We impose no Woodin-cardinal assertion on $\Hseq$.
The term symmetric ground has its set-forcing meaning. In a
$\ZFC$ ambient universe, it is equivalent to being an ordinary
set-forcing ground, which may fail choice
\cite[Corollary~5.9]{Usuba}.

\begin{theorem}[Woodin rank equiconsistency]\label{wd-main}
For every fixed first-order sentence $\sigma$,
\[
 \Con(W_1(\sigma))\ \longleftrightarrow\
 \Con(W_2(\sigma))\ \longleftrightarrow\
 \Con(W_3(\sigma)).
\]
\end{theorem}

The proof preserves the relevant membership structure $V_\alpha$.
It uses a relative inner model containing a given real and a proper
class of Woodins, together with a set-forcing presentation of that
inner model over its own HOD. The relative construction is invariant
under Turing-equivalent recodings in the same ambient universe.
These properties allow the collapse argument to descend to a model
with the required rank and dependent choice.

For the converse, retain the Woodin ground of the given symmetric
extension. Its old normal measures supply the local Prikry construction.
After the sequence generator has been coded, a bounded homogeneous
code makes enough of the original ground ordinal-definable to recover
the prescribed rank. This uses only set forcing over the Woodin
ground. This rank-realization argument works over the original
$W_2$ ground without an eventual-GCH hypothesis.

To obtain $W_3$, we instead use the particular ground supplied by
the reverse construction. It satisfies eventual GCH, and each of
its sets is internally definable from ordinals and one fixed set
parameter. A bounded $\diamondsuit^*$ code makes that parameter
and a ground-defining parameter ordinal-definable while preserving
GCH. The spectrum then recovers the whole sequence-generated model
as $\Hseq$, with the ambient universe a set-forcing extension of it.
This proves the consistency strengthening without a class preparation
preserving Woodin cardinals.

\subsection{Realization and rank preservation}

The construction first preserves a specified rank while passing to
a model generated by a set of ordinal sequences. It then realizes
that rank inside external HOD. Two presentation principles enter
the first step. Write $\SVC(S)$ when every nonempty set is a
surjective image of $\eta\times S$ for some ordinal $\eta$; $\SVC$
asserts that such a set $S$ exists. The assertion $\KW$ says that
every nonempty set is a surjective image of $\mu^\omega$ for some
ordinal $\mu$, with the sequence power computed in the model under
discussion. Write $T_{\mathrm{s}}=T_0+\SVC$.

The external realization step takes the following form.

\begin{theorem}[External realization]\label{thm:external-realization}
Suppose $M\models T_0+\SVC+\KW$. For every ordinal $\alpha$
there is an outer model $U\models\ZFC+\GCH+\CM$ of $M$ such that
\[
 \Hseq^U\models\SVC+\CM,\qquad V_\alpha^{\Hseq^U}=V_\alpha^M.
\]
If $M$ has a presentation as in Proposition~\ref{prop:secure-realize}
whose choice envelope satisfies eventual GCH, one may require
$\Hseq^U=M$.
\end{theorem}

The secure-ground case gives equality with the whole generated
model by further set forcing over its fresh choice envelope.
The general preparation uses class forcing, so the outer model in
Theorem~\ref{thm:external-realization} is not asserted to be a
set-forcing extension of $M$.

The next theorem supplies the required presentation while retaining
any chosen rank and its full power set. In particular, SVC is not an
assumption on the original model. Here $D(A)$ denotes the model
generated over a transitive base. It contains $D$ and the transitive
closure of $\{A\}$.

\begin{theorem}[Rank transfer without SVC]\label{thm:rank-transfer}
Suppose $V\models T_0$ and $\alpha\in\Ord$. In a set-generic extension of $V$ there is a model $M$ such that
\[
 M\models T_0+\SVC+\KW,\qquad
 V_\alpha^M=V_\alpha^V,\qquad
 \Pow(V_\alpha^V)^M=\Pow(V_\alpha^V)^V.
\]
Moreover, $M=D(A)$ for a choice symmetric ground $D\models\ZFC+\CM$ and a set $A\sq\mu^\omega$.
\end{theorem}

The model $M$ contains a rank-preserving SVC inner model of $V$,
but need not contain all of $V$. Once the starting model satisfies
SVC, the local construction also gives containment.

\begin{theorem}[Local rank-to-sequence transfer]\label{thm:transfer-intro}
Suppose $N\models T_{\mathrm{s}}$ and $\alpha\in\Ord$. In a set-generic extension of $N$ there is an inner model $M$ containing $N$ such that
\[
 M\models T_{\mathrm{s}}+\KW,\qquad V_\alpha^M=V_\alpha^N.
\]
More precisely, one can choose $\lambda>\alpha$ and $X=V_\lambda^N$ so that
\[
 \Pow(X)^M=\Pow(X)^N,\qquad V_\lambda^M=X.
\]
There are a choice symmetric ground $D\models\ZFC+\CM$, an ordinal $\mu$, and $A\sq\mu^\omega$ in $M$ such that $M=D(A)$.
\end{theorem}

The two transfer statements also give the following equiconsistency.

\begin{corollary}[Rank-sentence equiconsistency]\label{cor:ranksentence}
For every first-order sentence $\sigma$, the theories
\[
 T_0+\exists\alpha\,(V_\alpha\models\sigma)
 \quad\hbox{and}\quad
 T_0+\SVC+\KW+\exists\alpha\,(V_\alpha\models\sigma)
\]
are equiconsistent.
\end{corollary}
\begin{proof}
Apply Theorem~\ref{thm:rank-transfer} to a witnessing rank. The converse is immediate.
\end{proof}

Combining rank transfer with external realization gives the
implication from $E_3(\sigma)$ to $E_2(\sigma)$ at the level of
consistency. The implication from $E_2(\sigma)$ to $E_1(\sigma)$
is immediate. For the remaining implication,
Proposition~\ref{prop:one-three} uses core-model preparation and
homogeneous descent to obtain an inner model of the given
sequence-parameter HOD while retaining its witnessing rank.
This step does not assume that the original HOD already has a class
of measurables.

\subsection{Structural conservativity}

The rank-preserving construction also yields a structural
conservativity theorem. This result concerns full second-order
truth in arbitrary set-sized structures and allows a starting
theory without Foundation. Put
\[
 T_-=(\ZF-\mathsf{Foundation})+\DC+\CM.
\]

\begin{definition}[Structural $\Pi_2$ sentences]\label{def:structural-pi2}
A \emph{structural $\Pi_2$ sentence} is a set-theoretic assertion
of the form
\[
 \forall\mathcal M\,
 (\mathcal M\text{ is a set-sized }\mathcal L\text{-structure}
 \ \Longrightarrow\ \mathcal M\models_{\mathrm{full}}\varphi),
\]
where $\varphi$ is a fixed second-order sentence in a finite
signature $\mathcal L$. Satisfaction uses full second-order
semantics: if $A$ is the domain of $\mathcal M$, each $n$-ary
second-order relation variable ranges over all of $\Pow(A^n)$
in the ambient universe. This assertion has a $\Pi_2$ formulation
in the L\'evy hierarchy. The second-order quantifier complexity
of $\varphi$ is unrestricted.
\end{definition}

\begin{theorem}[Structural conservativity]\label{thm:main}
The theory $T_0+\SVC+\KW$ is conservative over $T_-$ for structural $\Pi_2$ assertions. That is, for every fixed full second-order sentence $\varphi$, the assertion that $\varphi$ holds in every set-sized structure has the same provability in the two theories.
\end{theorem}

Foundation removal preserves structural truth.
Theorem~\ref{thm:main} does not identify the non-well-founded
membership structure of an arbitrary starting universe with a rank
of a well-founded model. The argument represents a full second-order
counterexample by a structure on atoms. After the local forcing
construction, fibers of ordinal-sequence surjections give a pure
copy of that structure and of an SVC witness. The full power sorts
retain the truth of the original sentence.

\begin{remark}[Role of the core-model preparation]\label{rem:prior}
The core-model argument gives a passage to SVC that preserves each specified rank. We use this rank-preserving result directly. The proof constructs a uniform choice model over the coding real inside relative HOD, using the core model in the negative branch. Canonical refinement preserves its pure universe under finite recodings and gives a set-forcing presentation over its own HOD. The proof does not require $V=\HOD(r)$. The earlier $\SVC+\mathsf{KW}^{*,1}_1$ conservativity result is compatible with this preparation and is not an additional hypothesis of the structural conservativity theorem.
\end{remark}

\subsection{Limits of the conclusions}

The large-cardinal assumption cannot simply be deleted from
Theorem~\ref{thm:external-main}. Put $\mathbb R=\Pow(\omega)$ and
\[
 A_L\quad\equiv\quad
 \mathbb R=\mathbb R^L\ \wedge\ \omega_2=\omega_2^L.
\]
Over $\ZF$, this agreement implies that every countable ordinal
sequence belongs to $L$. To see this, fix such a sequence $s$ and
apply Jensen covering in the choice model $L[s]$. A constructible
cover of its range has size at most $\omega_1$ there. Agreement at
$\omega_2$ makes its constructible size at most $\omega_1$ as well;
agreement on reals then returns $s$ to $L$. Thus
\[
 \ZF+A_L+\KW\ \vdash\ \mathsf{AC}.
\]
Likewise, if $\Hseq^U\models A_L$ for $U\models\ZFC$, all its
allowed sequence parameters are constructible, and
$\Hseq^U=\HOD^U$. A symmetric extension of $L$ using countable
conditions and countable supports satisfies $\ZF+\DC+A_L$ while
$\Pow(\omega_1)$ is not wellorderable and $\DC_{\omega_1}$ fails.
Appendix~\ref{app:no-meas} gives the construction and a single
guarded rank sentence $\sigma$ that can hold in this model but in
no rank of any external $\Hseq^U$. Hence the rank-equiconsistency
conclusion itself fails when $\CM$ is removed.

There is also an obstruction below measurables to the structural
conservativity theorem. Let $\mathsf{CR}$ assert a class of Ramsey
cardinals, with Ramseyness understood as
$\kappa\longrightarrow(\kappa)^{<\omega}_2$. There is a
parameter-free $\Pi^2_3$ sentence $\Phi$ of third-order arithmetic
such that, for $T_{\mathrm R}=\ZF+\DC+\mathsf{CR}$,
\[
 T_{\mathrm R}+\KW\vdash\Phi,
 \qquad
 \Con(\ZFC+\mathsf{CR})\Longrightarrow\Con(T_{\mathrm R}+\neg\Phi).
\]
Its semantic form over this base is
\[
 \bigl[\neg\exists 0^\dagger\ \wedge\
       \mathbb R=\mathbb R^K\ \wedge\ \omega_2=\omega_2^K\bigr]
       \ \Longrightarrow\ \mathsf{Fodor}(\omega_1),
\]
where $K$ is the classical Dodd--Jensen core computed in HOD.
The Ramsey class and the absence of $0^\dagger$ exclude an inner
model with a measurable. Strong covering then returns every
countable ordinal sequence to $K$, so $\KW$ implies Choice under
the displayed agreement. A countably supported symmetric
extension of the classical core retains the agreement, DC, and a
class of Ramseys, but has a stationary regressive map with no
stationary fiber. Appendix~\ref{app:ramsey} proves the claim and
gives its third-order formulation. The $\Pi^2_3$ bound counts
quantifiers over sets of reals; it is not a claim of minimal
complexity. The same sentence has a structural formulation as in
Definition~\ref{def:structural-pi2}. Thus $\mathsf{CR}$ cannot
replace $\CM$ in Theorem~\ref{thm:main}.

The choice of inner model matters as well. Let
$C=L[\Ord^\omega]$ denote the Chang model, the least transitive
inner model of $\ZF$ containing every countable ordinal sequence.
Under the same ambient measurables, its failure of SVC prevents
the strengthened conclusion of Theorem~\ref{thm:external-main}.

\begin{proposition}[The Chang obstruction]\label{prop:chang-no-svc}
If $U\models\ZFC+\CM$, then $C^U\models\neg\SVC$.
\end{proposition}
\begin{proof}
Every member of $C$ is uniquely definable over $C$ from a countable
ordinal sequence, with finite ordinal parameters absorbed into
that sequence \cite[Section~1.2]{MitchellChang}. For a normal
ultrapower $j:U\to M$, countable closure gives $C^M=C^U$;
hence $j\restr C:C\to C$ is elementary. We also use Kunen's
finite-movement lemma: after choosing normal measures at distinct
measurable cardinals, each ordinal is moved by only finitely many
of the corresponding ultrapowers
\cite[Lemma~4.10]{CarlSchlichtWelch}. The stated form follows from
the countable-sequence version by selecting infinitely many
exceptions if they exist.

Suppose $S\in C$ witnesses SVC. Choose an uncountable set $I$ of
ambient measurables above $\operatorname{rank}(S)$, with a normal
measure at each, and let $\lambda>\sup I$. There is in $C$ a
surjection $f:\eta\times S\twoheadrightarrow\lambda^\omega$;
Chang contains the full ambient sequence space. Let $p\in\Ord^\omega$
define $f$ uniquely over $C$. Only countably many of the selected
ultrapowers move a coordinate of $p$. Choose distinct remaining
measurables $\seq{\kappa_n:n<\omega}$ and put
$z=\seq{\kappa_n:n<\omega}$. Take one preimage $f(\xi,s)=z$.
Some selected ultrapower $j$ with critical point $\kappa_n$ fixes
$\xi$. It fixes $s$ by rank and fixes $f$ by its definition from
$p$ over the unchanged Chang model. Therefore
\[
 j(z)=j(f)(j(\xi),j(s))=f(\xi,s)=z,
 \qquad j(z)(n)=j(\kappa_n)>\kappa_n=z(n),
\]
a contradiction. All choices here are made in $U$.
\end{proof}

Replacing $\Hseq$ by $C$ in $E_2(\sigma)$ therefore makes that
theory inconsistent for every $\sigma$. This already rules out
the three-way Chang replacement. The proposition uses internal
definability over Chang and its invariance under ambient normal
ultrapowers. It does not assert invariance of the ambiently computed
$\HOD_{\Ord^\omega}$, nor does global failure of SVC exclude SVC
from each individual Chang rank.

Even the comparison of theories (1) and (3) can fail after replacing
$\Hseq$ by $C$. Relative to $\ZFC$ with a proper class of strong
cardinals and an inaccessible limit of strong cardinals, there is
a sentence $\sigma$ for which theory (3) is consistent but the Chang
version of theory (1) is inconsistent. Appendix~\ref{app:chang}
gives the statement and proof. This explains why the choice of
$\HOD_{\Ord^\omega}$ matters even without the SVC requirement in
theory (2). Whether an anti-inner-model hypothesis restores the
two-way comparison remains open; Question~\ref{q:chang-anti-inner}
considers excluding inner models with a measurable of Mitchell
order at least two.

\subsection{Proof outline and formalization}

Sections~\ref{sec:prelim}--\ref{sec:transfer} prove the local
rank-to-sequence theorem. A high Prikry sequence and a countably
closed labeling forcing produce $Q\sq\kappa^\omega$ and
$F:Q\twoheadrightarrow X$. Paired comparison removes the labeling
coordinate from low-rank names while keeping their values fixed.
External countable domination gives DC in $N(Q,F)$.
Section~\ref{sec:remove-svc} supplies the initial passage to SVC
without changing the protected rank. The core-model and atomization
reductions originate in the research archive \cite{ResearchArchive};
Section~\ref{sec:foundation} gives the atomization and fiber
purification needed for structural conservativity.

Sections~\ref{sec:external-basics}--\ref{sec:finish} complete the
external-HOD argument. Sparse Cohen rows with evaluated terminal
caps code the sequence generator over a recognizable choice
envelope. The code makes the generator ordinal-definable, while
homogeneous descent bounds the resulting HOD by the intended model.
Repeated $\diamondsuit^*$ codes provide a secure ground with
eventual GCH. A countably closed collapse gives GCH in the choice
envelope, and the final coding preserves it.
The measurable consistency implications combine this realization with
the rank-preserving preparation.

Sections~\ref{wd-inner-model-section}--\ref{wd-consistency-section}
prove the Woodin theorem. The relative fully backgrounded construction
and its top-iteration alternative give a canonical choice inner model
with a proper class of Woodins, presented by set forcing over its own
HOD. Homogeneous descent produces the required symmetric model.
In the other direction, an old-ground form of the local Prikry
construction and bounded high coding realize its protected rank.
Section~\ref{wd-exact-section} uses the special reverse ground to
obtain GCH and exact equality with a symmetric ground. The proof
first codes a fixed approximation-and-cover parameter, then places
the spectrum above that code. This identifies the full choice
envelope in the final universe.
Every forcing used in the Woodin argument is set forcing.
Appendix~\ref{app:chang} gives the separate strong-cardinal obstruction
to the Chang replacement.

We use transitive-model notation for the forcing and inner-model
interpretations. This notation does not assume that consistency
supplies a transitive model. The set-forcing arguments use the
forcing theorem. Generated models use the definable transitive-base
construction, and the Foundation-free construction uses the
interpreted cumulative hierarchy over a set of atoms. These
constructions can equivalently be carried out over countable models,
using their internal notions of well-foundedness and satisfaction.
The KW and Foundation reductions in
Sections~\ref{sec:prelim}--\ref{sec:foundation} use only set forcing.
The measurable exact-realization preparation uses the progressively closed
$\diamondsuit^*$ coding of Section~\ref{sec:envelope}. Each of its set arguments takes
place in a set-sized initial segment.


\section{Small forcing and generated-model presentations}\label{sec:prelim}

\begin{lemma}[Small forcing in $\ZF$]\label{lem:ZFmeasure}
Let $V\models\ZF$, let $\bb\in V$ be a complete Boolean algebra, and let $\UU$ be a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, where $\kappa>h(\bb)$. For every $V$-generic $G\sq\bb$, the filter
\[
 \UU^G=\{a\in\Pow(\kappa)^{V[G]}:\exists b\in\UU\ (b\sq a)\}
\]
is a nonprincipal $\kappa$-complete ultrafilter in $V[G]$. In particular, arbitrary set forcing over $\ZF$ preserves a proper-class tail of measurable cardinals.
\end{lemma}
\begin{proof}
For a name $\dot a\sq\check\kappa$, the range of
\[
 \zeta\longmapsto\|\check\zeta\in\dot a\|_{\bb}
\]
is canonically well-orderable by least preimages. Its cardinality is below $h(\bb)$, hence below $\kappa$. The partition into Boolean-value fibers therefore has a unique $\UU$-positive fiber $b_{\dot a}$, with associated Boolean value $v_{\dot a}$. In the generic extension, this old fiber is contained in $a$ if $v_{\dot a}\in G$, and is contained in $\kappa\setminus a$ otherwise. Thus the generated filter decides every subset of $\kappa$.

To prove completeness, take a name for a sequence $\seq{a_i:i<\delta}$, where $\delta<\kappa$. Localize and give the name a fixed default value outside the localization. For every $i$, form the unique positive Boolean-value fiber $b_i$ in $V$. Uniqueness makes this construction uniform, so the sequence $\seq{b_i:i<\delta}$ belongs to $V$ without a choice of antichain witnesses. If each evaluated $a_i\in\UU^G$, its associated Boolean value belongs to $G$. Hence
\[
 b=\bigcap_{i<\delta}b_i\in\UU,
 \qquad b\sq\bigcap_{i<\delta}a_i.
\]
Nonprincipality is immediate. Completeness also prevents $\kappa$ from acquiring cardinality below $\kappa$. A short enumeration would express $\kappa$ as a union of fewer than $\kappa$ null singletons.
\end{proof}

For a normal measure in a $\ZFC$ ground and forcing of size below its measurable cardinal, we also use the usual normal generated lift and the downward L\'evy--Solovay theorem; see \cite{HamkinsGap}. If $D\sq W\sq D[G]$ and the forcing is small relative to a normal $D$-measure $\UU$ on $\kappa$, its restriction to $W$ is the set
\begin{equation}\label{eq:measure-restrict}
 \UU^W=\{a\in\Pow(\kappa)^W:\exists b\in\UU\ (b\sq a)\}.
\end{equation}
This definition belongs to $W$ when $D\sq W$ and $W\models\ZF$. Restricting the ambient lift gives normality and completeness. The argument requires this restriction construction; measurability need not be downward absolute to an arbitrary inner model.


\begin{lemma}[Bounded presentations]\label{lem:presentation}
Suppose $N\models\ZF$ and $M=N(Q,F)$, where $Q\sq\kappa^\omega$ and $F:Q\to X\in N$. Each set in $M$ is the value of a bounded, set-sized evaluator whose inputs are the whole sets $Q,F$, finitely many members of $Q$, parameters in $N$, and ordinal codes. The evaluator is absolute between transitive outer models containing the same inputs. For every nonempty $Y\in M$, there are an ordinal $\eta$, a nonempty set $Z\in N$, and a surjection in $M$
\[
 \pi:\eta\times Z\times Q^{<\omega}\longrightarrow Y.
\]
The whole sets $Q,F$ remain fixed parameters of this surjection; they are not replaced by finitely many of their members.
\end{lemma}
\begin{proof}
We use transitive-base constructibility. For a sufficiently large ordinal $\theta$, put
\[
 B_\theta=\TC\bigl(\{V_\theta^N,Q,F\}\bigr),\qquad
 J_0(B_\theta)=B_\theta,\quad
 J_{\xi+1}(B_\theta)=\operatorname{Def}(J_\xi(B_\theta)),
\]
and take unions at limits. The transitive base makes these levels transitive and increasing. Taking the union of these hierarchies over ground ranks and construction stages gives $N(Q,F)$. Thus each $Y\in M$ is contained in a set-sized level $J_\beta(B_\theta)$. This transitive-base construction includes the members of the adjoined sets, which predicate constructibility may omit; see \cite{Grigorieff,Usuba}.

At a fixed level, code each element by a finite expression. A base element is a leaf. An element first produced at a successor stage is specified by a formula over the preceding set-sized level and finitely many parameters from that level. Substitute descriptions of those parameters. Transfinite induction gives a finite expression for each element, with leaves in $B_\theta$ and ordinal stage labels. At a limit stage use an earlier stage. At a successor stage combine the finitely many earlier descriptions. A description of a member of $J_\beta(B_\theta)$ uses stages bounded by $\beta$, allowing a finite increase in $\beta$ if needed.

To evaluate an expression, use satisfaction for the indicated set structures $J_\xi(B_\theta)$. Set satisfaction, the definable-subset operation, and the recursion through a bounded list of levels are absolute between transitive outer models with the same base and ordinals. This proves the evaluator's absoluteness without a satisfaction predicate for the proper class $M$ or $N$.

We next reduce its leaves. Apart from the whole sets $Q,F$, an element of $\TC(\{Q,F\})$ is obtained by finitely many pairing and projection operations from individual members of $Q$ and ground elements. Indeed, values of $F$ and their hereditary members belong to $N$, and the members occurring in a sequence $q\in Q$ are built from natural numbers and ordinals below $\kappa$. The same statement holds for the ordered pairs coding the graph of $F$. Leaves from $\TC(\{V_\theta^N\})$ are already ground elements. Consequently every expression can be converted to one using the fixed whole parameters $Q,F$, a finite tuple from $Q$, and a finite tuple from one set $V_\rho^N$, where $\rho$ may be chosen uniformly from $\theta,X,\kappa$.

Let $Z=(V_\rho^N)^{<\omega}$, increasing $\rho$ so that $Z$ is nonempty and contains all the required fixed ground data. Finite syntactic expressions, finite ordinal labels below the chosen bound, and finite type information form a wellorderable set. Index it by an ordinal $\eta$. Evaluate the codes with all parameters in $Z$ and all finite tuples from $Q$. Keep an output exactly when the code is valid and its value belongs to $Y$. This defines a set-sized partial function in $M$ with range $Y$. Fix one $y_0\in Y$ and send every unused argument to $y_0$, obtaining the total map $\pi$. Using all codes avoids choosing a definition for each $y\in Y$ or a wellordering of the ground parameter set $Z$.
\end{proof}

\section{The choice symmetric ground from SVC}\label{sec:core}

\begin{proposition}[A fixed small presentation with a measurable tail]\label{prop:coretransfer}
If $N\models\ZF+\SVC+\CM$, there is a choice inner model $D$ with
\[
 D\models\ZFC+\CM,\qquad N=D(z)
\]
for some $z\in N$. There are fixed set forcings $P\in N$ and $T\in D$ and a common extension
\[
 E=N[H]=D[G]
\]
such that $N$ is a symmetric submodel of $D[G]$ for a set-sized symmetric system in $D$.
\end{proposition}
\begin{proof}
By \cite[Proposition~5.4]{Usuba}, SVC gives a choice symmetric ground $D$ and a set generator $z$ with $N=D(z)$. Use \cite[Lemma~4.2]{Usuba} to fix a common-cover presentation in which $N$ is a symmetric submodel. An arbitrary pair of forcings witnessing a common extension need not give this symmetric system; see \cite[Note~4.6]{Usuba}. A sufficiently large pair of collapses gives the required presentation.

It remains to show that this same $D$ has a proper class of measurables. Put
\[
 \theta=h^N(\RO^N(P)),\qquad \tau=|T|^D.
\]
These are fixed ordinals. For every $N$-measurable
$\kappa>\max\{\theta,\tau,\omega\}$, Lemma~\ref{lem:ZFmeasure} lifts its measure to $N[H]$. Thus $\kappa$ is measurable in the choice model $D[G]$. Downward L\'evy--Solovay for the $D$-forcing $T$, whose $D$-size is $\tau<\kappa$, puts a measure on $\kappa$ in $D$. The bounds and the ground were fixed independently of $\kappa$, so $D\models\CM$. No absoluteness of the Hartogs calculation between $N$ and $D$ is used.
\end{proof}

Now assume also $N\models\DC$. Fix an SVC witness $S\in N$, the generator $z$, the protected rank, and the ground symmetric-system data. Choose an uncountable regular $D$-cardinal $\rho$ above the size of $T$ and the rank bounds for this presentation. Then choose a sufficiently high beth fixed point $\lambda$ of $D$ with $\cf^D(\lambda)=\rho$. One obtains such a point as the supremum of a sufficiently high, strictly increasing continuous sequence of beth fixed points of length $\rho$. The forcing $T$ preserves this cofinality. Since
$D\subseteq N\subseteq D[G]$, it follows that $\cf^N(\lambda)=\rho>\omega$.

Choose $\lambda$ large enough that the transitive set
$X=V_\lambda^N$ contains $z$, $S$, and the protected data. Then
\[
 N=D(X),\qquad N\models\SVC(X).
\]
For the first equality, $D(X)\subseteq N$ and $z\in X\subseteq D(X)$; for the second, extend each given surjection on $\eta\times S$ by a default value on the rest of $\eta\times X$. Increase the bounds once more if necessary, and choose a measurable $\kappa$ of $D$ above them. The fixed forcing $T$ is small below $\kappa$, so $\kappa$ remains inaccessible in $D[G]$. All rank-bounded data below $\lambda$, including $X$ and the old countable-condition posets introduced below, have size below $\kappa$ there.

This construction preserves $N$ and its protected ranks. We use the common choice cover to choose subsequent generics and make the size calculations.

\section{The Prikry ambient universe}\label{sec:ambient}

For the next four sections fix a symmetric presentation
\[
 D\sq N\sq D[G],\qquad D\models\ZFC+\CM,
 \qquad N\models\ZF+\DC+\SVC.
\]
Write $\mathcal S=(\mathbb A,\mathcal G,\mathcal F)\in D$ for its symmetric system, and take $G$ to be $\mathbb A$-generic over $D$. 

Choose a transitive rank $X=V_\lambda^N$ containing a generator for $N$ over $D$, an SVC witness, and the structure to be protected. Choose $\lambda$ sufficiently closed, with uncountable cofinality in the full choice cover; one may take a ground beth fixed point of cofinality greater than the size of the fixed small presentation. Then choose a measurable $\kappa$ of $D$ above $\lambda+\omega$ and all presentation bounds. In particular, arrange
\begin{equation}\label{eq:setup}
 N=D(X),\qquad N\models\SVC(X),\qquad
 |X|^{D[G]}<\kappa,
 \qquad \mathcal S,\RO(\mathbb A)\in V_\kappa^D.
\end{equation}
The forcing $\mathbb A$ and all subsequent choice-envelope forcing below $\kappa$ have size less than $\kappa$. We can impose these bounds before choosing $\kappa$. The presentation is fixed, and a small forcing has fewer than $\kappa$ bounded-rank names below a sufficiently large ground inaccessible. The small cover preserves the chosen ground cofinality of $\lambda$, so $\cf^N(\lambda)>\omega$.

Let $\UU\in D$ be a normal measure on $\kappa$, and let $\pp_{\UU}$ be its \emph{old} Prikry forcing in $D$. Its ground wellordering belongs to $N$. The old measure base generates the normal lift in every small choice cover under consideration; therefore $\pp_{\UU}$ is dense in the Prikry forcing computed there from that lift.

Put $\nu=\omega_1^N$ and let
\[
 \cc=\{k:k\text{ is an }N\text{-countable partial function }
                   \nu\longrightarrow X\}^{N},
\]
ordered by reverse inclusion. We keep the poset computed in $N$ fixed. Every condition belongs to $X$, since its domain is bounded in $\nu<\lambda$ and its countable range has bounded rank below $\lambda$. We may use the dense initial-segment presentation $\bigcup_{\gamma<\nu}{}^\gamma X$ computed in $N$. In the full choice cover $|\cc|<\kappa$, after increasing the bound on the small presentation if necessary.

Choose $(c,K)$ generic for the actual product $\pp_{\UU}\times\cc$ over the \emph{full choice cover} $D[G]$, and set
\[
 W=N[c][K]=N[K][c].
\]
The product uses these fixed old posets. Genericity over the full cover gives the displayed genericities over $N$ and the stronger genericity needed in part (3) below. We do not assume that the old collapse remains countably closed after adjoining $c$.

In $N$, $\DC$ makes $\cc$ countably closed. Its generic union enumerates $X$ with domain $\nu$. By $\SVC(X)$, $N[K]$ can wellorder every old set, and the same is then true of every set of names and its evaluated range. Thus $N[K]\models\ZFC$. The full cover $D[G][K]$ is a small forcing extension of $D$, so its normal lift of $\UU$ restricts to the normal generated measure in $N[K]$. Product genericity makes $c$ Prikry generic over this choice model. Consequently $W\models\ZFC$.

\begin{lemma}\label{lem:ambientfacts}
In this setup:
\begin{enumerate}
\item ${}^{\omega}X\cap W={}^{\omega}X\cap N$;
\item every $W$-countable family of elements of $N$ is contained in an $N$-wellorderable set;
\item $V_\kappa^{N[c]}=V_\kappa^N$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (1), first force with $\cc$. The deciding-sequence argument uses $\DC$ in $N$ and the countable closure of $\cc$. It shows that $N[K]/N$ adds no countable sequence of elements of any fixed old set. In $N[K]$, $X$ has size at most $\nu<\kappa$. The Prikry property for the lifted normal measure implies that the subsequent forcing adds no bounded subset of $\kappa$. Use the enumeration of $X$ to code a countable sequence from $X$ in $W$. The sequence therefore already lies in $N[K]$, and countable closure returns it to $N$. The same argument gives equality of the reals. We use the Prikry property and infinite-subsequence criterion as stated in \cite[Section~3]{Benhamou}.

For (2), bound the ranks of the countably many ground elements by an ordinal and place them in one set $Z\in N$. Let $\seq{z_n:n<\omega}\in W$ be the resulting sequence. View it as the value of a $\cc$-name $\dot z\in N[c]$. In $W$, for each $n$ choose $k_n\in K$ which forces
\[
 \dot z(n)=\check z_n
\]
over $N[c]$. Such a condition exists by the truth lemma; a fixed localization can be included in all the $k_n$. Since $\cc\sq X$, (1) gives $\seq{k_n:n<\omega}\in N$. Inside $N[c]$, that array, $Z$, and $\dot z$ reconstruct $\seq{z_n}$ by the forcing relation: $z_n$ is the unique $z\in Z$ for which $k_n\forces_\cc\dot z(n)=\check z$. Hence the whole sequence belongs to $N[c]$.

Take a $\pp_{\UU}$-name $\dot v\in N$ for it and let
\[
 B=\{z\in Z:\exists n<\omega\ \exists p\in\pp_{\UU}
                         \ (p\forces\dot v(n)=\check z)\}.
\]
This is an $N$-set containing every $z_n$. Fix the old wellordering of $\omega\times\pp_{\UU}$. Send each $z\in B$ to its least deciding pair. Distinct values have distinct pairs, so the map induces an $N$-wellordering of $B$ without wellordering $Z$.

For (3), use the following generic-intersection fact, valid without Choice:
\[
 N_0\sq E_0,\quad \mathbb P\in N_0,\quad h\text{ generic over }E_0
 \quad\Longrightarrow\quad N_0[h]\cap E_0=N_0,
\]
where the transitive models satisfy $\ZF$ and have the same ordinals, and the forcing is the same old poset in both models.

We give the proof. First, if $h_0,h_1$ are mutually $\mathbb P$-generic over $N_0$, then
\[
 N_0[h_0]\cap N_0[h_1]=N_0.
\]
Induct on the rank of a common set $x$. Its members belong to $N_0$ by induction, so $x\sq V_\gamma^{N_0}$ for some ordinal $\gamma$. Choose names $\tau_0,\tau_1\in N_0$ for $x$ and a product condition $(p_0,p_1)$ forcing their equality and that common value to be a subset of $V_\gamma^{N_0}$. For each $y\in V_\gamma^{N_0}$, $p_0$ must decide $y\in\tau_0$: otherwise two extensions giving opposite decisions, paired with one extension of $p_1$ deciding $y\in\tau_1$, contradict the forced equality. Hence
\[
 x=\{y\in V_\gamma^{N_0}:p_0\forces_{\mathbb P}\check y\in\tau_0\}\in N_0.
\]
Separation and the forcing theorem suffice; no choice of deciding conditions for all $y$ is needed.

Now let $x=\tau^h\in N_0[h]\cap E_0$, where $\tau\in N_0$ is a forcing name. The forcing theorem over $E_0$ gives $p\in h$ with
\[
 p\forces_{E_0,\mathbb P}\tau=\check x.
\]
In a further auxiliary extension, choose $h'$ generic over $E_0[h]$ with $p\in h'$. The filters $h,h'$ are mutually generic over $E_0$, hence over $N_0$, and $\tau^{h'}=x$. The preceding intersection argument gives $x\in N_0$. This assertion concerns membership in the original ground, so the auxiliary extension does not change it. The forcing theorem for the further product gives an equivalent formulation that does not infer transitive models from consistency.

Apply this fact with $E_0=D[G]$, $N_0=N$, and $\mathbb P=\pp_{\UU}$. The old forcing is dense in normal Prikry forcing for the lifted measure over the choice model $E_0$. Small forcing preserves the inaccessibility of $\kappa$, and the Prikry property gives
\[
 V_\kappa^{E_0[c]}=V_\kappa^{E_0}.
\]
The absence of new bounded subsets suffices because $E_0$ satisfies Choice and each old rank below the inaccessible $\kappa$ has size below $\kappa$. Induction on rank proves the displayed equality. If $x\in V_\kappa^{N[c]}$, the inclusion $N[c]\sq E_0[c]$ puts $x$ in $E_0$. Generic intersection then puts $x$ in $N$. The converse inclusion is immediate. This proves (3) using the fixed old posets, without reducing to bounded symmetric names or recomputing the collapse after Prikry forcing.
\end{proof}


\section{Tagged caps and the labeling forcing}\label{sec:caps}

All sequences, carriers, records, and diagram witnesses in this section belong to $N$.
The word \emph{old} means ``in $N$,'' even when we work in an auxiliary extension.
Changing finitely many initial blocks does not affect any asymptotic requirement.
Fix $m_0\ge4$ and, for $m\ge m_0$, put
\[
 I_m=[2^m,2^{m+1}),\qquad
 w_m=2^{\lfloor\sqrt m\rfloor},\qquad
 G_m=2^{\lfloor m^{1/4}\rfloor},\qquad
 J_m=[2^m+w_m,2^m+2w_m].
\]
For $m<m_0$ use $I_0=\{0,1\}$ and $I_m=[2^m,2^{m+1})$ for $1\le m<m_0$, put $J_m=\{2^m\}$ and $G_m=w_m=1$, and give every row the value $2^m$ there.
Thus the blocks partition all of $\omega$, including the index $0$.
Include $E_*=[0,m_0)$ in every cap and diagram exceptional set.
The point $0$ and all data in the finite initial interval $[0,2^{m_0})$ contribute only finite carrier data.
Every core is therefore a subset of its block. All separation assertions concern the eventual tail.
An eventually positive sequence $d_m$ is \emph{super-$G$} if
\[
 \frac{d_m}{G_m^k}\longrightarrow\infty
 \qquad\text{for every fixed }k<\omega.
\]
By changing an initial segment, we may take every witness reserve below to be a positive integer sequence.
The core width $w_m$ is super-$G$.
A \emph{row} is an old function $f$ with $f(m)\in J_m$ and super-$G$ distances from the two core endpoints.
Write
\[
 A_f=\{f(m):m<\omega\},\qquad
 f\prec g\quad\Longleftrightarrow\quad g(m)-f(m)\text{ is super-}G.
\]
A \emph{row chain} is a set of rows linearly ordered by $\prec$.
Its pairwise order and separation requirements hold on every sufficiently late block, including exceptional blocks.

The ideal of label errors is
\begin{equation}\label{eq:Jideal}
 \mathcal J=\{a\sq\omega:|a\cap n|=o(\log n)\}^{N}.
\end{equation}
If $a\in\mathcal J$, its occupied block set $\{m:a\cap I_m\ne\varnothing\}$ has asymptotic density zero, and $|a\cap I_m|=o(m)$.
Indeed, the number of occupied blocks below $m$ is bounded by $|a\cap 2^{m+1}|=o(m)$. This also bounds the load of the $m$th block.
Since $m/G_m\to0$, this load is $o(G_m)$.

\begin{definition}[Cap records and diagrams]\label{def:cap}
A \emph{cap record} consists of an old carrier $a\sq\omega$, a countable row support $L_a$, an old density-zero set $Z_a\sq\omega$, and, for every $m\notin Z_a$, a labeling of the \emph{entire} finite set $a\cap I_m$ by members of $L_a$.
It satisfies:
\begin{enumerate}
\item $|a\cap I_m|=o(G_m)$ on all blocks, and $a\cap I_m\sq J_m$ whenever $m\notin Z_a$;
\item for each fixed $f\in L_a$, on all sufficiently late blocks outside $Z_a$, the canonical point $f(m)$ belongs to $a$ and has tag $f$.
\end{enumerate}
A \emph{finite diagram} consists of finitely many \emph{whole cap records} and finitely many specified canonical row points.
A witness for the diagram is an old density-zero exceptional set, containing the exceptions of its cap records, and a super-$G$ reserve $d_m$.
Outside that exceptional set, equal points have equal tags; the convex envelopes of distinct represented tags occur in row order; the gaps between consecutive distinct-tag envelopes and the two core margins are at least $d_m$.
A canonical point $f(m)$ has tag $f$.
\end{definition}

A single tag may occupy several widely separated pieces, with no bound on its diameter.
The finite-diagram requirement controls \emph{all} tags in each selected cap block, including tags whose canonical-point requirements have not yet stabilized.
Clause (2) stabilizes separately for each cap--row pair, not uniformly over all its tags.
On a record's exceptional blocks there is no tag or canonical-point requirement; only its all-block load bound remains.

\begin{definition}[The forcing $\rr$ and designated tops]\label{def:R}
A condition consists of a countable row chain $L$, a coloring $\ell:L\to X$, and a countable nonempty family of immutable cap records supported on $L$.
Fix once and for all the distinguished empty cap with empty support, exception set $E_*$, and empty labelings, together with the default witness $(E_*,\seq{w_m:m<\omega})$ for its one-cap diagram and for the empty diagram.
Require every condition to contain this record and these same default witnesses.
This family is \emph{finitely directed modulo finite inclusion}: for any finitely many of its carriers $a_1,\ldots,a_k$, it contains a carrier $b$ with $a_i\sm b$ for every $i$.
For every finite diagram of its caps and canonical rows, the condition includes a witness as in Definition~\ref{def:cap}.
Extension retains the rows, their colors, all cap records, and all previously assigned diagram witnesses.

A listed cap $b$ is a \emph{designated top} of this condition if its support is the entire current row chain and
\[
 a\sm b\quad\text{for every listed cap }a,
 \qquad A_f\sm b\quad\text{for every }f\in L.
\]
These containments hold modulo a finite set of \emph{points}, including on exceptional blocks.
They are additional properties of a designated top, not requirements on every cap record.
\end{definition}

With the fixed default witnesses, the condition with no rows and one empty cap is a largest condition.
All countable families and their assigned witness functions are actual old sets included in the condition.
Conditions admit uniform codes in $X$. The countably many colors have ranks bounded below $\lambda$, since $\cf^N(\lambda)>\omega$. The remaining countable data and the finite rank overhead of coding also stay below $\lambda$.
A countable decreasing chain of conditions has its literal union as a lower bound.
Every pair of rows and every finite diagram occurs at an earlier stage, so its requirements and witness persist. Finite directedness modulo finite inclusion persists for the same reason.
We use $\DC$ only to show that the union of the old countable families is countable. The diagram witnesses are already assigned.

\begin{lemma}[Dense tops and insertions]\label{lem:tops}
The following extensions are dense in $\rr$.
\begin{enumerate}
\item A condition can be extended by a designated top containing every previous cap and every current canonical row modulo finite sets.
\item A new row of any prescribed color can be inserted at any old cut in the countable current row chain.
\item Every old $a\in\mathcal J$ can be included in a cap, and every fixed integer translate of a listed cap can be included in a cap.
\end{enumerate}
\end{lemma}
\begin{proof}
\emph{Tops.}
Enumerate the caps as $a_i$ and the rows as $f_i$; when a list is finite, omit nonexistent entries in the formulas below.
Let $\mathcal D_n$ be the diagram of the first $n$ whole caps and first $n$ canonical rows.
Take increasing old density-zero sets $U_n$ containing their assigned diagram exceptions and cap exceptions.
Choose increasing thresholds $M_n$, $n\ge1$, such that, for $m\ge M_n$,
\[
 \sum_{i<n}|a_i\cap I_m|+n\le G_m/n,
\]
and, off $U_n$, the diagram $\mathcal D_n$ has reserve at least $4G_m^{n+4}$.
We may also require $w_m\ge4G_m^{n+4}$ and
\[
 |U_n\cap t|/t<1/n\qquad(t\ge M_n).
\]
Each step imposes only finitely many eventual requirements.
On $M_n\le m<M_{n+1}$ define
\[
 b\cap I_m=
 \bigcup_{i<n}(a_i\cap I_m)\ \cup\ \{f_i(m):i<n\}.
\]
Take arbitrary finite initial data, for instance empty blocks.
Let $E$ contain the initial block indices and consist thereafter of the $m$ which belong to $U_n$ on the $n$th band.
If $t$ is in that band, the exceptional indices from all earlier bands lie in $U_n$; consequently
\[
 |E\cap t|/t\le 1/n+O(1/t),
\]
so $E$ is old and density zero.

On a block outside $E$, use the labels of $\mathcal D_n$.
This copies every point of each selected cap, including points with transient high tags. Equal points receive consistent labels.
The load is at most $G_m/n$.
Every fixed old cap and row is included on \emph{all} sufficiently late blocks.
Give $b$ support $L$ and exception set $E$.
It is a valid record. A super-$G$ reserve is given, for example, by $G_m^{n+2}$ on the $n$th band.

For a finite diagram involving $b$ and specified old caps and rows, take the band index large enough to include all those old items.
Their points are then already in $b$, with the same tags outside $E$, and their union diagram is a subconfiguration of $\mathcal D_n$.
Thus $E$, enlarged by a finite initial segment, and the same bandwise reserve give a \emph{new} witness.
No old witness is changed.
The enlarged cap family is directed modulo finite inclusion because $b$ dominates all its members.
The same construction applies to any countable compatible family of records and diagram witnesses, even if it is not directed. Adjoining the constructed top makes it directed.

\smallskip\noindent
\emph{Insertion at a cut.}
First choose a designated top $b$.
Enlarge the construction's exceptional set to include the exception of $b$ and the witness exception for the diagram consisting of the whole cap $b$.
On a good block, partition its finitely many active tag packets according to the given cut.
Choose the new point in the gap between the last left packet and the first right packet, using a core endpoint if a side is empty.
If both sides are empty, use the core itself.
The gap or margin has a super-$G$ reserve. Its integer midpoint is separated from both bounding packets or endpoints by half that reserve minus one.

On exceptional blocks, use a separate slow interpolation of canonical rows.
Enumerate the old rows and activate the first $n$ only after their finite all-block order and margin bounds exceed, say, $4G_m^{n+4}$.
Between the last active row on the left and first on the right, with core endpoints at empty sides, choose an integer midpoint.
The activation thresholds tend to infinity, and every fixed row is eventually active.
This defines an old row with the required cut and super-$G$ margins on the exceptional blocks.
On good blocks, every fixed old row eventually has its canonical point in $b$ with the correct tag; the packet-gap construction gives the same required order and separation there.
Together these estimates give the eventual row inequalities on all blocks.

Give the new row the requested color.
For each new finite diagram, eventually its old cap points and canonical anchors lie in $b$ and agree with $b$'s tags outside the corresponding old diagram exceptions.
The chosen gap separates the new row from every packet.
Finite unions of these exceptions and the midpoint reserve supply new witnesses.
This extends the row chain without changing any old record. Apply the top construction again to obtain a new designated top.

\smallskip\noindent
\emph{Errors and translates.}
For $a\in\mathcal J$, use a record with carrier $a$, empty support, and exception set its occupied block set.
It has no good-block points requiring tags and has load $o(G_m)$.
It is compatible with every old finite diagram after adjoining this density-zero exception.
Apply the compatible-family version of the top construction.

For a fixed $j\in\mathbb Z$ and a listed cap $a$, take the new carrier
\[
 a\cup(a+j),\qquad a+j=\{k+j:k\in a,\ k+j\ge0\},
\]
with the same support, retaining the original tags and giving translated points their source tags.
On good sufficiently late blocks, core margins and distinct-tag gaps exceed $|j|$ by a super-$G$ amount.
Thus the translated points remain in the core and inconsistent distinct-tag overlaps cannot occur.
The original copy retains each fixed supported canonical point.
Adjoin the old exceptional block set, its adjacent block indices, and a finite initial segment to the new exception set.
This covers all boundary crossings.
The new load at block $m$ is bounded by a fixed sum of the old loads at $m-1,m,m+1$. It is therefore $o(G_m)$, since the ratios of these neighboring $G$-values are bounded.
For a new finite diagram, use the old diagram including $a$ and its other constituents; subtracting $|j|$ from its eventual gaps and margins leaves a super-$G$ reserve.
Use these reserves to assign consistent new witnesses without changing any old witness.
Finally apply the compatible-family top construction.
\end{proof}

The dense initial-segment presentation $X^{<\omega_1^N}$ of $\cc$ projects onto $\rr$.
Fix the uniform coding of conditions by elements of $X$ and start at the largest condition.
At a successor entry, adopt the decoded condition if it is a valid extension of the current one; otherwise keep the current one.
At a countable limit take the literal union.
This recursion is set-defined in $N$ and requires no global choice of lower bounds.
Every extension of the current decoded condition is reached by appending its code, which proves the projection lifting property.
By Lemma~\ref{lem:ambientfacts}(1), the relevant countable $X$-strings and condition codes are the same in $N[c]$. The old projection therefore works there as well.
Write $g$ for the projected generic.

\subsection{The finite matching lemma}\label{subsec:finite}

For an increasing bijection between finite frozen subsets of two integer intervals, including their endpoints, say it is \emph{$L$-balanced} if every pair of corresponding consecutive frozen gaps $d,d'$ satisfies
\[
 \min(d,L)=\min(d',L).
\]
Give the endpoints two distinct formal sentinel tags, one below and one above every genuine tag. These tags occur nowhere else.

\begin{lemma}[Finite tagged interpolation]\label{lem:finite}
Let $A_i$ be finite prescribed subsets of integer intervals $[u_i,v_i]$, including the endpoints, for $i=0,1$.
Let $h:F_0\to F_1$ be an increasing frozen bijection, with $F_i\sq A_i$, matching the endpoints.
Choose a common finite linear tag order $T$ amalgamating the two tag orders over the frozen tags, and suppose:
\begin{enumerate}
\item the tag maps on $A_i$ are weakly increasing, and frozen matched points have equal tags;
\item distinct-tag envelope gaps and the sentinel margins in each prescribed configuration are at least the positive integer $D$;
\item $h$ is $L$-balanced, where $L\ge D$.
\end{enumerate}
Take $T$ to consist of the tags occurring on at least one side, including the sentinels.
Let $n\ge |A_0|+|A_1|+|T|$ and suppose $D\ge100(n+1)^3$.
Then there are finite extensions $A_i'\supseteq A_i$ and an increasing tag-preserving bijection $h':A_0'\to A_1'$ extending $h$ such that, with
\[
 \varepsilon=\left\lfloor\frac{D}{100(n+1)^3}\right\rfloor,
\]
the map $h'$ is $\varepsilon$-balanced and every distinct-tag envelope gap and sentinel margin in the final configurations is at least $\varepsilon$.
Every tag occurs on both sides, and the total number of added final points is at most $|A_0|+|A_1|$, in particular $O(n)$.
For tags initially present on only one side, the construction supplies the partner packets. Partners are not prescribed independently.

There is also an untagged version. Under the balance hypothesis, the same bound on $D$ and the prescribed point count give an increasing $\varepsilon$-balanced extension retaining all points. The same linear addition bound holds, but there is no tag-separation conclusion.
\end{lemma}
\begin{proof}
We first give a two-pass untyped procedure.
Suppose matched endpoint intervals have lengths either equal and less than $R$, or both at least $R$, with at most $n$ prescribed interior points in total and $R\ge(n+1)^2$.
If the lengths are equal and short, translate the two patterns into one interval, take their union, and translate back.
Otherwise put $d=\lfloor R/(n+1)\rfloor$ and insert the prescribed source points, with successive positive gap lengths $g_0,\ldots,g_k$, where $k+1\le n+1$.
Assign target gaps the preliminary lengths $\min(g_j,d)$.
Their sum is at most $(n+1)d\le R$, no greater than the target interval length.
At least one source gap is at least $d$. Otherwise the source interval length, which is their sum, would be less than $R$.
Put all leftover target length in the first such gap.
The map copies short gaps exactly. Both members of every other gap pair have length at least $d$.

Now insert the originally prescribed target points, working backwards in each resulting gap.
Translate in equal gaps shorter than $d$; in long gaps repeat the allocation with threshold $\lfloor d/(n+1)\rfloor$.
The resulting map is balanced at
\[
 q=\left\lfloor\frac{R}{(n+1)^2}\right\rfloor.
\]
Every allocation is positive, so new interior points cannot collide with old matched endpoints.
At most one partner is added for each originally prescribed point.
This proves the untyped procedure.

Lower the frozen balance threshold from $L$ to $D$ and work separately in consecutive frozen gaps.
If the two endpoint tags of such a gap are equal, tag monotonicity forces every prescribed point in the gap to have that tag.
Use the untyped procedure with $R=D$ and give all added points that tag.
This handles every short frozen gap. Distinct endpoint tags, including a genuine tag next to a sentinel, would force the gap to have length at least $D$.

It remains to treat a frozen gap with distinct endpoint tags $s<t$.
Both corresponding gaps have length at least $D$.
The frozen tags determine the same list of required tags between $s$ and $t$ on both sides.
On one side, call the closed envelope of the prescribed points of a represented tag its \emph{packet}.
The packets are ordered, with gaps of length at least $D$ between consecutive packets.
Put
\[
 R=\left\lfloor\frac{D}{4(n+1)}\right\rfloor.
\]
For consecutive nonempty packets with last and first points $u<v$, let $h_0\le n$ be the number of intervening tags missing on this side.
Extend the left packet's zone to $u+R$, assign the $j$th missing tag the zone
\[
 [u+2jR,u+(2j+1)R]\qquad(1\le j\le h_0),
\]
and start the right packet's zone at $v-R$.
Every empty guard between successive zones has length at least $R$, because
\[
 (2h_0+3)R\le D\le v-u.
\]
At a frozen endpoint, begin or end the corresponding endpoint-tag zone at that endpoint.
Thus every zone contains its entire prescribed packet and has length at least $R$; an absent tag gets a zone of length $R$.
No packet is compressed, regardless of its diameter.
Apply this on both sides with the same ordered tag list and match the two endpoints of corresponding zones and guards.
The resulting skeleton extends the frozen endpoint map and is $R$-balanced.
Sentinel zones contain only the original outer endpoint and a temporary artificial inner endpoint.

In each matched same-tag zone apply the untyped procedure to the original prescriptions and label each added point by that tag.
Our bound on $D$ gives $R\ge(n+1)^2$.
Each active tag occurs on at least one side. If it is missing on the other side, the procedure adds a partner point there.
Leave the guards empty.
The refined skeleton is balanced at
\[
 \left\lfloor\frac{R}{(n+1)^2}\right\rfloor
 =\left\lfloor\frac{D}{4(n+1)^3}\right\rfloor
 \ge\varepsilon.
\]
Every artificial zone endpoint lies in a free interpacket gap and has an artificial partner.
Delete these endpoints in matched pairs, retaining the original frozen endpoints and all original prescriptions.
A resulting consecutive gap is a sum of corresponding skeleton gaps.
If a constituent pair is long at the displayed threshold, both sums are long; otherwise every constituent pair is equal, so the sums are equal.
Balance therefore survives deletion.
An entire empty guard remains between distinct final tags, including each sentinel and its neighboring genuine tag. Thus the final separation and margins satisfy the same bound.

After removing the artificial skeleton points, every added final point is the partner of an original prescribed point.
This gives the asserted linear bound.
Adjacent frozen gaps agree at their shared frozen endpoint, so their maps glue.
For the untagged version, apply the untyped procedure with $R=D$ separately in each frozen gap and join the resulting maps. No zones are needed.
\end{proof}

\begin{remark}\label{rem:shortgap}
Translation is used only in short equal same-tag gaps at the \emph{current} threshold.
After lowering the threshold from the original $L$ to $D$, requiring translation in every gap below the original threshold is unnecessary and may be incorrect.
The tag-preserving conclusion does not identify two separately prescribed same-tag canonical points unless the frozen map already contains that pair.
\end{remark}

To apply the lemma to caps, adjoin the core endpoints as sentinels. Take $D_m$ no larger than the diagram reserve, the frozen map's balance reserve, and $w_m$.
Each selected finite diagram uses entire prescribed cap blocks and only finitely activated canonical anchors.
The number of its active tags is bounded by its point count, so these data have $n_m=o(G_m)$ after sufficiently slow activation.
If $D_m$ is super-$G$, then so is
\[
 \left\lfloor\frac{D_m}{100(n_m+1)^3}\right\rfloor:
 \qquad
 \frac{D_m}{(n_m+1)^3G_m^k}
 \ge\frac{D_m}{G_m^{k+3}}\longrightarrow\infty
\]
for each fixed $k$, on all sufficiently late blocks.
Thus the linear load increase remains $o(G_m)$ and the polynomial reserve loss remains super-$G$.
A growing collection of diagram requirements must be activated by an old slow diagonal. The argument does not assert that an arbitrary growing minimum of reserves is super-$G$.
The finite construction is uniform. It uses the displayed leftmost allocations and assigns leftover length to the first long gap, so old finite input arrays produce old output arrays.
On exceptional blocks use the untagged version.
Any finitely many initial blocks can be handled separately when map extensions are required only eventually, while retaining the prescribed root points.

\section{Paired forcing and literal comparison}\label{sec:paired}

Throughout this section, all conditions, rows, cap records, and block maps
are old, meaning that they belong to $N$. A designated top of a condition is a
listed cap whose support is the entire row chain and which contains each
listed cap and each canonical row carrier modulo finite sets. The last
containment holds on all sufficiently late blocks, including exceptional
blocks. Cap families are finitely directed under modulo-finite
inclusion.

Fix a root condition $r_*$ with a designated top $b$. To fix finitely many
old carriers, first strengthen the root and pad a new top to contain them
literally. No existing cap record or assigned witness is changed.

\begin{definition}\label{def:paired}
A paired condition over $r_*$ consists of conditions $r_0,r_1\le r_*$,
an order- and color-preserving bijection $\pi:L_0\longrightarrow L_1$
between their row chains fixing the root rows, designated tops
$B_0,B_1$ containing $b$ literally, and old increasing bijections
\[
 h_m:B_0\cap I_m\longrightarrow B_1\cap I_m.
\]
The maps fix $b$ pointwise. Outside the core $J_m$ the two tops agree
and the maps are the identity. With core endpoints adjoined as sentinels,
the maps are $L_m$-balanced on every block, for an old super-$G$
sequence $\seq{L_m:m<\omega}$.

There is one old density-zero set $E$ containing the exceptional sets of
the two tops such that $h_m$ transports all top tags by $\pi$ whenever
$m\notin E$. Moreover, for every fixed $f\in L_0$ there is $M_f<\omega$
such that
\begin{equation}\label{eq:paired-canonical}
 h_m(f(m))=\pi(f)(m)\qquad(m\ge M_f,\ m\notin E).
\end{equation}
The designated-top requirement puts these canonical values in their respective tops on all
sufficiently late blocks.
There is no canonical-match requirement on blocks in $E$. Each side's
row chain still has super-$G$ margins and row inequalities on all
sufficiently late blocks.

An extension extends both side conditions and the row bijection,
retaining all their immutable data. For some $M<\omega$, its block map
extends the earlier map on its entire domain whenever $m\ge M$.
The extension may use a new comparison exception set and balance reserve.
\end{definition}

The exception in \eqref{eq:paired-canonical} is essential. Let
$Z\sq[m_0,\omega)$ be an infinite old density-zero set, let $p(m)$
be the midpoint of $J_m$ for $m\ge m_0$, and set $p(m)=2^m$
on the initial blocks. Give
$b=\{p(m)-1,p(m)+1:m\in Z\}$ empty support and exception set $Z\cup E_*$.
This is a valid root cap. Insert the row $p$ on both sides with different
colors. Matching its canonical point on every sufficiently late block,
while fixing $b$, would force its partner to equal $p(m)$ on all
sufficiently late $m\in Z$. The differently colored partner must be
distinct from the native row $p$ on that side, contradicting their
super-$G$ separation. Requiring matches only off $E$ avoids this obstruction.

\begin{lemma}[Paired-cap lifting]\label{lem:paired}
The paired forcing is countably closed in $N$. Both coordinate
projections have the lifting property for arbitrary one-sided
extensions. Any two extensions of $r_*$ can be strengthened to the two
coordinates of a paired condition.
\end{lemma}
\begin{proof}
We first specify a schedule and the treatment of canonical anchors, then
construct the extensions and check their records.

\smallskip\noindent\emph{Old diagonal scheduling.}
Consider countably many old density-zero sets, super-$G$ reserves,
eventual equalities and containments, and $o(G_m)$ load functions.
At stage $n\ge1$, select finitely many requirements. Increase the selected
families so that every fixed requirement is eventually selected. Requirements
concerning a particular old top or map selected at stage $n$ may also be included.
Let $U_n$ be the union of all exceptions selected up to stage $n$.
Choose strictly increasing $M_n$ such that, for $m\ge M_n$, all selected
eventual assertions hold, the selected total point load and number of
active anchors are at most $G_m/n$, and every selected reserve and $w_m$
are at least $C G_m^{n+10}$. Here $C$ is a fixed sufficiently large
constant for Lemma~\ref{lem:finite}. Also require
\[
 |U_n\cap t|/t<1/n\qquad(t\ge M_n).
\]
Each choice is possible because it involves only finitely many
asymptotic requirements. Set $s(m)=n$ on $[M_n,M_{n+1})$ and
\[
 E=\{m\ge M_1:m\in U_{s(m)}\},
\]
with finitely many initial blocks added if necessary. If $t$ is in the
$n$th band, all earlier exceptional blocks after $M_1$ belong to $U_n$,
so $|E\cap t|/t\le1/n+M_1/t$. Thus $E$ is old and density zero;
each fixed selected exception is contained in $E$ modulo finite.

The bandwise load is $o(G_m)$. The point-and-tag count is bounded
by a fixed multiple of the total whole-cap load, the active anchors
on both sides, and the sentinel count. The $G_m/n$ schedule
therefore gives $\nu_m\le G_m$ eventually. For this count
and input reserve $D_m\ge C G_m^{s(m)+10}$, enlarge the thresholds to
accommodate rounding and obtain
\[
 \left\lfloor\frac{D_m}{100(\nu_m+1)^3}\right\rfloor
       \ge G_m^{s(m)+5}.
\]
This is super-$G$ and absorbs the polynomial loss in
Lemma~\ref{lem:finite}. Only finite minima are used at each stage.
The exception set is a diagonal pseudounion, not a countable union.

\smallskip\noindent\emph{The row order and canonical anchors.}
For one-sided lifting use the stronger source row order as an abstract
order $T$, identifying old target rows by the existing row bijection.
For initial two-sided amalgamation, amalgamate the native orders over
the root by comparing their cuts in the root order. Within a single
cut, put the foreign left rows before the foreign right rows and preserve
each native order. Opposite non-root rows remain distinct abstract
labels even when their native old functions coincide. Retain their
inherited colors.

Every label is native on at least one side. Labels native on both sides are
already identified. Choose increasing finite $H_n$ exhausting $T$.
Include their native canonical-anchor diagrams, and the eventual
containments and matches of identified anchors, in the schedule.
On a good block in the $n$th band prescribe the \emph{entire} input top
blocks and the native anchors from $H_n$. Previously identified anchors
are frozen matched pairs after their stabilization bounds and off
their selected exceptions. Apply the tagged finite lemma. Define a
missing canonical value to be the image or inverse image of its native
anchor, with the intended tag.

On an exceptional block, retain each side's native anchors from $H_n$
and interpolate the missing labels independently in the order $T$,
using core endpoints as sentinels. Finite native chains have all-block
super-$G$ gaps and margins. Divide each gap among its finitely many
missing labels; the schedule makes the resulting integer gaps and
margins at least $G_m^{n+5}$. Then include \emph{all these canonical
values on both sides} in the prescribed point sets for the untyped
finite interpolation. This puts the canonical points in their tops
on exceptional blocks too, although their images need not be their
canonical partners there. The additional cost is linear in $|H_n|$.

Before a missing row becomes active, assign it any fixed interior
value on blocks $m\ge m_0$, and the required value $2^m$ on the
initial blocks. Each fixed row, and each fixed pair, is permanently active
eventually. The two block constructions therefore give old row
functions with all-block super-$G$ margins and pairwise inequalities
in the order $T$. A high-index tag in an input block
receives its partner \emph{packet} immediately because the whole block
is prescribed. Its canonical anchor is activated later.

\smallskip\noindent\emph{One-sided lifting.}
Given a paired condition with tops $B_0,B_1$ and map $h$, strengthen
its first side arbitrarily and extend it to have a designated
top $A_0$. On each sufficiently late block prescribe the entire
$A_0$ block, the entire $B_0$ block if not already included, the entire
$B_1$ block, and the active native anchors. Freeze $h$ and the core
sentinels. Outside the core add the union of the prescribed points on
both sides and use the identity.

The source diagrams involving $A_0,B_0$ and active native anchors, and
the target diagrams involving $B_1$ and its native anchors, supply
old exceptions and reserves. Include the old paired exception and the
selected canonical stabilization bounds. The old map's balance
reserve, these diagram reserves, the native-row gaps, and $w_m$
give the finite reserve minimum for the schedule. The prescribed
loads are $o(G_m)$, so the constructions for good and exceptional blocks
apply. The new maps eventually extend the entire
old maps, fix $b$, and have all-block super-$G$ balance. They define
the extended target chain and color-preserving row bijection without
changing any native row.

\smallskip\noindent\emph{Initial two-sided amalgamation.}
Extend the two given conditions to have tops $A_0,A_1$ containing $b$
literally. A previous top contains $b$ modulo finite, so padding a
new top by the finite remainder suffices. Amalgamate the abstract
row orders as above, freeze the identity on $b$ and the sentinels,
and apply the same construction to the two entire top blocks.
Include the finite diagrams comparing $b$ and the native input tops.
On exceptional blocks interpolate the active anchors separately,
insert all their values into the untyped point sets, and retain the
identity at $b$.

In both constructions use the identity on the union of all prescribed
points on finitely many initial blocks if necessary. Non-root map
extension is only eventual; the root remains fixed throughout.
Sentinels are auxiliary. For each matched pair of core endpoints,
retain both copies if either was a prescribed carrier point; otherwise
remove both copies from the new tops. This preserves the
bijection. The margin requirement keeps prescribed good-block points
away from the endpoints. Measure balance with the sentinels
adjoined. The numerical reserve requirements are asymptotic.

\smallskip\noindent\emph{Records and immutable witnesses.}
Let $C_i$ be the new top carriers, with full new row supports,
the good-block tags just constructed, and exception set $E$.
Their loads are $o(G_m)$ and their good-block distinct-tag and
margin reserves are super-$G$. Every fixed canonical row belongs
to its top on all sufficiently late blocks, including exceptional
ones, and has its own tag on good late blocks. Thus the new records
are valid designated tops.

Keep every native old record, labeling, exception set, and assigned
finite-diagram witness literally. Each native cap $a$ lies modulo
finite in its input top and hence in $C_i$. A transported record
on the other side, if included, has the exact carrier
\[
 a'=\bigcup_m h_m[a\cap C_i\cap I_m]
\]
(or the corresponding inverse image), and transported support.
Its exceptions include $E$, those of $a$, the old diagram exception
comparing $a$ to its input top, and a finite initial containment
error. Outside that finite union, its labels agree with the
top's labels, and each fixed supported row eventually has its canonical point.
Its load is at most that of $a$. Do not add canonical
corrections on exceptional blocks.

For a new finite diagram, unite the relevant finitely many
exceptions and enlarge its finite initial segment. Beyond it,
all prescribed points and correctly tagged canonical anchors lie
in $C_i$, and their tag envelopes are sub-envelopes of the new
top's packets. The top reserve witnesses the diagram. Assign the
new witnesses without changing any old one. Each new top dominates
all native and transported records modulo finite, so the cap
family is finitely directed. The constructions therefore give side
extensions in the stated forcing.

\smallskip\noindent\emph{Countable closure.}
Let $t_0\ge t_1\ge\cdots$ be an old decreasing sequence of paired
conditions. Take the unions of the side conditions, row bijections, and assigned
witnesses. Each finite diagram occurred at a stage, so the side
unions are conditions. Enumerate their caps, rows, finite diagrams,
earlier maps, and exceptions.

At stage $n$ choose a later $t_{k(n)}$ containing the first $n$
enumerated items, with $k(n)$ increasing, and use its \emph{whole}
designated tops and map. Include in the schedule their eventual
containment of the first $n$ caps and canonical rows on all blocks;
their agreement with the first $n$ earlier maps; the first $n$
canonical matches off their comparison exception; the diagrams
of the \emph{selected tops} together with the first $n$ old caps
and canonical anchors; their top and diagram exceptions;
their map and diagram reserves; and
their two top loads. Agreement comes from the extension order.
Choose $M_n$ by the schedule above.

On the $n$th band, copy the selected whole top blocks and map.
This eventually extends every earlier map and contains every old cap and
canonical row. Each fixed canonical
match holds eventually off the diagonal $E$. The root and the
identity off the core persist. Loads are $o(G_m)$, and all-block
balance reserves are super-$G$. Copy the selected top labels.
For every finite diagram involving a new top and specified old
records, the corresponding diagrams with the selected stage top
have been scheduled from some stage onwards. Their copied
blocks and reserves witness the new-top diagram off $E$ and a
finite initial segment. Add only the two new tops and their new witnesses.
They restore directedness and give a paired lower bound. Each block
copies a single selected top, rather than an infinite pointwise union of cap loads.
\end{proof}

\subsection{The objects retained in the inner model}
Let $g\sq\rr$ be generic and let $\ell_g$ be its union coloring. Define
\[
 \mathcal I_g=\{a\in\Pow(\omega)^N:
       a\sm b\text{ for some cap }b\text{ appearing in }g\}.
\]
By finite directedness and Lemma~\ref{lem:tops}, this ideal is hereditary
under old subsets, is closed under finite unions and integer translates,
and contains $\mathcal J$. Let $Q(c,g)$ consist of the increasing
enumerations of finite modifications, using arbitrary ordinals below
$\kappa$, of $c[a]$ for infinite old $a\in\mathcal I_g$.

Fix $x_0\in X$. For $q\in Q(c,g)$ set
\[
 \iota_c(q)=\{n:c(n)\in\ran(q)\}.
\]
If a generic row $f$ and an integer $j$ satisfy
$\iota_c(q)\sym(A_f+j)\in\mathcal J$, put $F(c,g)(q)=\ell_g(f)$.
If there is no such row and translate, put $F(c,g)(q)=x_0$.
The index carrier differs finitely from an old carrier, so is old.
Distinct rows have disjoint translated $\mathcal J$-clouds: their
super-$G$ separation gives distinct canonical points on every
sufficiently late block. One point on each such block is not
a $\mathcal J$-error, so the definition is unambiguous.
Lemma~\ref{lem:tops}(2) makes $F$ onto $X$.

A carrier already supported by a cap cannot acquire the cloud of
a later row outside its recorded support. The finite diagram of
the whole cap and that row separates the canonical point from all
cap points on a density-one set of blocks. The reserve eventually
exceeds any fixed translating integer. The difference from the
translated row therefore has a point on a density-one set of
blocks and is not in $\mathcal J$. Hence localized label
decisions remain unchanged by later extensions.

Every member of $Q$ is Prikry generic over $N$. To verify this for the ambient
generic, work in the collapse-first choice cover $N[K]$,
which contains every old carrier. There $c$ is generic for the
lifted measure, so Mathias's criterion applies, together with its
infinite-subsequence consequence and finite-change invariance. The old measure
base gives genericity for $\pp_{\UU}$ and hence over $N$; see
\cite[Section~3]{Benhamou}. Only the evaluations
of the fixed $N$-names $Q,F$ are retained. The row order, tags,
cap records, and master sequence are not separately adjoined.

\begin{lemma}[Literal common-model comparison]\label{lem:literal}
Let $r_0,r_1\le r_*$ and let $p_0,p_1\in\pp_{\UU}$ be compatible.
Fix finitely many supported $Q$-parameters whose old carriers lie
literally in the root top $b$, with their finite ordinal decorations.
In an auxiliary extension there are genuine
$\pp_{\UU}\times\rr$-generics $(c_0,g_0),(c_1,g_1)$ over $N$,
meeting the respective conditions, such that
\[
 Q(c_0,g_0)=Q(c_1,g_1),\qquad
 F(c_0,g_0)=F(c_1,g_1),
\]
and the supported parameters are literally equal. In particular
the two models $N(Q,F)$ are literally the same inner model there.
\end{lemma}
\begin{proof}
Work in the full choice cover of Section~\ref{sec:ambient}.
Lemma~\ref{lem:paired} gives a paired condition whose coordinates
strengthen $r_0,r_1$. Force below it with the \emph{old} paired
forcing. Its lifting assertions involve fixed old condition sets,
so they remain valid in this cover. Each coordinate is therefore
a projection below that condition. Given a dense open subset of
the old $\rr$ in the cover, strengthen the selected coordinate
into it and lift that extension. The resulting marginal
filters $g_i$ are genuinely $\rr$-generic over the cover and over $N$.

The old paired forcing and its data have size below $\kappa$
in the cover by \eqref{eq:setup}. A further forcing of size below
$\kappa$ makes its generic filter countable. Enumerate that filter
and use its directedness to take a cofinal decreasing sequence.
The sequence need not belong to $N$. Its maps eventually extend each fixed
earlier map. Copy one whole map per block beyond selected
agreement thresholds to obtain an increasing block-preserving
partial bijection $h$ of $\omega$, fixing $b$, which eventually
agrees with each old top map in the filter.

\smallskip\noindent\emph{Old localization of the carrier action.}
For $a\in\mathcal I_{g_0}$ choose an old paired condition $t$ in
the filter whose designated top contains $a$ modulo finite.
Directedness of the filter and the top property give such a $t$.
Then
\[
 a'=h^t[a\cap B_0^t]\in N,\qquad a'\sq B_1^t,\qquad
 h[a\cap\dom(h)]\eqf a',
\]
and $a\setminus\dom(h)$ is finite. Thus $a'$ is an old allowed
carrier on the other side. Apply the same argument to the inverse map.
Although the real $h$ may be new, its action on each specified carrier
is determined modulo finite by one old map.

For an old $d\in\mathcal J$ its localized image is again in
$\mathcal J$. If $n\in I_m$, then
\[
 |h^t[d\cap B_0^t]\cap n|
       \le |d\cap2^{m+1}|=o(m)=o(\log n).
\]
For a fixed generic row $f$ and integer $j$, choose an old paired
condition $t$ containing $f$ whose first top contains both $A_f$
and $A_f+j$ modulo finite. Translate absorption, dense tops,
and lifting make this a dense requirement. Write $q=\pi^t(f)$.
Once the balance reserve exceeds $|j|$,
\[
 h^t_m(f(m)+j)=h^t_m(f(m))+j.
\]
Indeed, each consecutive domain-top gap between these points is
shorter than the reserve and has its length copied exactly.
The core margins keep both endpoints in the same block eventually.
The fixed row's canonical match holds eventually off the old
$E_t$. Its image and $A_q+j$ therefore differ, modulo finite,
by at most two points in each block in $E_t$. Up to block $m$
there are at most $2|E_t\cap(m+1)|+O(1)=o(m)$ such points.
This is an old $\mathcal J$-error. Combined with the error-image
estimate, it shows that the map transports the entire translated cloud
and preserves its color. For a particular carrier in that cloud, choose
one paired condition whose top covers that carrier, the canonical
translate, and the specified error set. Directedness and dense tops
meet these finitely many requirements, so the same old map handles
the whole comparison. Apply the argument to the inverse map as well.
Only the error for the fixed row is used, not a countable union of errors. Whole exceptional
cap blocks are transported literally, not discarded.

\smallskip\noindent\emph{Genuine Prikry and product genericity.}
Choose a common strengthening $p_*=(s,A)$ of $p_0,p_1$, including
a common reservoir as well as a common stem. Normalize finitely
many initial blocks of $h$ to the identity on the whole block,
including all integers below $|s|$. The resulting map still eventually
agrees with each old map and fixes $b$.

Amalgamate two ordered copies of $\omega$, identifying $n$ on
the first with $h(n)$ on the second. Between identified points,
interleave the finite remainders in any fixed order preserving
the order of each side; do likewise in an empty block. Since
$h$ preserves finite blocks, the resulting order has type $\omega$.
Its increasing cofinal embeddings $e_i:\omega\longrightarrow\omega$
satisfy
\[
 e_0(n)=e_1(h(n))\quad(n\in\dom(h)),\qquad
 e_0(k)=e_1(k)=k\quad(k<|s|).
\]
The only cross-side identifications are the specified $h$-pairs.
Thus $e_0(n)=e_1(m)$ if and only if $n\in\dom(h)$
and $h(n)=m$.

All preparation precedes the Prikry forcing and has total
forcing size below $\kappa$ over the original choice ground.
The old normal measure generates its normal lift to this prepared
choice universe, with the old measure-one sets as a base.
Choose a master sequence $z$ Prikry generic over that entire
universe through $(s,A)$, and put $c_i=z\circ e_i$. The maps
$e_i$ are in the prepared universe, so Mathias's criterion makes
each $c_i$ Prikry generic there. The dense old presentation gives
genericity for the old $\pp_{\UU}$ as well. Each $c_i$ contains
$s$ and has all subsequent entries in $A$, so meets $p_i$.

Thus $c_i$ is $\pp_{\UU}$-generic over $N[g_i]$. Since $g_i$
is $\rr$-generic over $N$ and these are fixed old posets, the
product lemma makes $(c_i,g_i)$
$\pp_{\UU}\times\rr$-generic over $N$. No Prikry generic was
chosen before the comparison preparation.

\smallskip\noindent\emph{Literal retained equality.}
For corresponding carriers, $c_0[a]\eqf c_1[a']$.
Closure under arbitrary finite ordinal modifications in the fixed
definitions gives equality of the sets $Q$. For any $q$ in this
common set, the cross-side identification gives
\[
 \iota_{c_1}(q)\eqf
 h[\iota_{c_0}(q)\cap\dom(h)].
\]
Indeed, all but finitely many values of $q$ lie on its allowed
carrier in the first master, and that carrier is almost contained
in $\dom(h)$; the only common master values are the identified
pairs. Both index carriers are old. Applying the cloud correspondence
in both directions gives equality of the two functions $F$ on every
such $q$, including their default-label regions.

Because $h$ fixes $b$, the two subsequences on every protected
carrier $a\sq b$ agree literally. The same finite ordinal
decorations therefore give the same actual parameters. These
are equal evaluations of the fixed $N$-names $Q,F$ in one ambient
universe, so their generated models $N(Q,F)$ are literally equal.
The carrier and error ideals and the definition of $Q$ are never
recomputed using new auxiliary reals.
\end{proof}

\section{Rank preservation and dependent choice}\label{sec:transfer}

Put
\[
 M=N(Q,F),\qquad Q=Q(c,g),\qquad F=F(c,g),
\]
using the literal evaluations of the fixed names from the preceding section. Thus $N\sq M\sq W$, and $M\models\ZF$ by transitive-base constructibility.

\begin{lemma}[Eliminating the labeling coordinate]\label{lem:eliminate}
If $Y\in N$, $a\sq Y$, and $a\in M$, then $a\in N[c]$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:presentation}, fix a bounded set-sized evaluator for $a$. Its inputs are the whole sets $Q,F$, finitely many actual members $q_0,\ldots,q_{d-1}$ of $Q$, fixed ground parameters, and fixed ordinal codes. Each $q_i$ is the increasing enumeration of a finite ordinal modification of $c[a_i]$, for an old carrier $a_i$ supported by a generic cap. Choose a product condition $(p_*,r_*)$ in the actual generic that verifies these descriptions and the presentation.

By dense tops and finite padding, strengthen this condition \emph{within the actual generic} so that its designated top $b$ contains every $a_i$ literally. First cover the finitely many carriers modulo finite, then add their remaining finite points to the new top. Rename the strengthened condition $(p_*,r_*)$. Retain the old records and pad only the new top. Keep all finite ordinal decorations as fixed parameters.

Let $\dot a$ be the name obtained from this evaluator and the fixed names for its inputs, with a default value outside the chosen localization. For $y\in Y$, define in $N$
\begin{align*}
 E_y^+&=\{p\le p_*:\exists r\le r_*\ (p,r)\forces\check y\in\dot a\},\\
 E_y^-&=\{p\le p_*:\exists r\le r_*\ (p,r)\forces\check y\notin\dot a\}.
\end{align*}
Their union is dense below $p_*$. To see this, strengthen $(p,r_*)$ to decide the indicated statement. No member of $E_y^+$ is compatible in the old Prikry order with a member of $E_y^-$. Otherwise Lemma~\ref{lem:literal}, applied to their labeling witnesses, would give two genuine product-generic realizations with the same literal $N,Q,F$, the same supported $q_i$, and the same ground and ordinal parameters. The evaluator in Lemma~\ref{lem:presentation} uses satisfaction only for these same set-sized constructibility levels. It therefore has the same value in both realizations, contradicting the opposite membership decisions.

Form the old Prikry name
\[
 \dot b=\{(\check y,p):y\in Y,\ p\in E_y^+\}.
\]
Its actual value is $a$. For every $y$, the truth lemma supplies a deciding product condition in the actual generic below the localization. Its Prikry coordinate belongs to the appropriate $E_y^\pm$. The incompatibility just proved excludes the opposite decision from the Prikry filter. Thus $a=\dot b^c\in N[c]$. This argument selects no family of maximal antichains.
\end{proof}

\begin{corollary}\label{cor:rankpres}
$\Pow(X)^M=\Pow(X)^N$ and $V_\lambda^M=X$.
\end{corollary}
\begin{proof}
Every $M$-subset of $X$ belongs to $N[c]$ by Lemma~\ref{lem:eliminate}. Its rank is at most $\lambda+1<\kappa$, so it belongs to $N$ by Lemma~\ref{lem:ambientfacts}(3). This proves the power-set equality. If there were a new member of $V_\lambda^M$, choose one of least rank. All its members would belong to $V_\lambda^N=X$. It would then be a new subset of $X$, contrary to the power-set equality.
\end{proof}

\begin{lemma}[External countable domination]\label{lem:CD}
For every $\seq{q_n:n<\omega}\in W$ with $q_n\in Q$, there is an increasing $q^*\in Q$ such that
\[
 \ran(q_n)\sm\ran(q^*)\qquad(n<\omega).
\]
\end{lemma}
\begin{proof}
In $W$, choose old carriers witnessing the $q_n$ and conditions of $g$ containing their supporting caps. Since $g$ is a filter and $W\models\ZFC$, choose a decreasing sequence $\seq{r_n:n<\omega}$ in $g$, with each term below the witnesses chosen so far. The labeling conditions have codes in $X$, as verified in Definition~\ref{def:R}. By Lemma~\ref{lem:ambientfacts}(1), the whole code array belongs to $N$. Decode it there to obtain the old decreasing sequence of conditions of $\rr$.

For any old decreasing sequence $\seq{r_n}$, the set
\[
 D_{\vec r}=\{r\in\rr:(\exists n\ r\perp r_n)
                      \text{ or }(\forall n\ r\le r_n)\}
\]
is old and dense. Starting with any condition, recursively take common refinements with the successive $r_n$ as long as possible. If a step fails, the current condition is incompatible with that $r_n$. Otherwise $\DC$ in $N$ gives a descending sequence, and countable closure gives a lower bound below all its terms. Since all $r_n$ belong to $g$, a member of $g\cap D_{\vec r}$ must be below every $r_n$.

The dense-top lemma gives a further member of $g$ with a top $b$ containing all the carrier witnesses modulo finite. Let $q^*$ be the increasing enumeration of $c[b]$. It belongs to $Q$, and its range contains each $\ran(q_n)$ modulo the finite carrier errors and finite ordinal decorations. Genericity is used only after the descending sequence has been shown to be old.
\end{proof}

\begin{lemma}\label{lem:DCM}
$M\models\DC$.
\end{lemma}
\begin{proof}
We first prove that every $W$-countable family from $Q$ has an $M$-wellorderable cover contained in $Q$. For $\seq{q_n:n<\omega}$ choose $q^*$ as in Lemma~\ref{lem:CD}, and put
\[
 u_n=\{k<\omega:q^*(k)\in\ran(q_n)\}.
\]
The array $\seq{u_n:n<\omega}$ is coded by a single real of $W$, so it belongs to $N$ by Lemma~\ref{lem:ambientfacts}(1). Each $u_n$ is infinite. Let $v_n$ be the increasing enumeration of $q^*``u_n$. Equivalently, restrict $q^*$ to $u_n$ and reindex by the increasing enumeration of $u_n$. The sequence $\seq{v_n}$ is definable in $M$ from $q^*$ and the old mask array, and therefore belongs to $M$.

Take the increasing enumerations of all finite modifications of the ranges of the $v_n$, allowing arbitrary finite ordinal additions and deletions below $\kappa$. Intersect this family with $Q$ and call the resulting set $C$. It contains every $q_n$, since only finitely many values of $q_n$ lie outside $\ran(q^*)$. The set $C$ is an image of a wellorderable parameter set consisting of $n<\omega$, finitely many ordinals below $\kappa$, and finite operation codes. Least preimages give a wellordering of $C$ in $M$. This does not imply that the original external sequence $\seq{q_n}$ belongs to $M$.

Let $R\in M$ be a serial relation on a nonempty set $Y\in M$, and prescribe $y_0\in Y$. Choose a presentation in $M$
\[
 \pi:\eta\times Z\times Q^{<\omega}\twoheadrightarrow Y,
 \qquad Z\in N,
\]
as in Lemma~\ref{lem:presentation}. Seriality is absolute to $W$ because it quantifies only over the fixed sets $Y,R$. In $W\models\ZFC$, choose an $R$-chain starting at $y_0$ and an argument of $\pi$ for each entry. By Lemma~\ref{lem:ambientfacts}(2), an $N$-wellorderable set $B\sq Z$ covers the resulting countably many ground coordinates. Intersect the cover with $Z$ if necessary. Combine the finite $Q$-tuples into one countable family and apply the preceding paragraph to obtain an $M$-wellorderable cover $C\sq Q$. Then
\[
 Y_0=\pi[\eta\times B\times C^{<\omega}]
\]
belongs to $M$, is wellorderable there by least preimages, and contains the chosen external chain.

Inside $M$, form the tree $T$ of finite $R$-chains in $Y_0$ starting at $y_0$. Write $t\mathrel{\lhd}s$ when $t$ properly extends $s$. A rank function for this relation must assign smaller values to longer chains. The tree is wellorderable. If the extension relation were not well founded, a nonempty subset of $T$ would have no terminal member. Using the fixed wellordering, recursively choose the least proper extension within that subset. This gives an infinite branch in $M$ without any choice principle.

Thus, if there were no internal branch, the extension relation would be well founded and would have an ordinal rank function by well-founded recursion in $\ZF$. The same function would remain a rank function in $W$, contradicting the chosen external branch. Hence an $R$-chain starting at $y_0$ exists in $M$, proving $\DC$.
\end{proof}

\begin{proposition}[The sequence seed]\label{prop:seed}
The model $M$ satisfies $\ZF+\DC+\CM+\SVC+\KW$. There are an ordinal $\mu$ and a set $A\sq\mu^\omega$ with $M=D(A)$. Moreover, $D$ is a symmetric ground of $M$.
\end{proposition}
\begin{proof}
Because $N\models\SVC(X)$ and $F:Q\twoheadrightarrow X$, the presentation in Lemma~\ref{lem:presentation} gives, for every nonempty $Y\in M$, a surjection
\[
 \eta\times Q^{<\omega}\twoheadrightarrow Y
\]
after increasing the ordinal factor. To obtain it, cover the ground parameter set $Z$ by $\xi\times X$, replace the $X$-coordinate by $F(q)$, and combine the two ordinal factors and the finite $Q$-coordinates. Thus $Q^{<\omega}$ is an SVC witness in $M$.

For each such $Y$, choose an ordinal $\mu$ large enough to contain $\eta,\kappa$ and the finite coding symbols. To inject an ordinal coordinate and a finite tuple of members of $Q\sq\kappa^\omega$ into $\mu^\omega$, code the tuple length and ordinal in the initial positions, then interleave the finitely many sequences. Transfer the displayed surjection to the image of this injection. Extend it to all of $\mu^\omega$ by assigning one fixed element of $Y$ outside that image. This proves $\KW$ with the power $\mu^\omega$ computed internally in $M$.

To obtain one sequence seed for the whole model, define
\[
 R(p,q)\quad\Longleftrightarrow\quad F(p)\in F(q)
 \qquad(p,q\in Q).
\]
Transitivity of $X$ and surjectivity of $F$ give
\[
 F(p)=F(q)\quad\Longleftrightarrow\quad
 \forall t\in Q\,[R(t,p)\leftrightarrow R(t,q)].
\]
Indeed, distinct sets in the transitive set $X$ have a distinguishing member in $X$, and that member has an $F$-preimage. Equality of predecessor sets is therefore an equivalence relation. Its quotient relation is well founded and extensional, and the map $[q]\mapsto F(q)$ is an isomorphism to membership on $X$.

Use different finite tags for domain and relation codes, and interleave pairs of sequences. The resulting set $A\sq\mu^\omega$ codes the whole domain $Q$ and binary relation $R$, for one sufficiently large ordinal $\mu$. Both sets can be decoded in $D(A)$. Their quotient relation remains well founded and extensional. Its unique Mostowski collapse therefore recovers the same literal $X$ and the map $F$. Hence $D(A)$ contains $N=D(X)$ and then $M=N(Q,F)$. Conversely, $A\in M$ and $D\sq M$, so minimality of transitive-base constructibility gives $D(A)\sq M$. Therefore $M=D(A)$.

The full choice cover $D[G][c][K]$ is a set-forcing extension of $D$. Above the size of its forcing, every normal measure in $D$ has its normal generated lift in this cover. For each such ground measure $\UU$ on $\delta$, its restriction to $M$ is the set
\[
 \{a\in\Pow(\delta)^M:\exists b\in\UU\ (b\sq a)\},
\]
which belongs to $M$ by Separation. It is a nonprincipal $\delta$-complete ultrafilter. Every $M$-sequence of fewer than $\delta$ members is also a sequence in the full cover, where the generated lift is complete. Its intersection belongs to $M$ and satisfies the same ground-witness definition. Hence $M\models\CM$. Only a proper-class tail is needed, although the selected $\kappa$ was singularized.

Finally, $D\sq M\sq D[G][c][K]$ and $M\models\SVC$. By the intermediate symmetric-model theorem, in the form of \cite[Corollary~5.10]{Usuba}, $D$ is a symmetric ground of $M$. This uses the intermediate-model theorem; the notation $D(A)$ alone would not give the conclusion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:transfer-intro}]
Start with the $\ZF+\DC+\SVC+\CM$ model $N$ in the local statement. By Proposition~\ref{prop:coretransfer}, this same $N$ has a symmetric-ground presentation over a choice ground $D\models\ZFC+\CM$. Choose the protected rank $X=V_\lambda^N$ and the measurable $\kappa$ as in \eqref{eq:setup}, with $\lambda$ above the required rank. The construction gives $N\sq M$. Corollary~\ref{cor:rankpres} proves $\Pow(X)^M=\Pow(X)^N$ and $V_\lambda^M=X$, while Lemma~\ref{lem:DCM} and Proposition~\ref{prop:seed} give the required theory and the presentation $M=D(A)$ with $D$ a symmetric ground. This proves the local theorem. Sections~\ref{sec:remove-svc}--\ref{sec:foundation} remove SVC and Foundation from the starting assumptions of the structural conservativity theorem.
\end{proof}


\section{The core-model preparation removes SVC}\label{sec:remove-svc}

\subsection{The standard real-kukri and core alternatives}

All mouse constructions in this section take place in a choice
background. The real is an amenable bottom predicate in the usual
relative $J$-hierarchy, which starts from the empty set. It is not
adjoined with its transitive closure as a separate initial base.
Every measure critical point is above $\omega$, and every comparison
map fixes the bottom real. We use the standard Jensen--Zeman kukri
for the positive alternative and the classical Mitchell true core
for the negative alternative. We do not identify their fine indexing
or their resulting models.

An $r$-kukri is the least sound active $r$-mouse whose top normal
measure has critical point $\kappa$ and whose lower measurable
cardinals are unbounded below $\kappa$. It projects to $\omega$ and
is countable in the background. The standard least-sound extraction
is the rank-one instance of \cite[Definition~1.4 and Proposition~1.6]{MiceMachetes}.
The corresponding top iteration is described in
\cite[Section~2, Definition~2.1 and Facts~(i)--(v)]{WelchHartig}.
Only lower measurables will be used as witnesses in the resulting
class model.

\begin{lemma}[Noncreation of the least real-kukri]\label{lem:kukri-noncreation}
Set forcing over a choice background does not create an $r$-kukri
for a ground real $r$.
\end{lemma}
\begin{proof}
Use the standard initial-segment comparison for sound mice projecting
to their common real base, and fixed-countable-code absoluteness of
linear iterability. These are the real-bottom forms of the ordinary
comparison and absoluteness arguments; see
\cite[Lemmas~3.32 and~3.39]{MitchellCovering} and
\cite[Chapters~4--5]{ZemanInnerModels}. All structures in each
comparison use one chosen standard category.

Suppose a condition $p$ forces that the least such mouse exists,
and let $\dot P$ name its whole transitive fine structure, including
its predicates. In the product forcing below $(p,p)$ the two
interpretations $P_0,P_1$ are sound iterable mice over the same real.
Comparison makes them equal or makes one a proper fine initial
segment of the other. If $P_0\lhd P_1$, the factor containing the
whole $P_1$ also contains the whole fine initial segment $P_0$,
including its active record, obtained from $P_1$ and the ordinal
height of $P_0$. It is countable in that factor. Choose a real code
there. Its iterability in the product returns to the factor by
fixed-code $\Pi^1_2$ absoluteness. Soundness, projection and the
top-limit property are absolute tests on the same set structure.
This contradicts the factor's minimality of $P_1$. The reverse
proper-segment alternative is excluded in the other factor.
No assertion of minimality in the product is required.

Thus $(p,p)$ forces the two copies of $\dot P$ equal. The product-name
lemma gives a ground set $P$ with $p\forces\dot P=\check P$.
This applies to the whole fine structure, not just to a real code.
Its sound first-projectum-$\omega$ presentation gives a canonical
Skolem surjection from $\omega$, using its finite standard parameter.
Satisfaction for the whole set structure belongs to the ground, so
this surjection belongs there too. Hence $P$ is already countable
in the ground. Choose a ground real code and use fixed-code
absoluteness once more to obtain its iterability there. This
contradicts the supposed nonexistence. Conversely an existing
countable iterable eligible mouse persists, so the existence test
is invariant under set forcing.
\end{proof}

\begin{lemma}[The fixed-real classical core]\label{lem:fixed-core}
Let $B\models\ZFC$ and let $r\in B$ be a real. If $B$ has no
$r$-kukri, the classical true core $K(r)^B$ is a uniformly
$r$-definable inner model of $\ZFC$ containing $r$. It is unchanged
by set forcing over $B$. In this same core, every ambient singular
cardinal that is regular in the core is measurable.
\end{lemma}
\begin{proof}
We apply the classical core construction, generic-invariance theorem
and full-covering theorem over the fixed real. The construction and
covering framework are in \cite[Sections~1, 3 and~4]{MitchellCovering};
the singular-cardinal consequence is its Theorem~2.5. The classical
true-core generic-invariance application is also recorded in
\cite[Section~4, Lemma~4.2]{GitikHayutKaragila} and
\cite[proof of Theorem~3.11]{HayutKaragila}. This theorem concerns
the true core, not clausewise absoluteness of its preliminary
countably complete construction.

The standard obstruction is excluded by the absence of the
$r$-kukri. In the usual ordering of mice, an order-one upper
measure supplies an eligible order-zero active mouse with
cofinally many lower measurables; taking its least sound core
gives the kukri. The stronger $\mu$-measurable
obstruction in the full-covering statement supplies the same
configuration. Here the source convention is a $P$-internal elementary embedding
$i:P\to M$ with $M$ an inner target of $P$. If its critical point
is $\kappa>\omega$ and its whole derived normal measure $U$ belongs
to $M$, then
\[
 \Pow(\kappa)^P
 =U\cup\{\kappa\setminus a:a\in U\}\sq M.
\]
Thus $P$ and $M$ have the same $\kappa$-subsets; normality and
completeness restrict to $M$, which regards $\kappa$ as measurable.
Hence
\[
 \{\xi<\kappa:P\models\text{``$\xi$ is measurable''}\}\in U.
\]
The standard relative mouse extraction and least-sound extraction
therefore supply an $r$-kukri. Here $i$ fixes the real in $P$.
This argument excludes the relative obstruction; it makes no
claim about unrelativized mice that omit $r$.

For completeness, the fixed-real application keeps the real as
one amenable predicate throughout the rudimentary hierarchy,
fine codes and comparisons. All hulls used for covering or for
the true-core construction contain $\omega$ and $r$. Their
collapses fix the real, and all comparison critical points lie
above it. Including this fixed countable base leaves the
covering cardinal bounds and thick-hull argument unchanged.
The failure configuration is consequently a mouse over that
same real. This is the ordinary real-bottom instance of the
classical theorems, not a comparison of varying arbitrary
oracles. Lemma~\ref{lem:kukri-noncreation} keeps the obstruction
absent in the required set-forcing extensions. The true-core
theorem gives literal equality of the ground and extension cores.
\end{proof}

\begin{lemma}[Measurable capture by singularization]\label{lem:core-capture}
In the negative alternative of Lemma~\ref{lem:fixed-core}, every
measurable cardinal of the choice background is measurable in
its fixed core.
\end{lemma}
\begin{proof}
Let $\kappa$ be measurable in the choice background $C$, and
choose a normal measure there. Ordinary Prikry forcing preserves
cardinals and makes $\cf(\kappa)=\omega$ \cite{Benhamou}.
Before forcing, $\kappa$ was regular in the inner model
$K=K(r)^C$. In the Prikry extension $C[h]$, generic invariance
gives $K(r)^{C[h]}=K$. Thus $\kappa$ is an ambient singular
cardinal regular in this same core. Apply the full-covering
consequence in Lemma~\ref{lem:fixed-core}. Its witnessing measure
belongs to the unchanged ground class $K$.

This is an internal forcing proof in $\ZFC$ of an assertion about
the unchanged core. It assumes no actual generic filter in the
original ambient universe. It uses neither a trace of the
original measure on $K$ nor an extra weak-amenability assertion.
This auxiliary Prikry argument adds no stage to the main
Prikry--labeling construction.
\end{proof}

\begin{lemma}[Measurables in one fixed relative core]\label{lem:fixed-measurables}
In a $\ZF$ universe with unbounded measurables, put $B=\HOD(r)$.
If $B$ has no $r$-kukri, then $K(r)^B\models\CM$.
\end{lemma}
\begin{proof}
Fix one ambient measurable $\kappa$ and one ambient
$\kappa$-complete nonprincipal ultrafilter $\mu$. Set
\[
 C=\HOD(r,\mu),\qquad u=\mu\cap\Pow(\kappa)^C.
\]
The parameterized Vop\v{e}nka theorem applies in $\ZF$: take
$Y=r$, $X=V_\theta\ni\mu$ and $x=\mu$ in
\cite[Lemma~3.6]{GoldbergChoiceless}. Thus $C$ is a set-generic
extension of $B$ and satisfies $\ZFC$.

The whole $u$ belongs to $C$, since it is ordinal definable from
$r,\mu,\kappa$ and every member is already in $C$. This does
not require $\mu\in C$. Every $C$-subset of $\kappa$ is decided
by $u$. Every whole $C$-sequence of fewer than $\kappa$ members
of $u$ is also an ambient sequence, so its intersection belongs
to $u$ by ambient completeness. Nonprincipality also transfers.
Consequently $C$ regards $\kappa$ as measurable.

The negative alternative persists by
Lemma~\ref{lem:kukri-noncreation}. Apply
Lemma~\ref{lem:core-capture} inside $C$ and use
$K(r)^C=K(r)^B$. This makes the original ambient $\kappa$
measurable in the one fixed core. Repeat the existential
argument separately for each ambient measurable. No simultaneous
choice of measures, ambient normality or ambient Dependent
Choice is needed.
\end{proof}

\subsection{The positive choice model and recoding}

\begin{lemma}[The top-iteration choice model]\label{lem:kukri-class}
If a choice background contains an $r$-kukri, its canonical top
iteration gives a uniformly $r$-definable inner model $S_r$
containing $r$ and satisfying $\ZFC+\CM$.
\end{lemma}
\begin{proof}
Use the least sound mouse and its standard top-only iteration
$(P_i,\kappa_i)$, with the real predicate fixed. The construction
in \cite[Section~2, Facts~(ii)--(v)]{WelchHartig} gives the
inner model
\[
 S_r=\bigcup_{i\in\Ord}H_{\kappa_i}^{P_i},\qquad
 |P_i|=H_{(\kappa_i^+)^{S_r}}^{S_r}.
\]
Its critical points are unbounded, and its displayed lower cuts
are coherent elementary cuts. The theorem applies with the
fixed amenable real predicate included in every hull and fixed
by every map; see the fine-structure convention in
\cite[Chapter~1]{ZemanInnerModels}.

The whole $H$-cut identification supplies all final subsets and
short graphs below $\kappa_i$. Top normality makes $\kappa_i$
inaccessible in $S_r$, so the lower cut is $V_{\kappa_i}^{S_r}$.
Given any bound below $\kappa_i$, choose a lower measurable
$\delta$ above it in $P_i$. Its whole normal measure and all
$S_r$-subsets of $\delta$, regressive graphs and short families
of such subsets have rank below $\kappa_i$. They occur in the
same unchanged cut. The measure therefore remains full and
normal in $S_r$. These lower witnesses are unbounded, giving
$\CM$. No measurability of the moving top itself is used.

The published construction supplies $\ZFC$ and canonical
set-length iterates. Its defining recursion takes place inside
the choice background, so it defines one class without class
choice. The bottom real is retained throughout.
\end{proof}

\begin{lemma}[Recoding covariance]\label{lem:recoding}
Mutually recursive reals $r,s$ in the same choice background give
the same pure output in each of the two alternatives above.
\end{lemma}
\begin{proof}
We first check the pure carriers in the declared predicate
convention. The first $J$-block is $\mathrm{HF}$ for either
real. The second is the appropriate rudimentary closure of
$\mathrm{HF}\cup\{\mathrm{HF}\}$. In $J_2[s]$, both
$\omega$ and the whole set $\mathrm{HF}$ are present, as is
the whole trace $s$. If program $e$ computes $r$ from $s$, then
\[
 r=\{n\in\omega:\exists c\in\mathrm{HF}\;
       c\text{ is an accepting $s$-oracle computation of $e(n)$}\}.
\]
This is bounded Separation with the old set $\mathrm{HF}$ as
the computation bound. Thus $r\in J_2[s]$. Since intersection
with this old $r$ is rudimentary, $J_2[s]$ is closed under the
$r$-predicate operations. Reverse the programs and use minimality
of rudimentary closure to obtain the same $J_2$. Induction gives
the same subsequent $J$-carriers when the measure predicates
are held fixed. The finite auxiliary $S$-stages need not agree.

The inverse formula translations preserve $\Sigma_1$ truth
with each fixed ordinal parameter tuple. Above $\mathrm{HF}$,
computation witnesses are bounded by the old set
$\mathrm{HF}$; its uniform $\Sigma_1$ name can be witnessed
by the whole pure finite-stage construction tower, whose graph
is a bounded subset of $\mathrm{HF}$ at $J_2$. At an
$\mathrm{HF}$ reduct, bounded quantifier domains are finite,
so one finite evaluation tree collects the needed finite
computation witnesses. This supplies the same translations
without assuming that $\omega$ is a set in that reduct.

Consequently missing-subset projecta and good parameter classes
agree. Each canonical Skolem graph is a uniquely defined partial
$\Sigma_1$ function in the other language. The two Skolem
hulls are closed under one another's such functions and hence
are identical. Very-good parameter classes agree; the unchanged
ordinal ordering gives the same least standard parameters.
Repeat this argument on the theory-code reducts. Their formula
indices translate recursively while tuple coordinates stay fixed.
Whole bounded theory witnesses translate by bounded image or
preimage under the old recursive translation graph. At an
$\mathrm{HF}$ reduct the finite witnesses are hereditarily
finite in both presentations. Thus the code recovery and pure
sound-core collapses agree. These are the standard fine-code
operations of \cite[Chapter~1]{ZemanInnerModels}, in each
chosen category separately.

The transport is invertible on the eligible mice and commutes
with their comparisons, measure quotients and direct limits.
It therefore preserves the least kukri and its top-iteration
cuts. In the negative alternative, inverse transport preserves
the class of relative candidate mice, their comparison relation
and the ambient thick-class tests in this same background.
Apply the standard true-core definition and uniqueness theorem
to these corresponding data. The pure true cores are consequently
equal; no literal equality of preliminary auxiliary constructions
is required.
\end{proof}

\subsection{Canonical refinement inside a choice model}

We now obtain the own-HOD presentation directly from a choice model
with unbounded measurables. This construction uses ordinary normal
measures and their iterated ultrapowers. It applies inside either
choice model supplied above.
For a set selection $F$ of measures, $L[r,F]$ denotes construction
with its domain and filter-incidence predicates, rather than with the
whole vector and its transitive closure as an initial seed.

\begin{lemma}[Bounded uniqueness and exhaustion]\label{lem:bounded-refinement}
Work in a choice universe $B$. Let $d$ be a set of measurable cardinals
above $\omega$ such that $d\cap\kappa$ is bounded below $\kappa$
for every $\kappa\in d$. Let $F$ and $G$ be whole selections of full
normal $B$-measures on $d$, and let $r$ be a real. Then
\[
 L[r,F]=L[r,G],
\]
and the two predicates have the same restricted kernels in this model.
The model's only normal measures are these listed restricted measures.
\end{lemma}
\begin{proof}
The restricted kernel of a bounded selection is a whole set in its
predicate model: form the model's bounded powerset and tuple carrier,
take one later construction-stage trace of the incidence and domain
predicates on that carrier, and regroup the trace by Replacement.
Each restricted measure is full and normal there, since all its
internal test sequences and functions are background sets.

For uniqueness put $T=\ran(F)\cup\ran(G)$. If $U\in T$ is on
$\kappa$, then
\[
 \kappa\in j_U(d)\quad\Longleftrightarrow\quad d\cap\kappa\in U,
\]
which is false by spacing. Thus $j_U(T)$ has no measure at the old
$\kappa$. The whole earlier family has rank below $\kappa$ and is
fixed. Consequently $T$ is a closed set of measures in the sense of
\cite[Lemma~19.35]{Jech}, and every member has relative $T$-rank zero.
This permits several different measures at one critical point.

Apply that lemma to two copies of $(B,T)$, obtaining ordinary
iteration maps $i:B\to M$ and $j:B\to N$, with, say, $i(T)$ an
initial segment of $j(T)$. Write $\ell(e)=\sup\{\delta+1:\delta\in e\}$.
The initial-segment test compares \emph{all} cross-pairs of measures
at a common critical point and relative rank on all common subsets
\cite[(19.20)]{Jech}. Both $i(F)$ and $i(G)$ have domain $i(d)$,
including any new positions in this image domain. The restriction
$j(F)\restr\ell(i(d))$ has the same domain. Thus simultaneous
induction through the three relative hierarchies gives
\[
 L[r,i(F)]=L[r,i(G)]=L[r,j(F)\restr\ell(i(d))].
\]
At each common level every predicate argument belongs to both terminal
models, so the incidence answers agree. The first two predicates also
agree on every query in their common construction. These are
first-order assertions inside $M$ about its two set parameters
$i(F),i(G)$. Pulling them back through the single map $i$ proves
both assertions of uniqueness. All critical points fix $r$.

For clarity, this comparison terminates at a set stage. Choose
$\gamma$ above the original domain and put
$\mu=2^{|\gamma|}$ and $\Theta=\mu^+$ in $B$. Finite-support
representations bound each current image of $\gamma$ and
$\Pow(\gamma)$, at stage $\nu<\Theta$, in size by
$\mu\cdot|\nu|^{<\omega}<\Theta$. All choices can therefore be
made in one wellordered set carrier. If comparison continued through
$\Theta$, Fodor's lemma at limit stages, followed by stationary
thinning, would make the earlier representatives of the disagreement
ordinal constant on both sides. In the two-measure case it also makes
the representatives of a common distinguishing subset constant.
Slot erasure makes comparison critical points strictly increase.
At a one-sided step, the padded side consequently fixes the old seed
while the other moves it. At a two-sided step, normality puts the old
seed into exactly one image of the distinguishing subset, and later
maps preserve that disagreement. Both conclusions contradict the
equal later representatives. This is the set version of the
comparison argument in \cite[Lemma~19.35]{Jech}.

For exhaustion, suppose a predicate model has an extra normal measure.
Choose its first critical prefix with this property, allowing the
whole final prefix. Replace that selection by its whole internal
kernel $E$ and write $M_0=L[r,E]$. Every proper critical prefix of
$E$ has no extra normal measure in its own predicate model. These
prefix constructions are literal inside $M_0$, so $M_0$ satisfies
this assertion. Choose there the least $\kappa$ carrying an extra
normal measure, then an extra $W$ of least Mitchell rank, breaking
ties by the canonical $L[r,E]$ order. This choice is definable from
$r,E$ alone; the Mitchell order is well founded
\cite[Lemma~19.32]{Jech}.

Let $j_W:M_0\to N_0=\operatorname{Ult}(M_0,W)$ and $E_1=j_W(E)$.
Compare $(M_0,E)$ and $(N_0,E_1)$ by the same closed-family lemma,
with terminal maps
\[
 i:M_0\to P=L[r,I],\qquad k:N_0\to Q=L[r,J].
\]
If $I$ were a proper critical prefix of $J$, literal hierarchy
induction would give $P=L[r,J\restr\ell(\dom I)]\sq Q$.
Its restricted kernel is exactly $I$. The transferred minimal-prefix
assertion in $Q$ forbids an extra normal measure in this very model
$P$, whereas $i(W)$ is such a measure. No fullness over $Q$ is being
inferred here. The reverse orientation is excluded in the same way.
Thus $P=Q$ and $I=J$ as whole restricted kernels.

Put $l=k\circ j_W$. The maps $i,l$ have the same target and the
same images of $r,E$. The canonical definitions give
\[
 i(\kappa)=l(\kappa),\qquad i(W)=l(W).
\]
The normal ultrapower $N_0$ has the same subsets of $\kappa$ as
$M_0$. If $E_1$ has a measure $T$ at the old $\kappa$, then
$T\in N_0\sq M_0$. Codes as $\kappa$-subsets for short families
and regressive graphs show that $T$ is a full normal $M_0$-measure.
It is a Mitchell predecessor of $W$, so minimality of $W$'s rank
forces $T$ to be the sole listed measure at $\kappa$.

The two initial sequences agree below $\kappa$. If they agree at
$\kappa$ as well, all comparison critical points exceed $\kappa$;
then $i(\kappa)=\kappa<j_W(\kappa)\leq l(\kappa)$, a
contradiction. A target-only slot is an unlisted predecessor and
has already been excluded. The remaining case is a source slot
$U=E(\kappa)$ and no target slot there. The first source step uses
$U$, the target pads, and every later critical point exceeds
$\kappa$. Choose the first subset $x$ distinguishing $U$ from
$W$ in the canonical $L[r,E]$ order. The images of $\kappa,W,U$
under $i,l$ agree, so this definition gives $i(x)=l(x)$. But
normality and the fixed old seeds give
\[
 x\in U\ \Longleftrightarrow\ \kappa\in i(x),\qquad
 x\in W\ \Longleftrightarrow\ \kappa\in l(x),
\]
contradicting the choice of $x$. This proves exhaustion. All
iterations here use images of a whole set family of full normal
measures inside a ZFC model, so ordinary set-family iterability
applies \cite[Lecture~2]{SteelUltrapowers}.
\end{proof}

\begin{lemma}[Uniform canonical refinement]\label{lem:canonical-refinement}
There is one first-order formula $\varphi(x,r)$ such that in every
$B\models\ZFC+\CM$, for each real $r\in B$, the class
\[
 C_r^B=\{x\in B:B\models\varphi(x,r)\}
\]
is an inner model containing $r$ and satisfying
$\ZFC+\CM+V=\HOD(r)$. If $r,s$ are recursively interdefinable
in this same $B$, then $C_r^B=C_s^B$. Moreover $C_r^B$ is a
set-forcing extension of its own HOD.
\end{lemma}
\begin{proof}
Work in $B$. Recursively let $\kappa_\alpha$ be the least measurable
strictly above
\[
 \max\bigl(\omega+1,\sup_{\beta<\alpha}(\kappa_\beta+1)\bigr),
 \qquad d_\alpha=\{\kappa_\beta:\beta<\alpha\}.
\]
Each set-length recursion is uniquely definable. Every earlier listed
domain is bounded strictly below the next listed cardinal, including
at limit steps. For each set $\alpha$, Replacement and Choice in
$B$ supply a whole selection $F$ of full normal measures on
$d_\alpha$. Set $N_\alpha^r=L[r,F]$. By
Lemma~\ref{lem:bounded-refinement} this model is independent of the
selection. We use the single formula
\[
 \varphi(x,r)\quad\Longleftrightarrow\quad
 \exists\alpha\,\exists F\,
 \bigl(F\text{ is such a selection on }d_\alpha
       \ \wedge\ x\in L[r,F]\bigr).
\]
All quantifiers and measure tests in this formula are evaluated in
$B$. No proper-class selection of measures is made.

For $\alpha<\beta$ choose one $F$ on $d_\beta$. Inside
$N_\beta^r$ take the whole prefix kernel on $d_\alpha$, using
the entire bounded powerset and tuple carrier. Its domain is the
old domain cut at the ordinal $\kappa_\alpha$. Literal hierarchy
induction identifies construction from this kernel with
$L[r,F\restr d_\alpha]$. Hence $N_\alpha^r\sq N_\beta^r$.
The kernels agree on every older positive and negative decision.

Write $C=\bigcup_\alpha N_\alpha^r$. This uniformly definable
transitive class contains all ordinals and is closed under the
G\"odel operations. For any $B$-set of its elements, Replacement
bounds their least appearance stages by one ordinal; a rank set
in a later $N_\alpha^r$ covers them. Thus $C$ is almost universal
and satisfies $\ZF$ \cite[Theorem~13.9]{Jech}. Whole wellorders
from the prefix models give Choice in $C$.

For a listed $\kappa$, the entire set
$X_\kappa=\Pow(\kappa)^B\cap C$ exists in $B$. Bound the
appearance stages of all its members by $a_\kappa$ already listing
$\kappa$. Every $N_\beta^r$ with $\beta\geq a_\kappa$ then has
exactly this final power set. Compatibility identifies their whole
restricted filters as one set $W_\kappa$. Every short family or
regressive graph belonging to $C$ belongs wholly to a later prefix,
where that same $W_\kappa$ is normal and complete. Thus it is a
full normal $C$-measure. The listed cardinals are unbounded, so
$C\models\CM$.

Choose $a_\kappa$ least. Replacement collects these ordinals on
each bounded listed domain. At one later prefix all its powersets
and measures have stabilized, and its whole kernel collects that
entire stabilized prefix. Consequently the class predicate
\[
 E(\kappa,x)\quad\Longleftrightarrow\quad x\in W_\kappa
\]
is uniformly definable in $B$, and each bounded prefix is a $C$-set.
Include its listed domain as part of the predicate.

We claim $C=L[r,E]$. A set-length construction queries only bounded
critical indices, where $E$ agrees with one original selection on
every query in its prefix model. Literal induction gives
$L[r,E]\sq C$. The global relative constructor is itself uniformly
definable, closed under G\"odel operations, and almost universal:
bound the appearance stages of a background set of its elements
and use a later construction level. Hence it satisfies $\ZF$.
Whole later traces reconstruct its stage wellorders, giving Choice.

For the reverse inclusion fix a prefix and a construction length
$\gamma$. In $L[r,E]$ take a whole rank carrier $V_\rho$ high
enough for the levels and tuple queries of a recursion of length
$\gamma$. Choose $\rho$ from the elementary rank bounds for a
relative construction of length $\gamma$, before identifying any
internally produced level with an external prefix level.
A later global level supplies the whole $E$ and domain
trace on that carrier. Restrict this set trace at the ordinal cut
for the desired prefix and run the entire prefix recursion internally.
Every produced level is old and has the chosen rank bound. External
induction identifies it with the actual prefix level. Thus every
prefix is contained in $L[r,E]$, proving the claim.

Now let $W$ be any whole normal $C$-measure on $\kappa$, and
choose $N_\alpha^r$ containing $W$. Every $C$-subset of
$\kappa$ is in $W$ or is the complement of a member of $W$.
Transitivity and Separation therefore give
$\Pow(\kappa)^{N_\alpha^r}=\Pow(\kappa)^C$.
Normality restricts to the prefix. Bounded exhaustion identifies
$W$ with its listed kernel, and compatibility identifies it with
$W_\kappa$. Therefore the internal pure formula ``$U$ is the
unique normal measure on $\kappa$'' recovers exactly $E$.
Since $C=L[r,E]$, its relative stages and their canonical wellorders
are internally definable from $r$ and ordinals. Every set, and every
hereditary member, is consequently ordinal definable from $r$ in
$C$. Thus $C=\HOD(r)^C$.

Apply Vop\v{e}nka's theorem inside $C$, with its own HOD as ground
and $r$ as the parameter \cite[Lemma~3.6]{GoldbergChoiceless}.
It gives set forcing and a generic $g$ such that $C=\HOD^C[g]$.
Finally, if $r,s$ are recursively interdefinable in the same $B$,
the measurable recursion is identical, and for every same bounded
selection $F$ the pure universes $L[r,F]$ and $L[s,F]$ agree.
Their canonical prefix models and their unions therefore agree.
\end{proof}

\begin{corollary}[Refinement of a uniform choice envelope]
\label{cor:choice-envelope}
Suppose one formula in a $\ZF$ universe defines, for each real $r$,
an all-ordinal inner model $E_r$ containing $r$ and satisfying
$\ZFC+\CM$. Suppose $E_r=E_s$ whenever $r,s$ are recursively
interdefinable in that same universe. Then there is a uniform
assignment $r\mapsto C_r$ with the same properties such that
$C_r\models V=\HOD(r)$ and $C_r$ is a set-forcing extension
of its own HOD.
\end{corollary}
\begin{proof}
Relativize the fixed formula $\varphi$ to the definition of $E_r$.
Apply Lemma~\ref{lem:canonical-refinement} inside this actual choice
model. When $E_r=E_s$, both evaluations take place in the same
background containing both reals and their recursive interdefinitions.
The lemma's same-background recoding conclusion gives $C_r=C_s$.
\end{proof}


\begin{lemma}[Core-model preparation]\label{lem:earlier-core}
In a $\ZF$ universe with a proper class of measurables there is a uniform assignment, for every real $r$, of a choice inner model
$C_r$ such that:
\begin{enumerate}
\item $r\in C_r$ and $C_r\models\ZFC+\CM$;
\item the pure universe of $C_r$ is unchanged by the finite recursive
recodings induced by permuting a collapse enumeration;
\item $C_r$ is a set-forcing extension of $D_r=\HOD^{C_r}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Work in $B_r=\HOD(r)$. If there is an $r$-kukri, let
$E_r=S_r$ be the top-iteration choice model of
Lemma~\ref{lem:kukri-class}. If there is no $r$-kukri in $B_r$,
let $E_r=K(r)^{B_r}$. Lemmas~\ref{lem:fixed-core}
and~\ref{lem:fixed-measurables} give the required $\ZFC+\CM$
choice envelope in the negative branch. Both envelopes contain $r$.

For mutually recursive $r,s$, we have $B_r=B_s$.
Lemma~\ref{lem:recoding} identifies the branch and the pure
output envelope: $E_r=E_s$. All recipes are uniformly definable
in the background, without selecting a real code for a mouse or
an additional wellorder. Apply Corollary~\ref{cor:choice-envelope}
inside these envelopes. It supplies $C_r$ with all three properties,
including its presentation over $D_r=\HOD^{C_r}$.
\end{proof}

\subsection{Rank and dependent choice}


\begin{proposition}[Rank-preserving passage to SVC]\label{prop:remove-svc}
Suppose $V\models T_0$. For every ordinal $\beta$ there is an inner model $N\sq V$ satisfying $T_0+\SVC$ such that
\[
 V_\beta^N=V_\beta^V,\qquad
 \Pow(V_\beta^V)^N=\Pow(V_\beta^V)^V.
\]
One can arrange a choice ground $D\models\ZFC+\CM$ and a choice set-forcing cover $C=D[g]$ with $D\sq N\sq C$.
\end{proposition}
\begin{proof}
Choose $\lambda>\beta+\omega$ with $\cf^V(\lambda)>\omega$, and let $B=V_\lambda^V$. DC supplies such a limit ordinal. For example, start sufficiently high and add $\omega_1^V$. Collapse $B$ to be countable, and let $r$ code membership and equality of entries in its generic enumeration. By Lemma~\ref{lem:ZFmeasure}, this fixed collapse preserves a proper-class tail of measurables. Apply Lemma~\ref{lem:earlier-core} in the collapse extension and put $C=C_r$.

Inside $C$, first quotient the coded enumeration by its equality relation and then take its well-founded extensional collapse. This recovers the same transitive set $B$. Define
\[
 N=\HOD^{C}_{B\cup\{B\}},
\]
allowing individual elements of $B$ as parameters as well as the whole set $B$. This relative hereditary-definability construction satisfies $\ZF$ and contains $B$ and all its elements.

Finite permutations of the $\omega$-domain witness weak homogeneity of the collapse by moving the finite domain of one condition away from that of another. They replace $r$ by exactly the recodings covered by Lemma~\ref{lem:earlier-core}(2) and fix the actual ground set $B$ and its members. Thus the group preserves every fixed ordinal-and-$B$ definition over $C$. The usual hereditary rank induction gives $N\sq V$. Consequently
\[
 V_\lambda^N=B=V_\lambda^V.
\]
In particular, both the rank $V_\beta$ and all its subsets have been retained.

For every nonempty $Y\in N$, the relative-HOD definition gives a surjection in $N$ from $\eta\times B^{<\omega}$ onto $Y$, for some ordinal $\eta$. The surjection enumerates all bounded definition codes and finite parameter lists, without choosing a definition for each element of $Y$. Finite lists from the limit rank $B$ can be coded into $B$, so $N\models\SVC(B)$.

For DC, let $R\in N$ be serial on $Y$, fix $y_0\in Y$, and take $f:\eta\times B\twoheadrightarrow Y$ in $N$. At a current $y$ put
\[
 \xi(y)=\min\{\xi<\eta:\exists b\in B\ (y\,R\,f(\xi,b))\}.
\]
Use DC in $V$ to choose successive $b_n$ at these least ordinal coordinates. The array $s=\seq{b_n:n<\omega}$ belongs to $B$ because $\cf^V(\lambda)>\omega$ and $\lambda$ is a limit. Hence it belongs to $N$. Within $N$, recursion from $s$, $y_0$, and the least-coordinate rule reconstructs the entire chain. Thus $N\models\DC$.

Finally, $D=\HOD^C\sq N\sq C=D[g]$. Since the forcing is fixed, downward L\'evy--Solovay gives a measurable tail in $D$ from $C\models\CM$. For every sufficiently high ground measure $U$, its generated lift in $C$ restricts to the set
\[
 \{a\in\Pow(\delta)^N:\exists b\in U\ (b\sq a)\}
\]
in $N$. This is the complete nonprincipal ultrafilter required for $\CM^N$. We fix the ground $D$ and cover $C$ before taking this tail. If desired, the intermediate symmetric-model theorem and $\SVC^N$ make $D$ a symmetric ground of $N$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:rank-transfer}]
Apply Proposition~\ref{prop:remove-svc} above
$\alpha+1$ to obtain $N\sq V$. Then apply the local Theorem~\ref{thm:transfer-intro} to $N$, protecting the same rank and its power set. The resulting $M$ has the displayed rank and power-set equalities with $V$, together with the stated sequence-seed presentation. Choose all required generics over the original collapse extension. The construction then takes place in a set-generic extension of $V$. The inclusion asserted is $N\sq M$; the argument does not assert $V\sq M$.
\end{proof}

\section{Removing Foundation by atomization and fibers}\label{sec:foundation}

We now reduce the Foundation-free starting theory to the preceding case. The atomization and fiber purification follow the earlier argument of \cite[23 September, 06:07 UTC]{ResearchArchive}. We first prove the version of the local forcing theorem needed here. In this section, $\mathsf{ZFA}$ has Foundation above a fixed set $\mathcal A$ of atoms, and $\operatorname{Ker}$ denotes the pure kernel.

\subsection{The local construction with atoms}

\begin{lemma}[The choice-envelope version]\label{lem:atom-local}
Let $N\sq E$ be $\mathsf{ZFA}$ models with the same ordinals and the same set $\mathcal A$ of atoms. Suppose:
\begin{enumerate}
\item $N\models\DC+\SVC(X)$, where $X=V_\lambda^N(\mathcal A)$ is a sufficiently high limit rank, $\cf^N(\lambda)>\omega$, and $\mathcal A\sq X$;
\item $E$ satisfies Choice, $\mathcal A$ is countable in $E$, and $\operatorname{Ker}(E)=D[g]$ for a pure $D\sq N$ satisfying $\ZFC+\CM$ and fixed set forcing;
\item a measurable $\kappa>\lambda+\omega$ of $D$ is chosen above the size of this fixed forcing and the sizes in $E$ of $X$, the old collapse, and the cap and comparison posets.
\end{enumerate}
Then the local Prikry--labeling construction produces $N\sq U$ with
\[
 U\models\mathsf{ZFA}+\DC+\CM+\SVC+\KW,\qquad
 \Pow(X)^U=\Pow(X)^N,\qquad V_\lambda^U(\mathcal A)=X.
\]
\end{lemma}
\begin{proof}
This version requires only the pure $D$ and the choice envelope $E$. Use the old pure Prikry poset from $D$ with its old wellordering. The old normal measure base generates the normal lift in $E$ and its small choice extensions. Compute the countably closed collapse in $N$ and choose the product generic over $E$, as in Section~\ref{sec:ambient}. Internal countable closure and condition coding use $\cf^N(\lambda)>\omega$. No requirement is imposed on $\cf^E(\lambda)$.

The small-forcing assertions with atoms reduce to their pure counterparts. Fix an enumeration of the atoms in $E$. Recursively code each atom by its tagged index and each set by the tagged set of its members' codes. Every resulting code is pure and belongs to $\operatorname{Ker}(E)$. Choice in $E$ codes any set forcing by an ordinal condition set and a pure order. Translate names using the same atom indices. Decoding a name for a pure value uses no atoms. The pure kernel after forcing is therefore the corresponding set-forcing extension of the old pure kernel. We use this coding only in the envelope and do not adjoin its atom enumeration to $N$ or $U$.

The collapse-first proof of Lemma~\ref{lem:ambientfacts}(1)--(2) therefore applies to atomful $X$ and arbitrary ground values. SVC makes $N[K]$ a choice model with atoms. Since Prikry forcing adds no bounded subsets, we obtain the same preservation of countable sequences from $X$ and wellorderable covers of ground families.

For part (3), we need preservation of atom-ranks. Since $|\mathcal A|^E<\kappa$ and $\kappa$ remains inaccessible in the choice envelope, every old atom-rank below $\kappa$ has size below $\kappa$. Prikry forcing preserves these ranks because it adds no bounded subsets. A new set of least rank would be a subset of a small old rank; its pullback by an old wellordering would be a new bounded subset of $\kappa$. The atoms are unchanged. The generic-intersection proof proceeds by rank over the common atom set, starting with the atoms, which already belong to $N$. Thus
\[
 V_\kappa^{N[c]}(\mathcal A)=V_\kappa^N(\mathcal A).
\]

Use transitive-base constructibility with atoms retained at every successor stage, for example
\[
 J_{\xi+1}=\mathcal A\cup\operatorname{Def}
       (J_\xi;\in,\mathsf{Atom}),
\]
so that taking definable subsets does not discard the atoms. Lemma~\ref{lem:presentation} still gives finite expressions. Atoms are ground leaves, and the atom predicate has the same interpretation in the compared universes. The presentation lemma does not use an equality $N=D(X)$.

All cap and paired constructions concern integer carriers and their literal colors in $X$. A color may be an atom. Since the constructions never recover a color from its members, the proofs in Sections~\ref{sec:caps}--\ref{sec:paired} are unchanged. The common atom predicate also gives bounded-evaluator absoluteness. Labeling elimination and the atom-rank equality just proved yield $\Pow(X)$ preservation.

The old condition-array argument, the real-coded masks, and the wellorderable-tree argument prove DC as before. Bounded presentations and $F:Q\twoheadrightarrow X$ give SVC and KW for every nonempty set of $U$, including sets containing atoms. Finally the full choice envelope after these set forcings has pure kernel a set-forcing extension of $D$, by the coding paragraph above. A sufficiently high tail of ground measures has generated lifts there, and the old-base definition used in Proposition~\ref{prop:seed} restricts those lifts to measures belonging to $U$. This proves $\CM^U$.

The atom lemma does not include the pure sequence-seed and symmetric-ground conclusions of Proposition~\ref{prop:seed}. The predecessor-set quotient does not distinguish distinct atoms, and a pure $D(A')$ cannot contain them. Neither conclusion is needed below.
\end{proof}

\subsection{Preparation of the protected atom structure}

Start with $V\models T_-$ and a full second-order counterexample $\mathcal M$ to a fixed sentence. Take its relational multisorted presentation, with a first-order sort $S_0$ and, for each arity $k$ in the sentence, a sort representing all $V$-subsets of $S_0^k$ with the incidence relation. The two-sorted presentation is the unary or tuple-coded case. Replace the combined domain by a set $\mathcal A$ of formal atoms, with distinct tags for the sorts.

Form the full interpreted cumulative hierarchy $\mathcal H(\mathcal A)$ using the original universe's power sets. Treat its atoms as opaque labels whose original members do not become members of the atoms. Non-well-founded membership in $V$ therefore introduces no cycles in the interpreted hierarchy. The hierarchy satisfies $\mathsf{ZFA}$. It also satisfies DC, since an ambient chain through a set of its coded objects has bounded atom-rank and belongs to the hierarchy. Its pure kernel has the original ordinals and their full power sets. It therefore retains the original measures and the ordinal-indexed families testing their completeness. Hence
\[
 \mathcal H(\mathcal A)\models\mathsf{ZFA}+\DC+\CM.
\]
The atomized structure remains full because inverse images under the atom-label bijection identify its subsets and finite-arity relations with all the original ones.

Choose a limit $\lambda$ of uncountable cofinality in this hierarchy, high enough to contain the structure and the full power sorts, and set
\[
 B=V_\lambda^{\mathcal H(\mathcal A)}(\mathcal A).
\]
Collapse $B$ to be countable. Its real code $r$ records membership, equality of repeated enumeration entries, and which entries are atoms. The small-forcing proof of Lemma~\ref{lem:ZFmeasure} works with fixed atoms, so a measurable tail remains. Apply the earlier core-model Lemma~\ref{lem:earlier-core} in the pure kernel of this collapse extension, obtaining $C_r=D[g]$.

Construct the choice companion $E$ as follows. Let $I$ be the least enumeration indices of the distinct atoms, as determined by $r$. Inside $C_r$, form the hierarchy over the formal atom set indexed by $I$. Use the collapse enumeration to relabel those formal atoms by the actual atoms. The resulting $E$ satisfies Choice, has pure kernel $C_r$, and contains the literal $B$. To prove the last assertion, quotient the code by its equality predicate and take its well-founded collapse with the specified atom labels. These labels distinguish the atoms from one another and from the empty set. Uniqueness of the collapse gives the same $B$.

The finite permutation group used in Proposition~\ref{prop:remove-svc} preserves this companion. The pure $C_r$ is unchanged, and the induced bijection between the two atom-index sets belongs to the same pure model. Transporting the hierarchy along this bijection gives the identical actual companion. Therefore
\[
 N=\HOD^E_{B\cup\{B\}},
\]
with the atom predicate retained, descends to $\mathcal H(\mathcal A)$ by the same hereditary homogeneity argument. It contains $B$ and has $V_\lambda^N(\mathcal A)=B$. Relative-HOD presentations give $\SVC(B)$, and the least-ordinal-coordinate DC proof in Proposition~\ref{prop:remove-svc} applies using the countable closure of $B$ in $\mathcal H(\mathcal A)$.

The pure $D=\HOD^{C_r}$ is contained in $N$, because definitions over the definable pure kernel of $E$ are available over $E$. Ground measure lifts restrict to $N$, giving $\CM^N$. We have a pure ground inside $N$ and a choice envelope above it. This does not assert a choice ground with atoms inside $N$, and it does not apply the pure SVC symmetric-ground theorem to an atom model.

Lemma~\ref{lem:atom-local} now applies with $X=B$. In particular, $\cf^N(\lambda)>\omega$ because $N\sq\mathcal H(\mathcal A)$. The envelope may see $B$ and $\lambda$ as countable, which the lemma permits. The old condition sets have codes in $B$ and are small in $E$. Once these set-sized bounds are fixed, choose a sufficiently high $D$-measurable $\kappa$. The lemma yields
\[
 U\models\mathsf{ZFA}+\DC+\CM+\SVC+\KW
\]
with the protected power sets unchanged. The original multisorted counterexample is consequently still full in $U$.

\subsection{Fiber purification and the final theorem}

\begin{lemma}[Fiber purification]\label{lem:fiber}
If $U\models\mathsf{ZFA}+\DC+\CM+\SVC+\KW$, then its pure kernel $P$ satisfies $T_0+\SVC+\KW$. Every full second-order structure in $U$ has a full isomorphic copy in $P$.
\end{lemma}
\begin{proof}
For the nonempty combined domain $S$ of a relational presentation, take a surjection $f:\kappa^\omega\twoheadrightarrow S$ in $U$ and put
\[
 j(s)=f^{-1}\{s\}.
\]
The nonempty disjoint fibers make $j$ injective. Each fiber is a set of ordinal sequences and is therefore pure. Thus the whole range $j[S]$, its sort predicates, and all transported finite-arity relations are pure sets in $P$. The graph of $j$ may contain atoms and need not belong to $P$. Empty domains can be treated directly.

To check fullness at each relevant arity, let $Y\in P$ with $Y\sq j[S_0]^k$. Its pullback under $j^k$ is a $U$-subset of $S_0^k$, represented by an element of the full power sort in $U$. The image of that element represents exactly $Y$. Conversely, each represented subset in the transported structure is pure and belongs to $P$. In the application, preservation of the protected power sets identifies these $U$-relations with the relations of the original counterexample. The copied structure therefore has the same full second-order truth.

The pure kernel satisfies $\ZF$, including Foundation. A DC chain through a pure set is itself pure. Measures on ordinals and all families testing their completeness are pure as well, so $\CM$ descends. The models have the same $\kappa^\omega$; a surjection from it onto a pure set has pure graph. Hence KW descends.

Finally, fix one SVC witness $C$ in $U$ and fiber-code it as above to obtain a pure copy $C^\flat$. For any nonempty pure $Y$, compose an SVC surjection $\eta\times C\twoheadrightarrow Y$ with the inverse of this fiber bijection. The resulting map $\eta\times C^\flat\twoheadrightarrow Y$ has pure graph and belongs to $P$. The same $C^\flat$ works for every $Y$, proving SVC in $P$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Given a full second-order counterexample in a model of $T_-$, atomize it, carry out the core-model preparation, and apply the local atom theorem as above. Lemma~\ref{lem:fiber} gives a pure model of $T_0+\SVC+\KW$ containing a full counterexample to the same sentence. The countermodel contrapositive proves structural conservativity. The reverse implication of provability follows because the target theory extends $T_-$. All constructions are interpretations and set-forcing constructions over the starting model. No external transitive-model hypothesis is used.
\end{proof}

\section{External HOD and homogeneous descent}\label{sec:external-basics}

\begin{lemma}\label{lem:HDC}
If $U\models\ZFC$, then $\Hseq^U\models\ZF+\DC$. It contains every ordinal $\omega$-sequence of $U$ and is closed under $U$-countable sequences of its elements.
\end{lemma}
\begin{proof}
The usual hereditary-definability proof of $\ZF$ applies to the definable class of allowed parameters. For example, the internal power set of $x\in\Hseq^U$ is
\[
 \{y\sq x:y\in\Hseq^U\},
\]
which is definable from a definition of $x$ and has only hereditary members of the class. The same argument gives Separation and Replacement.

Let $\seq{x_n:n<\omega}\in U$, with each $x_n\in\Hseq^U$. In $U$, choose a definition of $x_n$ from an ordinal sequence and ordinal parameters. Evaluate each definition at a rank level where its formula has the required unique value. Interleave the sequence parameters, the rank levels, the ordinal parameters, and the formula codes. This gives one $q\in(\Ord^\omega)^U$ from which the entire sequence $\seq{x_n}$ is definable in $U$. Its hereditary members already belong to $\Hseq^U$, so the sequence belongs to $\Hseq^U$. Every $q\in(\Ord^\omega)^U$ is itself hereditarily definable from $q$. To prove DC, choose a dependent-choice chain in $U$ through a serial relation belonging to $\Hseq^U$ and apply the countable closure just proved.
\end{proof}

\begin{lemma}[Homogeneous descent]\label{lem:homdesc}
Let $M\models\ZF$, and suppose $U=M[G]$ is a weakly homogeneous set-forcing extension with
\[
 (\Ord^\omega)^U=(\Ord^\omega)^M.
\]
Then $\Hseq^U\sq M$. No ordinal-definability assumption on the forcing is needed.
\end{lemma}
\begin{proof}
Induct on hereditary rank. Suppose $x\in\Hseq^U$ and every member of $x$ is in $M$. Choose $\beta$ with $x\sq V_\beta^M$. A definition of $x$ in $U$ uses an allowed parameter $q$ belonging to $M$ and ordinal parameters. By weak homogeneity, the largest condition decides the corresponding uniqueness assertion and, for each $y\in V_\beta^M$, whether $y$ belongs to its unique value. The forcing relation therefore defines $x$ as a subset of $V_\beta^M$ inside $M$.
\end{proof}

\begin{proposition}\label{prop:one-three}
For every $\sigma$, $\Con(E_1(\sigma))$ implies $\Con(E_3(\sigma))$.
\end{proposition}
\begin{proof}
Let $U\models E_1(\sigma)$, put $H=\Hseq^U$, and choose $\kappa$ with $V_\kappa^H\models\sigma$. Choose $\lambda>\kappa+\omega$ of uncountable cofinality in $U$, and set $X=V_\lambda^H$. This $X$ is ordinal-definable in $U$, belongs to $H$, and is closed under $U$-countable sequences of its elements by Lemma~\ref{lem:HDC}.

Collapse $X$ to be countable over $U$, coding both equality and membership and quotienting repetitions before collapse. Apply Lemma~\ref{lem:earlier-core}, and again set $W=\HOD_{X\cup\{X\}}^{C_r}$. The collapse-homogeneity argument gives $W\sq U$ and controls the parameters in each descended definition. The collapse forcing is definable from the OD set $X$, so these definitions use $X$, finitely many elements of $X$, and ordinals. The set $X$ is OD in $U$, and each of those finitely many elements is hereditarily definable from an ordinal $\omega$-sequence. Combining their parameters gives $W\sq H$.

Since $X\in W$, we have $V_\lambda^W=X$. The presentation $\SVC(X)$ and the ambient countable closure of $X$ give $\DC$ in $W$. The same generated-measure argument gives $\CM$. Therefore $W\models E_3(\sigma)$.
\end{proof}


\section{Sparse evaluated Cohen caps}\label{sec:spectrum}

We work over a choice model $B$ and code a \emph{family of coherent rows}. The code records which bit positions belong to each row, rather than only their union.

We use the convention $\Add(\gamma,1)=2^{<\gamma}$, ordered by extension; the dense partial-function presentation gives the same forcing. Throughout this section, the superscript specifies the model in which each poset is computed.

\begin{theorem}[Fixed-base spectrum]\label{thm:spectrum}
Let $B\models\ZFC$ have a proper class of inaccessible cardinals, and let $\varnothing\ne F\sq2^\tau$ belong to $B$, where $\tau$ is infinite. There are a schedule of inaccessible coding cardinals, uniformly defined forcings $Q_e$ for $e\in2^\tau$, and a cap cardinal $\rho$, all definable from $B$ and ordinal parameters independently of $F$, such that the weakly homogeneous forcing $P_F$ defined below has the following spectrum in $B[G]$:
\begin{equation}\label{eq:spectrum}
 e\in F\quad\Longleftrightarrow\quad
 \begin{gathered}
 \text{there is a $B$-generic }h_e\sq Q_e,\
 \text{and an }\Add(\rho,1)^{B[h_e]}\text{-generic over }B[h_e].
 \end{gathered}
\end{equation}
The first coding cardinal can be placed above any prescribed bound, and the resulting forcing is correspondingly distributive.
\end{theorem}

\subsection{The schedule and the actual forcing}
Order the physical sites $s=(\xi,i)$, for $\xi<\tau$ and $i<2$, lexicographically with $\xi$ first. Distinct rows have independent copies at a shared physical site. Choose inaccessible cardinals $\gamma_s$ recursively, with the first above $2^{|\tau|}$ and the prescribed protection bound, and with
\begin{equation}\label{eq:sparse}
 |\RO(S_{<s}^{\mathrm{univ}})|^B<\gamma_s.
\end{equation}
The universal prefix includes copies for every row in $2^\tau$, including those outside $F$. The schedule is therefore independent of the family being coded. At each step, choose the least inaccessible above the displayed bound.

For $e\in2^\tau$, let
\[
 Q_e=\prod_{\xi<\tau}^{\mathrm{full}}
       \Add(\gamma_{(\xi,e(\xi))},1)^B.
\]
Let $\rho$ be an inaccessible above the Boolean completion of the entire universal lower product. Define
\begin{equation}\label{eq:PF}
 P_F=\prod_{d\in F}^{\mathrm{full}}
       \left(Q_d*\dot{\Add}(\rho,1)^{B[\dot h_d]}\right).
\end{equation}
Compute the terminal forcing in its \emph{own row extension}. Form the actual extension from the evaluated terminal generics, without adjoining the larger raw termspace generics.

The full lower product
\[
 S_F=\prod_{d\in F}^{\mathrm{full}}Q_d
\]
can be grouped by physical sites. Denote the aggregate generic at $s$ by $G_s$. Since $|F|<\gamma_0\le\gamma_s$, each site forcing is a product of fewer than $\gamma_s$ copies of $\Add(\gamma_s,1)^B$. It is $<\gamma_s$-closed in $B$ and adds no shorter ordinal sequences over $B$.

For each $d$, let $T_d$ be a full set of total names for cap
conditions: the largest condition of $Q_d$ forces each member to
be a cap condition, and every total name has an equivalent member.
Use all bounded Boolean-name codes for graphs on $\rho\times2$
with this property, allowing equivalent codes. The resulting set is defined
canonically from $Q_d$ and $\rho$; the forcing has no auxiliary wellorder
as a parameter. Smallness of $\RO(Q_d)$ and regularity of $\rho$
bound the possible lengths of each name below $\rho$. Normalize
forced unions and Boolean mixes to these Boolean-name codes.
Lower automorphisms act on their Boolean values and preserve this
full name system. The product
\[
 S_F\times\prod_{d\in F}T_d
\]
with the ground termspace orders projects onto $P_F$ and preserves
the lower coordinate. To verify the lifting property, mix
each desired stronger term on its strengthened lower cone with
the old term elsewhere. Thus this projection commutes with the
projection to $S_F$. Each ground termspace is $<\rho$-closed. After the lower forcing, which has size below $\rho$, the full product is $<\rho$-distributive by Easton's lemma. The actual cap quotient therefore adds no ordinal sequences of length below $\rho$ over the lower extension. Here a closed \emph{dominating} forcing proves preservation; the actual extension need not equal the larger termspace extension. See \cite[Section~3]{ReitzCohen} for the term-forcing facts.

This also proves preservation. If the first site is above a regular $\theta$, the dominating forcing is $<\theta$-closed in $B$, so the actual forcing is $<\theta$-distributive. Taking $\theta>|V_\eta^B|$ preserves $V_\eta^B$ by rank induction. The canonical lower-row automorphisms lift through the name for the recomputed Cohen forcing, which is forced homogeneous. Coordinatewise automorphisms then witness weak homogeneity of the full product.

\subsection{Localization to physical sites}

\begin{lemma}\label{lem:site}
Suppose a $B$-generic $h_e\sq Q_e$ belongs to the $P_F$-extension. Then
\[
 B[h_e]\sq B[G_s:s\text{ is selected by }e].
\]
\end{lemma}
\begin{proof}
The whole lower generic $h_e$ can be coded by a subset of a ground ordinal below $\rho$. The cap quotient adds no such subsets, so $h_e$ is in the lower extension. Let $k_s\sq\gamma_s$ be its component at an $e$-selected site.

If there is a site after $s$, its first cardinal exceeds the
Boolean size of the prefix through $s$. The ground full tail is
closed up to that cardinal. Easton's lemma over the smaller prefix
therefore makes it add no subsets of $\gamma_s$; at a final site
there is no tail. Thus the required tail distributivity follows
from \eqref{eq:sparse}, and
\[
 k_s\in B[G_s][G_{<s}].
\]
The forcing producing $G_{<s}$ has size below $\gamma_s$. Consequently the pair
\[
 (B[G_s],B[G_s][G_{<s}])
\]
has the $<\gamma_s$-approximation property. To check that $k_s$ is approximated over $B[G_s]$, take a $B[G_s]$-set of ordinal positions of size below $\gamma_s$. It belongs to $B$, since $G_s$ adds no short ordinal sequences. The restriction of the $B$-Cohen generic $k_s$ to that set also belongs to $B$. Therefore
\begin{equation}\label{eq:ks}
 k_s\in B[G_s].
\end{equation}

For each site, fix in $B$ a wellordering of a bounded set of names for subsets of $\gamma_s$. Choose the least name whose value is the actual $k_s$. The indices form an ordinal sequence of length $|\tau|$. The lower forcing adds no such sequence, so the whole vector of indices belongs to $B$. Hence the vector of all $k_s$ belongs to the extension by the selected sites alone.

This vector determines the full product generic $h_e$. A ground product condition belongs to a generic filter exactly when it is compatible with the union of the generic bits at every coordinate. Otherwise, the filter's meeting the dense set deciding compatibility with that condition gives a contradiction. Hence $h_e$ belongs to the selected-site extension.
\end{proof}

\subsection{The quotient and its term order}

The spectrum depends on the following quotient calculation. Replacing this order by raw term forcing would destroy the spectrum.

Let $E=B[h_e]$. Use the old vector of names in
Lemma~\ref{lem:site} to name $h_e$ over the selected sites.
Restrict to an actual positive ground cone which forces that this
name is $Q_e$-generic over $B$. This is a forcing
statement: the ground dense subsets of $Q_e$ form a set.
The membership Boolean values of the named filter generate a
complete subalgebra of that selected-site completion, on the
cone, whose induced extension is exactly $E$. Its size is below
$\rho$; injectivity of the map from the whole $Q_e$ completion
is not required. The omitted sites remain independent product
directions over this algebra. Quotient positivity depends only
on the selected lower projection, since every terminal name has
been totalized. Restricting to this cone leaves the actual
extension unchanged. Let $P$ be a dense old-condition presentation of the quotient. Let
\[
 T=\left(\prod_{d\in F}T_d\right)^B
\]
be the old set of vectors of row terms. The quotient of the \emph{actual} forcing \eqref{eq:PF} is represented by $P\times T$ with order
\begin{equation}\label{eq:quotient-order}
 (s',t')\le(s,t)\quad\Longleftrightarrow\quad
 s'\le s\ \text{and}\
 \forall d\in F\;
 s'_d\forces_{Q_d}^{B} t'_d\le t_d.
\end{equation}
The right side uses the original ground forcing relation. It does not quantify over all new $E$-names for terminal conditions.

We apply Jakob's criterion \cite[Definition~3.5 and Theorem~3.8]{Jakob}: an iteration-like order with a square-$\rho$-cc base and a strongly $<\rho$-distributive term order has the $<\rho$-approximation property. Strong $<\rho$-distributivity means that every $\rho$-long sequence of dense open sets has a descending sequence meeting them. A final lower bound is not required. We check these hypotheses for \eqref{eq:quotient-order}.

\begin{lemma}\label{lem:quotient}
If $e\notin F$, the quotient order \eqref{eq:quotient-order} has the $<\rho$-approximation property over $E$.
\end{lemma}
\begin{proof}
The quotient base and its square have size below $\rho$. For a fixed old lower condition $s$, the section order on $T$ is $<\rho$-closed in $B$. In each row, take a name for the union of the descending cap conditions below $s_d$ and use a default value off that cone. The extension $E/B$ is by forcing of size below $\rho$. By the preservation observation preceding Jakob's criterion, this chain-condition forcing preserves strong $<\rho$-distributivity of the fixed ground section orders. Their disjoint union is the term order, so it is strongly $<\rho$-distributive in $E$.

The order is based, and the forcing relation gives its refinement property. For the projection property, suppose $(s',t')\le(s,t)$. In row $d$, mix the name $t'_d$ below $s'_d$ with $t_d$ off that Boolean cone. The resulting old vector $t''$ is a section extension below $s$, and below $s'$ it is equivalent to $t'$. All inputs to the mixing belong to the ground, and the mixing takes place in $B$.

To verify mixing, take two section extensions $t^0,t^1$ below $t$ at a fixed $s$. For every $d\in F$, choose the least $\xi_d$ with $d(\xi_d)\ne e(\xi_d)$. The site
\[
 s_d=(\xi_d,d(\xi_d))
\]
is omitted by $e$ and active in row $d$. Keep the selected sites unchanged, and refine $s$ in two ways by assigning opposite values to a fresh bit in this $d$-copy, simultaneously for every $d\in F$. This gives legal old conditions because $|F|$ is below the first coding cardinal and the product has full support. Both refinements remain positive in the quotient because only omitted sites changed. Their projections are incompatible in every actual row.

In row $d$, Boolean-mix $t^0_d$ and $t^1_d$ below the incompatible row projections, retaining $t_d$ elsewhere. The resulting old vector $t^*$ is a section extension of $t$ at $s$. Below the respective quotient refinements it specializes to $t^0$ and $t^1$, as required for mixing. Jakob's theorem gives the approximation property.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:spectrum}]
If $e\in F$, its planted row and cap witness the right side of \eqref{eq:spectrum}. Conversely, suppose that side holds and $e\notin F$. Lemma~\ref{lem:site} and Lemma~\ref{lem:quotient} give the $<\rho$-approximation property for $(B[h_e],B[G])$. A recomputed $\Add(\rho,1)^{B[h_e]}$-generic would be a new subset of $\rho$ whose every $B[h_e]$-small restriction belongs to $B[h_e]$, contradicting approximation. Preservation and homogeneity were proved after \eqref{eq:PF}.
\end{proof}

\begin{corollary}[Coding an orbit of presentations]\label{cor:ODcoding}
Let $x\in B$. There is an arbitrarily highly distributive, weakly homogeneous forcing, uniformly definable in $B$ from $x$ and ordinal parameters, which makes $x$ OD relative to the class $B$. If $B$ is ordinal-definable in the final extension, then $x$ is OD there.
\end{corollary}
\begin{proof}
Choose an infinite cardinal $\tau$ large enough to code $(\TC(\{x\}),\in,x)$. Let $F_x\sq2^\tau$ be the family of all pointed well-founded extensional presentation codes of this structure. The codes include domain and root data and allow unused positions. This nonempty family is defined from $x,\tau$ without choosing a presentation.

Apply Theorem~\ref{thm:spectrum}. Relative to the class $B$, the first-order capped-history predicate recovers exactly $F_x$. Each member has pointed Mostowski collapse $x$, so $x$ is OD relative to $B$. The schedules use the universal family $2^\tau$ for their bounds. Thus the decoder can define the candidate forcings without knowing $F_x$.
\end{proof}

\begin{lemma}[GCH preservation]\label{lem:spectrum-gch}
If $B\models\mathsf{GCH}$ in Theorem~\ref{thm:spectrum}, then the
actual evaluated forcing $P_F$ preserves all cardinals and GCH.
Consequently the coding forcing in Corollary~\ref{cor:ODcoding}
can be chosen to have these properties over a GCH base.
\end{lemma}
\begin{proof}
Write $S=S_F$ for the lower product and let $\nu$ be the cardinality
of the physical-site set. The schedule gives
$\nu,|F|<\gamma_0$, where $\gamma_0$ is its first inaccessible.
At an active site $\gamma$, the aggregate forcing has the dense
partial-function presentation $\Add(\gamma,m_\gamma)$ for some
$m_\gamma<\gamma_0$. It has size $\gamma$ and is
$<\gamma$-closed in $B$. Thus the full product $S$ is
$<\gamma_0$-closed and adds no ordinal sequences of length less
than $\gamma_0$.

All arithmetic and cofinalities in the lower-product argument are
computed in $B$. For an infinite cardinal $\lambda$, let
$S_{\le\lambda}$ contain the sites at most $\lambda$. If
$\lambda\ge\gamma_0$ and $\cf(\lambda)\ge\gamma_0$, then GCH gives
\[
 |S_{\le\lambda}|\le\lambda^\nu=\lambda.
\]
This includes every regular $\lambda\ge\gamma_0$. If a tail remains,
its first site $\eta$ is greater than $\lambda$, and the old tail
is $<\eta$-closed. Easton's lemma makes it $<\eta$-distributive
over the head extension. Hence every ordinal sequence of length
$\lambda$ in the full extension already belongs to the head
extension. The same conclusion is immediate when the head is all
of $S$.

We first prove cardinal preservation. Successors below $\gamma_0$
are preserved by closure. For $\lambda\ge\gamma_0$ with
$\cf(\lambda)\ge\gamma_0$, the head has size at most $\lambda$
and preserves $\lambda^+$; the tail adds no $\lambda$-sequences
and cannot collapse it. Now suppose that $\lambda$ is singular
and $\epsilon=\cf(\lambda)<\gamma_0$. Fix a cofinal sequence
$\seq{\lambda_j:j<\epsilon}$ of infinite regular cardinals between
$\gamma_0$ and $\lambda$. If $f:\lambda\to\lambda^+$ belongs to
the full extension, then $f\restr\lambda_j$ belongs to the
$S_{\le\lambda_j}$-extension. A name for this restriction has at
most $\lambda_j$ possible values at each argument, so its range
is bounded below the ground $\lambda^+$. In the full extension,
let $\beta_j<\lambda^+$ be the least strict bound on that range.
The whole sequence $\seq{\beta_j:j<\epsilon}$ belongs to $B$ by
$<\gamma_0$-closure. Ground regularity gives
$\sup_{j<\epsilon}\beta_j<\lambda^+$, so $f$ is bounded and cannot
collapse $\lambda^+$. Every ground successor cardinal is therefore
preserved. A ground limit cardinal cannot collapse while all its
smaller successor cardinals remain cardinals, so $S$ preserves
all cardinals.

When $|S_{\le\lambda}|\le\lambda$, Boolean-name counting gives
at most $(2^\lambda)^\lambda=\lambda^+$ names for subsets of
$\lambda$, and tail distributivity returns every such subset to
the head extension. This proves GCH for the cases already covered
by that size bound. Below $\gamma_0$ no subsets are added.
For singular $\lambda$ with $\epsilon<\gamma_0$, use the same
regular sequence $\seq{\lambda_j:j<\epsilon}$. In $B$, fix for
each $j$ a list of at most $\lambda_j^+$ head names covering all
subsets of $\lambda_j$ in its head extension. For each actual
$x\sq\lambda$, choose the least index representing
$x\cap\lambda_j$ under the actual head generic. The resulting
index vector has length $\epsilon<\gamma_0$, so the whole vector
belongs to $B$. Its entries are below $\lambda$, and GCH and
K\"onig's theorem give $\lambda^\epsilon=\lambda^+$ in $B$.
A vector determines at most one $x$, by union of its evaluated
restrictions. Cardinal preservation now gives
$2^\lambda=\lambda^+$ in the full lower extension.

Finally, $|S|<\rho$ by the choice of the inaccessible cap cardinal,
so $\rho$ remains inaccessible in the lower extension. For each
row $d$, the Boolean algebra $\RO(Q_d)^B$ has size below $\rho$.
The possible lengths of a total cap-condition name are therefore
bounded below $\rho$. For each fixed length bound there are fewer
than $\rho$ bounded Boolean graph codes, by strong limitness.
Taking all bounds gives $|T_d|^B\le\rho$, and hence
\[
 \left|\left(\prod_{d\in F}T_d\right)^B\right|
 \le\rho^{|F|}=\rho.
\]
Together with $|S|<\rho$, this gives a dense old-condition
presentation of the actual cap quotient of size at most $\rho$.
The termspace projection established after \eqref{eq:PF} makes
this quotient $<\rho$-distributive over the lower extension.
It preserves all smaller power sets and all cardinals at most
$\rho$; its $\rho^+$-chain condition preserves higher cardinals.
For $\lambda\ge\rho$, Boolean-name counting in the lower
extension gives at most
\[
 (2^\rho)^\lambda=2^\lambda=\lambda^+
\]
names for subsets of $\lambda$. Thus the actual evaluated quotient
also preserves GCH, completing the proof.
\end{proof}

\begin{remark}
Theorem~\ref{thm:spectrum} uses no continuum-function comparison, large cap width, or singularization. Sparse site localization supplies the omitted quotient directions. A universal Cohen source is not retained. The ground termspaces prove preservation, and the quotient argument applies to the evaluated forcing \eqref{eq:quotient-order}.
\end{remark}

\section{A fresh envelope and a secure ground}\label{sec:envelope}

\begin{lemma}[A fresh choice envelope]\label{lem:envelope}
Suppose $M\models\ZF+\DC+\SVC$. There is a weakly homogeneous set-forcing extension $B=M[K]\models\ZFC$ with
\[
 (\Ord^\omega)^B=(\Ord^\omega)^M.
\]
\end{lemma}
\begin{proof}
Choose an SVC witness $S$, and force with countable partial functions from $\omega_1^M$ to $S$. Countable closure and ground $\DC$ imply that no countable sequences of ground elements are added. Permutations of $\omega_1$ moving two countable domains apart witness weak homogeneity. The generic union maps $\omega_1$ onto $S$, so least preimages wellorder $S$. The SVC surjections then wellorder every old set. Each new set has an old name whose collection of possible named elements is now wellorderable. Evaluating that collection gives a wellordering of the new set. Thus the extension satisfies Choice.
\end{proof}

\begin{definition}[Secure grounds]
A choice model $D$ is \emph{secure} if, in every set-forcing
extension $U$ of $D$, we have $D\sq\HOD^U$ and $D$ is the mantle
of $U$. The mantle is the intersection of the set-forcing grounds
and is uniformly definable.
\end{definition}

We obtain a secure ground by coding with $\diamondsuit^*$.
For an uncountable regular $\lambda$, this principle asserts the
existence of a sequence $\seq{\mathcal D_\xi:\xi<\lambda}$ such that
$\mathcal D_\xi\sq\Pow(\xi)$ has size at most $|\xi|$ and, for each
$x\sq\lambda$, there is a club $C\sq\lambda$ on which
$x\cap\xi\in\mathcal D_\xi$. Write \emph{eventual GCH} for GCH
at every cardinal above some bound.

\begin{lemma}[Secure preparation with eventual GCH]\label{lem:secure}
Suppose $D_0\models\ZFC+\CM$. Given a regular cardinal $\chi$
and an ordinal $\alpha$ with $|V_\alpha^{D_0}|<\chi$, there is a
progressively closed class-forcing extension $D$ such that
\[
 D\models\ZFC+\CM+\text{eventual GCH},\qquad
 V_\alpha^D=V_\alpha^{D_0},
\]
and $D$ is secure. Every set-sized initial segment of the
preparation can be taken $<\chi$-closed in $D_0$.
\end{lemma}
\begin{proof}
Use the $\diamondsuit^*$ option of
\cite[Appendix, Lemma~20]{CFH}, starting above $\chi$.
The cited iteration combines GCH forcing with coding on blocks
of triple successors of beth fixed points. It preserves measurables;
using $\diamondsuit^*$ for the codes permits GCH. We record the
effect of omitting its bounded initial part.

Make every stage through $\chi$ trivial and retain the stated
iteration above it, including its Cohen stages at inaccessible
cardinals. All remaining factors are $<\chi$-closed. A union of
fewer than $\chi$ Easton supports still satisfies the support
bounds, since all active locations exceed $\chi$. Thus every
initial segment is $<\chi$-closed. The progressively closed
set-name construction gives $\ZFC$, with every set in an initial
extension. It adds no ordinal sequences of length less than
$\chi$ and hence preserves the indicated rank.

The GCH calculation applies above the omitted segment. For
measurables $\kappa>\chi$, the cutoff is fixed by the normal
ultrapower embedding. The iteration and its image agree below
$\kappa$ and have the same stage-$\kappa$ Cohen preparation and
closed upper tail used in the cited lift. Removing the bounded
initial stages introduces no further lifting requirement.
Consequently a class of measurables survives. These conclusions
give eventual GCH and $\CM$; they impose no GCH requirement on
the protected rank.

The coding density argument applies above each prescribed bound.
Every ordinal set appearing in an initial extension can be
written on a fresh block by choosing the appropriate coding
factors. Subsequent tails preserve that block's truth pattern.
It follows that every ordinal set of $D$ has arbitrarily high
$\diamondsuit^*$ codes. We next verify that these codes make
$D$ secure.

First, if $W[g]/W$ is set forcing and $\lambda$ is regular above
its size, then $\diamondsuit^*_\lambda$ has the same truth value
in the two models, by \cite[Lemma~19]{CFH}. The locations of
sufficiently high codes also agree. Indeed, if the forcing has
size $\nu$, cardinal preservation and the bound of
$(2^\kappa)^W$ on names for subsets of $\kappa$ show that the
cardinal-valued continuum functions agree for cardinals
$\kappa>\nu$. Let $b_\xi$ and $b'_\xi$ be the two beth
sequences. Above a fixed initial bound, monotonicity gives fixed
ordinal shifts $\theta,\theta'$ with
\[
 b_\xi\leq b'_{\theta+\xi},\qquad
 b'_\xi\leq b_{\theta'+\xi}
 \quad(\xi\in\Ord).
\]
For example, choose the shifted initial value beyond the
disturbed continuum segment and induct using their common
continuum function there.
If $\delta$ is a beth fixed point in either model above these
shifts, then $\delta$ is an infinite cardinal and
$\theta+\xi,\theta'+\xi<\delta$ for every $\xi<\delta$.
The inequalities put all values of the other beth sequence below
$\delta$. Continuity, together with $\beth_\delta\geq\delta$,
makes it a fixed point there too. Thus the high beth fixed
points and their triple successors agree. Describe a coding
block by its actual first beth fixed point and its ordinal
length. This avoids any change in the enumeration below the
agreement bound.

Now let $U$ be a set-forcing extension of $D$. For any ordinal
set $x\in D$, choose one of its codes above the forcing size and
the bound for agreement of locations. The unchanged truth
pattern defines $x$ in $U$ from the starting point and length.
Ordinal codes for transitive closures then give $D\sq\HOD^U$.

The same reasoning gives the Ground Axiom in $D$. If $D=W[g]$
were a nontrivial set extension, a sufficiently high code for
each ordinal set of $D$ could be read in $W$. Separation in $W$
would reconstruct every such set. Choice then gives $D=W$,
a contradiction. Finally, let $W$ be any ground of $U$.
Downward directedness of grounds gives a common ground below
$D$ and $W$; see \cite{UsubaGrounds,Usuba}. The intermediate-model
theorem makes it a ground of $D$. The Ground Axiom makes this
common ground equal to $D$. Since $D$ itself is a ground of $U$,
its mantle is exactly $D$.
\end{proof}

The Continuum Coding Axiom also gives secure grounds by the
same high-code and Ground Axiom argument \cite{ReitzGA}.
Its continuum codes entail unbounded failures of GCH. The
$\diamondsuit^*$ preparation is what allows the later choice
envelope to satisfy eventual GCH.

\begin{lemma}[A fresh GCH envelope]\label{lem:gch-envelope}
Suppose $M\models\ZF$ and $B=M[K]\models\ZFC+\CM$ is a
weakly homogeneous set-forcing extension with no new ordinal
$\omega$-sequences. If $B$ satisfies eventual GCH, there is a
further set-forcing extension $B'=B[H]$ such that
\[
 B'\models\ZFC+\GCH+\CM,\qquad
 (\Ord^\omega)^{B'}=(\Ord^\omega)^M.
\]
Moreover $B'/M$ is a weakly homogeneous set-forcing extension.
\end{lemma}
\begin{proof}
In $B$, choose an inaccessible $\delta$ above all GCH exceptions.
Use the following presentation of $L_\delta=\Col(\omega_1,<\delta)$.
A condition is a countable partial function $p$ on
$\delta\times\omega_1$ with $p(\beta,\xi)<\beta$, ordered by
reverse inclusion. All ordinals below $\delta$ are coordinates;
the definition does not use a model's list of cardinals.

In $B$ this forcing is countably closed, has size $\delta$, and
is $\delta$-cc. For the last assertion, the generalized
$\Delta$-system lemma gives a common root for $\delta$ many
countable domains. Inaccessibility bounds the number of value
assignments on that root below $\delta$. Two conditions therefore
agree on the root and are compatible. Each nonzero coordinate
adds a surjection from $\omega_1$ onto that ordinal. Hence
$\omega_1$ is preserved, $\delta$ becomes $\omega_2$, and all
cardinals at least $\delta$ are preserved.

There are no new reals, and the old continuum has size less than
$\delta$, so CH holds. A nice name for a subset of $\omega_1$
uses $\omega_1$ antichains, each of size less than $\delta$.
The entire name has size less than $\delta$, so
$\delta^{<\delta}=\delta$ bounds their number by $\delta$.
Thus $2^{\omega_1}=\delta=\omega_2$. For each old cardinal
$\lambda\geq\delta$, a name for a subset of $\lambda$ is coded
by a subset of $\lambda\times L_\delta$. There are at most
$(2^\lambda)^B$ such codes, and that cardinal is preserved.
The old subsets give the opposite bound, so the continuum
cardinal at $\lambda$ is unchanged. This proves global GCH.
Small forcing preserves all $B$-measurables above $\delta$.

The models $M$ and $B$ have the same countable ordinal sequences,
so they agree on $\omega_1$ and on this literal presentation of
$L_\delta$. An enumeration of a condition is coded by one
ordinal $\omega$-sequence and therefore belongs to $M$.
The same observation applies to the countable graphs of the
permutations used below. No inaccessibility assumption on
$\delta$ in $M$ is needed.

Permute the second coordinate $\omega_1$ to move the countable
domain of one condition away from that of another. A permutation
with countable support suffices, and belongs to $M$. It witnesses
weak homogeneity of $L_\delta$ there. If $\mathbb C\in M$ is
the old forcing for $K$, the product lemma identifies
$B'=M[K,H]$ as an extension by the fixed product
$\mathbb C\times L_\delta$. This product is weakly homogeneous.
Countable closure is used in the choice model $B$, so it adds
no ordinal $\omega$-sequences. This proves all assertions.
\end{proof}


\begin{lemma}[Recognizing an envelope]\label{lem:recognize}
Suppose $D$ is secure and $B=D(A,K)$ is a set-forcing extension of $D$ satisfying $\ZFC$. Choose $\eta$ with $A,K\in V_\eta^B$. If $U$ is a further set-forcing extension preserving $V_\eta^B$, then the actual class $B$ is definable in $U$ from the ordinal $\eta$ by
\[
 B=D(V_\eta^U).
\]
\end{lemma}
\begin{proof}
The right-hand model is constructed inside $B$ when $V_\eta^U=V_\eta^B$, and contains $D,A,K$. Thus it equals $B$. Security identifies $D$ inside $U$ as its mantle.
\end{proof}

\begin{remark}\label{rem:ODnotHOD}
Definability of the \emph{class} $B$ does not imply $B\sq\HOD^U$. Similarly, $V_\eta^U$ is OD but need not belong to HOD. The decoder can therefore recognize the choice envelope without placing its new wellorderings in the target inner model.
\end{remark}

\section{Exact external realization}\label{sec:realize}

\begin{proposition}[Realization over a secure ground]\label{prop:secure-realize}
Suppose
\[
 D\sq M=D(A)\sq B=M[K],
\]
where $D\models\ZFC+\CM$ is secure, $A\sq\mu^\omega$,
$B$ is a set-forcing extension of $D$ satisfying $\ZFC$, and
$B/M$ is weakly homogeneous with no new ordinal $\omega$-sequences.
Then there is a further set-forcing extension $U$ of $B$ such that
\[
 U\models\ZFC+\CM,\qquad \Hseq^U=M,\qquad
 M\models\SVC+\CM.
\]
If $B$ satisfies eventual GCH, one may also require $U\models\GCH$.
\end{proposition}
\begin{proof}
If $B$ satisfies eventual GCH, first apply
Lemma~\ref{lem:gch-envelope} to obtain $B'=B[H]$ with GCH.
Otherwise put $H=\varnothing$ and $B'=B$. In either case $B'/M$ is
weakly homogeneous and adds no ordinal $\omega$-sequences.
Since $B$ is a set extension of $D$, so is $B'$, and we have
$B'=D(A,K,H)$.

Choose $\eta$ in this actual envelope $B'$ so that $A,\seq{K,H}$ belong
to $V_\eta^{B'}$. Apply Corollary~\ref{cor:ODcoding} over $B'$,
starting above $|V_\eta^{B'}|$, to code $A$, and call the extension
$U$. Lemma~\ref{lem:recognize}, with the pair $\seq{K,H}$ as its
second generator, gives
\[
 B'=D(V_\eta^U).
\]
Thus the capped-history spectrum defines $A$ in $U$. In the
GCH case, Lemma~\ref{lem:spectrum-gch} gives $U\models\GCH$.

The combined forcing from $M$ to $U$ is weakly homogeneous. The name for the coding forcing is uniformly defined from the old $A$ and ordinal parameters. Automorphisms of the first forcing carry it to a forced equal name and therefore lift to the iteration. The second forcing is forced weakly homogeneous. For a local name for a second-stage automorphism, use the identity off a condition forcing it to be an automorphism. This requires only localized Boolean mixing, without a choiceless maximum principle.

By hypothesis, the first extension adds no ordinal $\omega$-sequences; the coding extension is highly distributive over the choice model $B'$. Lemma~\ref{lem:homdesc} gives $\Hseq^U\sq M$. Conversely, $D\sq\HOD^U\sq\Hseq^U$ and $A\in\Hseq^U$. The latter holds because it is OD in $U$ and every member of $A$ is an allowed ordinal-sequence parameter. Minimality of $D(A)$ gives $M\sq\Hseq^U$.

The entire extension $U/D$ is by set forcing. All sufficiently high normal $D$-measures lift to $U$, and their restrictions \eqref{eq:measure-restrict} belong to $M$. Thus both models have a proper class of measurables.

Finally, $B=M[K]\models\ZFC$ is a set-forcing extension of $M$.
The truth lemma gives a condition forcing Choice. Restricting
the forcing to that condition shows that Choice is forceable
over $M$. Blass's equivalence of this property with SVC,
as stated in \cite[Theorem~3.14]{Usuba}, gives $M\models\SVC$.
\end{proof}

The next preparation allows us to apply this proposition without assuming that class forcing preserves $\DC$ in arbitrary choiceless intermediate models.

\begin{proposition}[Installing the secure ground]\label{prop:install}
Suppose $M=D(A)\models\ZF+\DC+\CM+\SVC$, where $D\models\ZFC+\CM$ is a symmetric ground and $A\sq\mu^\omega$. Given an ordinal $\alpha$, there are
\[
 D^*\sq M^*=D^*(A)\sq B^*=M^*[K]
\]
satisfying the hypotheses of Proposition~\ref{prop:secure-realize},
with $B^*$ satisfying eventual GCH and
\[
 V_\alpha^{M^*}=V_\alpha^M.
\]
\end{proposition}
\begin{proof}
Take a fresh envelope $B=M[K]$ from Lemma~\ref{lem:envelope}. The symmetric-ground relation composes with set-forcing extensions. Since $D$ and $B$ satisfy Choice, $D$ is a set-forcing ground of $B$ by \cite[Lemma~5.8]{Usuba}. Write $B=D[b]$ using a forcing presentation in $D$.

Let $\mathbb P\in D$ be the presentation with generic $b$.
Choose in $D$ a regular $\delta>\max(|\mathbb P|,\omega)$, above a uniform
$\mathbb P$-name bound for the cardinality of the protected
$B$-rank. Such a bound is available before evaluating $b$,
using a set of names for that rank. Choose the preparation
$\mathbb Q$ of Lemma~\ref{lem:secure} in $D$, independently of
$b$, with every set-sized initial segment
$\mathbb Q_\beta$ $<\delta$-closed in $D$.
Take a coherent class generic $J$ over $B$ for this fixed
$D$-iteration in the set-name framework just specified.
For every $\beta$, the ordinary product lemma gives
\[
 D[b][J_\beta]=D[J_\beta][b].
\]
Since $\mathbb P$ is $\delta$-cc, Easton's lemma makes the
fixed $\mathbb Q_\beta$ $<\delta$-distributive in $B$.
This does not assert that its ground closure persists in $B$.
This commutation of small forcing with class preparation is used in
\cite[Appendix, Lemma~121]{ReitzGA}.

Put $D^*=D[J]$. Every dense subset of the fixed $\mathbb P$
in $D^*$ appears in some $D[J_\beta]$, where $b$ is generic
by the product lemma. Thus $b$ is generic over $D^*$, and
\[
 B^*=D^*[b]=\bigcup_\beta B[J_\beta]\models\ZFC,
 \qquad M^*=D^*(A).
\]
Every short ordinal sequence in $B^*$ appears at a set stage.
Distributivity at each such stage therefore shows that $B^*/B$ adds none of
length less than $\delta$. It also preserves the protected
rank: a first new set in that rank would be a new subset of an
old set of cardinality less than $\delta$, coded by its old
wellorder as a short ordinal sequence. Thus, by freshness of
$B/M$ and the inclusions $M\sq M^*\sq B^*$,
\begin{equation}\label{eq:same-sequences}
 (\Ord^\omega)^{M^*}=(\Ord^\omega)^{B^*}=(\Ord^\omega)^M.
\end{equation}

We claim
\begin{equation}\label{eq:intersection}
 M^*\cap B=M.
\end{equation}
Every set of $D^*(A)$ belongs to $M[J_\beta]$ for some set-sized initial segment of the class iteration. Indeed, take a bounded transitive-base construction witnessing membership, then a stage at which the required rank of $D^*$ has stabilized. The old set forcings for $K$ and $J_\beta$ are mutually generic over $M$, so
\[
 M[J_\beta]\cap M[K]=M.
\]
The usual product-intersection proof works over $\ZF$: compare two names for a first new set of a given rank and use product conditions deciding membership. This proves \eqref{eq:intersection}. Together with preservation of the protected $B$-rank, it gives $V_\alpha^{M^*}=V_\alpha^M$.

The same filter $K$ is generic over $M^*$ for the \emph{fixed} old collapse forcing. Each dense set in $M^*$ appears in some $M[J_\beta]$, where mutual genericity makes $K$ meet it. Also $B^*=M^*[K]$. One inclusion holds inside $B^*$. For the reverse inclusion, $D^*\sq M^*$ and $b\in B=M[K]\sq M^*[K]$ give $D^*[b]\sq M^*[K]$. The old collapse automorphisms still witness weak homogeneity, and \eqref{eq:same-sequences} shows that there are no new allowed parameters.

By Lemma~\ref{lem:secure}, $D^*$ is secure and satisfies
$\ZFC+\CM$ and eventual GCH. Its set-forcing extension $B^*$
also satisfies eventual GCH: above the forcing size, the
cardinals and continuum cardinal values are unchanged.
Thus the GCH clause of Proposition~\ref{prop:secure-realize}
applies. The set-forcing identity $B^*=M^*[K]$ also yields
$M^*\models\SVC$ by Blass's equivalence. We have not assumed
that the class preparation preserves $\DC$ in $M^*$.
Once $M^*$ is identified with $\Hseq^U$,
Lemma~\ref{lem:HDC} gives $\DC$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:external-realization}]
From $\SVC$, take a choice symmetric ground $D$ and a transitive generator $T$ with $M=D(T)$; see \cite[Proposition~5.4]{Usuba}. The choice ground may be taken to have a proper class of measurables: make a set-generic choice envelope, use Lemma~\ref{lem:ZFmeasure}, and apply downward L\'evy--Solovay to the resulting choice-ground presentation. Alternatively, in the application from Theorem~\ref{thm:rank-transfer}, the ground $D$ already has this property.

By $\KW$, choose a surjection $f:(\mu^\omega)^M\to T$. Code its domain and the relation $f(p)\in f(q)$ by a single set $A$ of ordinal $\omega$-sequences. Equality of predecessor sets and the well-founded extensional collapse, as in Proposition~\ref{prop:seed}, recover $T$ and $f$. Hence $M=D(A)$. Apply Proposition~\ref{prop:install} and then Proposition~\ref{prop:secure-realize}.
\end{proof}

\section{The three consistency implications}\label{sec:finish}

\begin{proof}[Proof of Theorem~\ref{thm:external-main}]
The implication $\Con(E_2(\sigma))\Rightarrow\Con(E_1(\sigma))$ follows directly from the definitions. If $U\models E_2(\sigma)$, Lemma~\ref{lem:HDC} gives $\Hseq^U$ as a model of $E_3(\sigma)$. Hence $\Con(E_2(\sigma))\Rightarrow\Con(E_3(\sigma))$.

Proposition~\ref{prop:one-three} gives $\Con(E_1(\sigma))\Rightarrow\Con(E_3(\sigma))$ without assuming that the original external HOD already contains a proper class of measurables.

For the remaining direction, let $V\models E_3(\sigma)$ and choose $\kappa$ with $V_\kappa^V\models\sigma$. Apply Theorem~\ref{thm:rank-transfer} above $\kappa$ to obtain a sequence-generated model $M$ with $V_\kappa^M=V_\kappa^V$. Apply Theorem~\ref{thm:external-realization} while protecting that rank. The resulting $U$ satisfies $\ZFC+\GCH+\CM$, and
\[
 \Hseq^U\models\SVC+\CM,
 \qquad V_\kappa^{\Hseq^U}=V_\kappa^M=V_\kappa^V.
\]
Hence $U\models E_2(\sigma)$, completing the cycle of implications.
\end{proof}

\subsection*{Why this is an external theorem}
The original Prikry ambient universe contains the master sequence $c$. Every external $\Hseq$ contains all ambient ordinal $\omega$-sequences, including $c$. We therefore cannot simply take the final external HOD in that universe. The master can expose real-coded information omitted from the intermediate model. Lemma~\ref{lem:envelope} provides a \emph{fresh} choice envelope with no new ordinal $\omega$-sequences.

The high witnesses in Section~\ref{sec:spectrum} need not belong to the final $\Hseq$. Their ambient existence makes $A$ definable, and weak homogeneity controls hereditary descent. The proof requires neither tree-Prikry completions of the Cohen caps nor a unique high witness for each code.

\subsection*{Proof dependencies and scope}
The construction uses two nonstandard forcing results, Lemma~\ref{lem:paired} and Theorem~\ref{thm:spectrum}. The first uses tagged finite interpolation and a countable diagonal argument. The second applies an existing approximation criterion to the displayed quotient order. Both are needed, for different parts of the proof. Lemma~\ref{lem:earlier-core} supplies the separate fine-structural reduction. The following qualifications describe their scope.
\begin{enumerate}
\item The global rank-preservation proof uses literal equality of $Q,F$ and of supported parameters, not merely isomorphism of their indexing structures.
\item The $\DC$ proof uses external countable domination and wellorderable search spaces; it does not assert that every ambient countable sequence belongs to the inner model.
\item The cap spectrum is computed over the actual recognizable envelope. It requires no classification of arbitrary incomparable grounds.
\item The secure preparation uses repeated $\diamondsuit^*$ codes
and retains a proper class of measurables. A subsequent countably
closed set collapse gives global GCH in the choice envelope;
the final evaluated cap forcing preserves it. The protected rank
belongs to the target inner model and need not satisfy GCH.
\item The conclusion concerns first-order sentences true in a rank segment. It neither identifies the complete theories of the three ambient universes nor proves the internal assertion $V=\HOD_{\Ord^\omega}$ in the original model.
\end{enumerate}

\section{A canonical relative inner model with unbounded Woodin cardinals}
\label{wd-inner-model-section}

The reverse consistency argument needs a choice model containing a
prescribed real, with unbounded Woodin cardinals, whose definition is
unchanged by a finite recursive recoding of that real. It also needs this
model to be a set-forcing extension of its own HOD. We establish these
requirements together. The fine structure used below is the classical
Mitchell--Steel short-extender fine structure of pure-extender premice. The model we
construct has no strategy predicate.

For a real $r$, put
\[
 a(r)=\HYP(r)=L_{\omega_1^{\mathrm{CK},r}}[r].
\]
Here $\HYP(r)$ denotes the entire transitive admissible set, including its
ordinals, rather than its collection of reals. It is countable,
transitive, and self-wellordered in the usual relative-premouse sense.
The normalization by $a(r)$ will remove the choice of a real code from
the construction.

\begin{convention}\label{wd-relative-convention}
We use classical premice over the named transitive base $a$.
Every relative Skolem hull contains $a\cup\{a\}$, and every extender has
critical point above $\operatorname{rank}(a)$. Cores, projecta, standard
parameters, and ultrapowers are taken in this one convention throughout.
An enumeration or wellorder of $a$ may be used to prove a property of a
premouse; it is not an additional predicate of that premouse.

For a class premouse $C$, \emph{countable iterability} means that every
countable elementary pullback of a set initial segment, with the base
fixed pointwise, has the ordinary Mitchell--Steel stack iterability in
the applicable degrees. The game allows taking an initial segment at
the start of a round. This is the countable-hull convention of
\cite[Section~1.1]{SteelCoreBook}; the relative fine-structural
\L o\'s and copying conventions are those of
\cite[Section~2.5]{SteelTutorial}.
\end{convention}

\begin{theorem}[Canonical Woodin envelope]\label{wd-envelope}
There is a uniform first-order class definition $F(a)$ such that
$\ZFC+\PCW$ proves the following, with assertions of $\ZFC$ understood
schematically. For every real $r$, if $a=\HYP(r)$ and $C=F(a)$, then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $C$ is a transitive inner model containing $a$ as a set, and
      $C\models\ZFC+\PCW$;
\item the definition uses only the pure set $a$, with no chosen
      enumeration, background witness, or iteration strategy;
\item $C$ is a set-forcing extension of $\HOD^C$.
\end{enumerate}
Consequently, in the same ambient universe,
\[
 r\equiv_T s\quad\Longrightarrow\quad
 F(\HYP(r))=F(\HYP(s)).
\]
\end{theorem}

The equality in the last assertion is literal equality of the resulting
classes. It is a same-universe assertion; no generic absoluteness of
$F$ is asserted or needed. We prove the theorem in four steps.

\subsection{The standard background construction and its stopping rule}
\label{wd-background}

We use the fully backgrounded, minimum-support Mitchell--Steel
construction, with its usual core and limit clauses
\cite{MSFine}. The explicit version in
\cite[Section~2.3]{LNSBackground} fixes the transition rule we need.
Suppose that the current sound premouse $M_\xi$ is passive and has
height $\alpha$. An active extension with fine extender $E$ and support
$\nu<\alpha$ is eligible when $(M_\xi,E)$ is a premouse and there is a
total background extender $E^*$ over the ambient universe such that
\begin{equation}\label{wd-background-predicate}
 V_{\nu+\omega}\subseteq\operatorname{Ult}(V,E^*),
 \qquad
 E\restr\nu=E^*\cap
       \bigl([\nu]^{<\omega}\times M_\xi\bigr).
\end{equation}
Among eligible extensions, the construction takes one of least support.
It first forms the raw active level $N_{\xi+1}$, verifies reliability,
and then takes its $\omega$-core. If there is no active extension, or the
current level is active, it instead forms the usual raw passive Jensen
successor before the same reliability and core operations.

At a limit $\lambda$, the raw level has height
\[
 \eta_\lambda=\liminf_{\xi\longrightarrow\lambda}
       (\rho_\omega(M_\xi)^+)^{M_\xi};
\]
its bounded initial segments are the stabilized bounded initial
segments of the earlier levels. Reliability is again checked before
taking the core. We retain the raw levels and the decoded cores as
distinct objects. This distinction matters when a core is represented
by a type~III code.

We add diagnostic stopping tests, without changing a transition which
passes those tests:
\begin{enumerate}[label=\textup{(\alph*)}]
\item Test each raw level for nontameness before testing reliability
      or taking a core. Test the decoded core as well.
\item At a passive core, stop if any eligible active head is nontame,
      before selecting a minimum-support head.
\item Stop if a core already has a sound tame active type~I proper
      initial segment whose top critical point is a limit of Woodin
      cardinals above the base.
\end{enumerate}
At a stop, we will choose the least index of a segment as in (c) on the
already determined sequence. The extraction argument below proves that
such a segment exists in every genuine stopping case.

We record precisely the classical fine-structure inputs to this
construction. The tame extension of Mitchell--Steel fine structure
gives reliability, solidity and universality of the relevant cores,
and countable-hull stack iterability for tame backgrounded levels and
their associated comparison objects
\cite{SchimmerlingSteelTame}. In particular, it applies to the associated
bicephali used in next-extender uniqueness; this scope of the tame
theory is also explained in \cite[p.~9]{SteelCMIP}. The assertion is
about tame levels of the construction and their associated objects.
It does not require that every premouse in the universe be tame.

Before we continue at a passive core, test (b) has excluded every
nontame eligible head. Two eligible heads of minimum support therefore
give an associated tame bicephalus to which the joint iterability and
next-extender uniqueness theorems apply. The classical category check
is useful here: type~II has successor natural support, whereas types~I
and~III have limit natural support. Equal minimum support places the
two heads in the same category covered by the classical uniqueness
theorem. Thus the fine extender is unique. We use joint iterability of
the associated bicephalus, not an inference from separate iterability
of its two heads.

The witness $E^*$ in \eqref{wd-background-predicate} is only existential
data. Once the fine head is unique, different witnesses do not change
the next premouse. Jensen successor, core-taking, the liminf clause,
and the diagnostic tests are uniform from the current set structures.
The resulting recursion is therefore
definable from $a$ alone. No wellorder of the possible background
witnesses is used.

Since $a$ is countable in the ambient universe, the critical point of
any total background extender, being measurable there, exceeds
$\operatorname{rank}(a)+\omega$. This bound may be used in the relative
fine-structure calculations without adding an eligibility restriction
to \eqref{wd-background-predicate}.

\begin{lemma}\label{wd-never-stops}
If the construction never stops, its stabilized class limit is a
countably iterable inner model of $\ZFC+\PCW$ containing $a$.
\end{lemma}

\begin{proof}
Every raw level and decoded core is tame. The tame fine-structure
theorem therefore supplies reliability at the point where the ordinary
construction requires it, so there is no breakdown. The standard
core-stabilization theorem gives the class premouse limit.

We now apply the Woodin-inheritance theorem for this very construction:
the fully backgrounded Mitchell--Steel construction, when it does not
break down, inherits the ambient Woodin cardinals. This is the theorem
recalled at the opening of \cite{SteelLocalKc}. In the relative version,
the Woodins above the base are inherited, which suffices for $\PCW$.
All extender critical points fix $a$, so the usual reflection and
background-restriction proof uses exactly the same bounded data above
the base. In particular, we have not imposed an extra inaccessible-
length or certificate requirement on
\eqref{wd-background-predicate}. Such a restriction would require a
separate inheritance argument.

The classical fine structure and stabilization give $\ZFC$ for the
proper-class limit. To check the required countable iterability, fix a
set initial segment and a countable pullback. A sufficiently late tame
stage contains its stabilized segment. Include the segment and the
pullback map in the ambient hull used for that stage's iterability
theorem, and use the initial-segment move and copying. This supplies
the required stack strategy on the pullback.
\end{proof}

\subsection{Extracting a tame top measure at the first overlap}
\label{wd-extraction}

The stopping alternative does not require an iterable nontame mouse.
It uses a measure fragment already present on a determined sequence.
We give the endpoint details because the Woodin overlapped by the new
extender can equal its natural support.

\begin{lemma}[First-overlap extraction]\label{wd-first-overlap}
Suppose a first nontame active level $Q=(P,E)$ is encountered in the
construction above, either as an eligible head or inside a decoded
core. Then the already determined sequence below that head has a
sound tame active type~I segment $S$ whose top critical point is a
limit of Woodin cardinals. The countable elementary pullbacks of $S$
have the inherited stack iterability. The least index of such a
segment is determined by the preceding core, without choosing a bad
head or a countable hull.
\end{lemma}

\begin{proof}
Choose the first offending extender index in the indicated premouse.
Write
\[
 \kappa=\operatorname{crit}(E),\qquad
 \nu=\nu(E),\qquad
 \alpha=\operatorname{lh}(E),\qquad P=J_\alpha[\vec E,a].
\]
There is a $\delta\geq\kappa$ which $P$ regards as Woodin and which
is overlapped by $E$. All earlier active initial segments are tame.
Let $i$ be the degree-zero ultrapower map by $E$, and put
$R=i(J_\kappa^P)$. The coherence and indexing axioms give
\begin{equation}\label{wd-overlap-coherence}
 J_\alpha^R=P,
 \qquad \alpha=(\nu^+)^{\operatorname{Ult}_0(P,E)},
 \qquad \alpha<i(\kappa).
\end{equation}
Only the wellfounded part required by these axioms is used at this
point; no iterability of the newly adjoined nontame extender is assumed.

Acceptability gives the following useful endpoint calculation. If
$\tau<\alpha$, every subset of $\tau$ in $R$ is already in
$J_\alpha^R$. Otherwise the first later level at which such a subset
appeared would give a surjection from $\tau$ onto $\alpha$, contradicting
that $\alpha$ is a cardinal of $R$. Also $P$ has no cardinals above
$\nu$ in this indexed active situation
\cite[Remark~2.5]{SteelOutline}. Hence $\delta\leq\nu<\alpha$.
The two structures agree on subsets of $\delta$ and on the bounded
extender witnesses used to test its Woodinness. Thus $R$ regards
$\delta$ as Woodin, including when $\delta=\nu$.

The restriction of $i$ to the coded set $J_\kappa^P$ is fully elementary
as a map of set structures. Indeed, for each fixed formula, relativizing
all its quantifiers to $J_\kappa^P$ gives a bounded formula; the
degree-zero map preserves that formula and sends the displayed set
to $R$. For every $\beta<\kappa$, $R$ has a Woodin cardinal between
$\beta$ and $i(\kappa)$. Reflecting this assertion to $J_\kappa^P$
shows that $\kappa$ is a limit of Woodin cardinals. Acceptability
transfers these bounded witnesses to $P$ and to the relevant proper
initial segments.

We must have $\delta>\kappa$. If $\delta=\kappa$, the proper segment
through $(\kappa^+)^P$ would have the same subsets of $\kappa$ and
would regard $\kappa$ as a Woodin limit of Woodin cardinals. That
segment is tame and has iterable countable pullbacks. This contradicts
the tame-smallness fact recorded immediately after
\cite[Definition~6.15]{SteelOutline}: a tame mouse cannot have a
Woodin limit of Woodin cardinals.

Set $\eta=(\kappa^+)^P$. Then
\[
 \kappa<\eta<\delta\leq\nu.
\]
Apply the initial-segment condition to $E\restr\eta$. Its natural
support is $\eta$, so this is not the type~Z exceptional fragment.
The condition puts its trivial completion either earlier on the
sequence of $P$, or on the one-step ultrapower using a nonempty
extender indexed at $\eta$
\cite[Definition~2.4(3)]{SteelOutline}. The second alternative is
impossible: $\eta$ is a cardinal of $P$, whereas a nonempty extender
at that index makes it noncardinal at the next Jensen stage, which
still lies below $\alpha$. Thus the completion occurs at an earlier
index $\gamma<\alpha$ on the sequence of $P$.

The active segment at $\gamma$ is the desired $S$. Its extender is
the normal-measure fragment with critical point $\kappa$, hence is
type~I. It agrees on the bounded sets below $\kappa$, so its top
critical point remains a limit of Woodin cardinals. It is tame by the
choice of the first offending index, and it is sound as a proper
initial segment.

Here is the iterability accounting. When $P$ is a passive decoded
core of a tame background level $N$, a countable hull of $P$ maps
weakly into $C_\omega(N)$. The tame level theorem supplies its stack
iterability, and the initial-segment choice before the first round
supplies that of the corresponding pullback of $S$.

If instead the decoded core itself is nontame, take an ambient
countable hull containing the \emph{unsquashed} core, its code, the
chosen overlap witness, and the relevant set satisfaction relations.
Collapse this data together. The overlap witness survives even when
it uses a type~III top extender and the endpoint $\delta=\nu$ lies
outside the ordinal domain of the squashed code. The restriction on
the codes is the weak core embedding to which the tame predecessor's
iterability theorem applies. We may therefore take the first bad
active initial segment and then the proper segment $S$ just found.
This uses iterability inherited from the preceding tame level, not
an assumed strategy on a newly adjoined bad head.

Finally, the fragment belongs to the fixed preceding sequence. The
existence of a bad eligible head is used only to prove that a suitable
index exists there. Taking the least suitable index makes $S$
independent of the choice of that head. Countable hulls are used to
verify iterability, never to select the output.
\end{proof}

For completeness, the raw-level tests create no further stopping
case. A first nontame passive successor would already have its local
overlap witness on the previously tested indexed extender sequence.
At a first nontame limit, that witness is bounded and occurs in a
stabilized earlier segment. Thus neither can be the first new failure.
The tests are nevertheless made before reliability, so the proof never
takes a core whose existence has not been justified.

\subsection{Iterating the top measure out of the model}
\label{wd-top-iteration}

\begin{lemma}[Top-measure envelope]\label{wd-top-envelope}
Let $S$ be a sound tame active type~I premouse over $a$, with inherited
countable-hull stack iterability, whose top critical point $\kappa$
is a limit of Woodin cardinals above the base. There is a class
premouse $C_S$, uniformly definable from $S$, containing $a$, whose
pure universe satisfies $\ZFC+\PCW$. Its set initial segments have
the countable iterability of Convention~\ref{wd-relative-convention}.
\end{lemma}

\begin{proof}
Form the linear degree-zero iteration of the top measure. Every
set-length history is wellfounded. For if a first bad history existed,
take a countable elementary hull of a sufficiently large ambient
$H_\Theta$ containing the history, its maps and direct-limit data,
and an actual descending $\omega$-sequence in the illfounded last
structure. The collapse is a bad countable linear history of a
countable pullback of $S$, contradicting its inherited stack strategy.
This argument also covers an uncountable $S$: the starting model in
the collapsed history is its iterable pullback, rather than $S$
itself. The unique compatible set recursions therefore define the
linear iteration through all ordinals.

Write $S_\xi$ for the iterates, $i_{\xi\zeta}$ for their maps, and
$\kappa_\xi=i_{0\xi}(\kappa)$. We will show that
\begin{equation}\label{wd-rank-chain}
 B_\xi=J_{\kappa_\xi}^{S_\xi}
       =V_{\kappa_\xi}^{S_\xi},
 \qquad B_\xi\prec B_\zeta\quad(\xi<\zeta),
\end{equation}
where the elementary map in the second assertion is literal inclusion.
Some care is needed because the entire active premouse $S$ has not
been assumed to satisfy $\ZFC$.

First, $\kappa$ is internally regular in the sense required below.
If $S$ contained a cofinal $f:\lambda\longrightarrow\kappa$ with
$\lambda<\kappa$, the next degree-zero map $j$ would send it to a
cofinal map from the same $\lambda$ to $j(\kappa)$. This statement is
bounded in its displayed parameters. But
$j(f)(\xi)=j(f(\xi))=f(\xi)<\kappa$ for every $\xi<\lambda$,
a contradiction. The same argument rules out a bijection from a
smaller ordinal onto $\kappa$.

The cofinal Woodin cardinals and acceptability give strong-limit
closure below $\kappa$. Given $\tau<\kappa$, choose a Woodin between
$\tau$ and $\kappa$. Its bounded inaccessible-rank calculation locates
the power set of $\tau$ and a bound on its size below that Woodin;
acceptability excludes new subsets at later levels. Consequently the
usual acceptable-hierarchy calculation above a self-wellordered base
gives $J_\kappa^S=V_\kappa^S$. Each of its elements has an internal
wellordering of order type below $\kappa$. All its power sets agree
with those computed in $S$ and belong to $J_\kappa^S$.

Put $B=J_\kappa^S$. The type~I top index lies beyond $\kappa^+$, so
$B$ itself and the finite Jensen successor closures used next are
sets in $S$. Fix a formula $\varphi$ and parameters from $B$, and
suppose that, in $B$, it defines a function on $x\in B$. Relativize
all quantifiers of this one formula to the set $B$. The resulting
graph is a bounded-definable subset of $x\times B$, and belongs to
a finite Jensen successor closure above $B$, still inside $S$.
There is no appeal to Replacement in $S$ in forming this graph.

Assign to each value its least construction level below $\kappa$.
The hierarchy through $\kappa$ and this least-level graph also lie
in the finite successor room above $B$. If these levels were
unbounded in $\kappa$, composition with an internal enumeration of
$x$ of length $\lambda<\kappa$ would give, in $S$, a cofinal map
$\lambda\longrightarrow\kappa$. Internal regularity excludes this.
Thus the range is bounded in some $J_\eta^S$, $\eta<\kappa$.
Coding that level by a set of size below $\kappa$ and using the
power-set calculation puts the range in $B$. This proves each
Replacement instance. The same bounded-graph argument followed by
the subset calculation proves Separation. The remaining elementary
axioms and Choice follow from the acceptable relative hierarchy and
the wellordering of the base. Hence $B\models\ZFC$.

The Woodins cofinal in $\kappa$ remain Woodin in $B$: their subsets,
functions, and bounded extender witnesses agree. Therefore
$B\models\PCW$.

Fine ultrapower coherence gives, for $\xi<\zeta$,
\[
 i_{\xi\zeta}(J_{\kappa_\xi}^{S_\xi})
       =J_{\kappa_\zeta}^{S_\zeta}.
\]
These hierarchy sets are actual elements of the premice. Bounded
relativization to them proves full elementarity of the maps between
the set structures $B_\xi$ and $B_\zeta$, and transfers
$\ZFC+\PCW$ from $B_0$ to every $B_\xi$.

The maps fix $B_\xi$ pointwise. To see this, argue by rank and take
an internal enumeration $e:\lambda\longrightarrow x$ with
$\lambda<\kappa_\xi$ for $x\in B_\xi$. By induction its values are
fixed, while bounded elementarity says that $i_{\xi\zeta}(e)$ maps
the same domain onto $i_{\xi\zeta}(x)$. Thus
$i_{\xi\zeta}(x)=x$. The base is fixed by convention. This proves
\eqref{wd-rank-chain} with inclusion as the elementary map.

The critical sequence is strictly increasing and unbounded in the
ordinals. Define
\[
 C_S=\bigcup_{\xi\in\Ord} B_\xi.
\]
It is transitive and contains all ordinals. The elementary-chain
argument applies to each formula: an existential witness in the
union lies in a later stage and can be reflected to any earlier
stage containing its parameters. Thus $B_\xi\prec C_S$, and
$C_S\models\ZFC+\PCW$. The stabilized extender sequence presents
it as the class premouse obtained by iterating out the top measure.

For countable iterability of a set initial segment of $C_S$, include
that segment, its pullback map, a sufficiently high top iterate,
and the preceding linear history in an ambient countable hull.
The collapsed starting premouse has its given stack strategy.
Following the collapsed top history, and then taking its tail and
the indicated initial segment, supplies the required strategy.
This proves the countable-hull assertion. It does not assert a
single strategy on the entire class premouse, and none is needed.
\end{proof}

\subsection{The fixed-base own-HOD argument}
\label{wd-own-hod}

The own-HOD step requires more than choosing a canonical extender
sequence externally. We must define that sequence over the pure
universe of the resulting premouse from bounded set parameters.
We use the direct condensation-stack argument of
\cite[Theorem~4.1 and Remark~4.3]{SchlutzenbergDefinability}, with
the trivial forcing. The following proof gives the fixed-base and
local-iterability details of that application.

\begin{proposition}\label{wd-relative-own-hod}
Let $C$ be a proper-class classical pure-extender premouse over a
transitive self-wellordered base $a$. Suppose that its pure universe
satisfies $\ZFC$, that $a$ is countable both in $C$ and in the ambient
universe, and that $C$ has the countable iterability of
Convention~\ref{wd-relative-convention}. Then, for some set of ordinals
$t\in C$,
\[
 C=\HOD_{\{t\}}^C.
\]
In particular $C=\HOD^C[h]$ for set forcing in $\HOD^C$.
\end{proposition}

\begin{proof}
Choose in $C$ an enumeration $e$ of $a$. Fix cardinals
$\gamma<\theta$ of $C$, with $\theta$ regular, so far above the base
that $a$, $e$, and a hierarchy segment coding them are bounded below
$\gamma$. Put $m=C\restr\theta$. These are parameters for this
internal proof; they do not change the external definition of $C$.

\smallskip\noindent
\emph{Local condensation.}
The ordinary finite-degree solidity, universality, and sound-source
condensation theorems hold with the base fixed
\cite{MSFine,SteelCoreBook}. Each required instance on an uncountable
set segment follows by reflection to a countable ambient hull
containing the structures, the map, their set satisfaction relations,
and $a$ pointwise. Countable iterability supplies the target strategy
there. This derives local condensation without selecting a class
strategy for $C$.

For the sound-hull construction we also need the stronger condensation
theorem for a $\rho$-sound source, which need not yet be fully sound.
We use the passive degree-zero case of
\cite[Theorem~5.2]{SchlutzenbergCondensation}. Retain its lifting,
projectum, cardinal, and $\rho$-soundness hypotheses, place $a$ in
each hull and fine theory, and require the maps to fix $a$ pointwise.
The proof gives the same alternatives in this relative setting.
We describe why its stronger conclusion, rather than just ordinary
sound-source condensation, is available.

The solid-core parameters and fine theories are formed using tuples
from $a$ as well as the ordinal parameters. Relative \L o\'s and the
classical hull calculations preserve them. In the equal-successor-
cardinal case of that proof, the required agreement of power sets
and generalized solidity witnesses is unchanged, since the base
is fixed and has its enumeration below our cardinal floor. The
condensation used for already sound proper segments is the ordinary
theorem just established; the stronger conclusion is not assumed
at this point.

In the other case, the bicephali and cephalanxes have the same named
base. The iterability proof starts with the source embedding and
the identity into one iterable target, as in Claim~5 of the proof
of the cited theorem. The Shift maps send
$[u,f]$ to $[\pi(u),\pi(f)]$ and fix every element of $a$.
Model and degree drops take initial segments over the same base;
direct limits retain the fixed base. If a type~III structure occurs
in a comparison, the usual squashed copying and unsquashing are
performed with this same fixed-base \L o\'s calculation.

The subsequent comparison analysis
\cite[Theorems~4.3, 4.5, 4.17 and the proof of Theorem~5.2]{SchlutzenbergCondensation}
uses least disagreement, extender lengths and generators, cardinal
cutpoints, and the solid-core calculations. All these computations
take place above the base. Replacing an ordinal hull by its hull
together with $a$ does not repair an omitted-generator contradiction:
the maps already have every base element in their range. The same
terminal alternatives follow. In the passive degree-zero application
these are the initial-segment and active-ultrapower possibilities,
including the type~I normal-measure exception. The active type~II
final alternative is absent. This supplies the stronger local
condensation needed below. The argument has used classical relative
premice throughout; no conversion to an operator-premouse convention
is required.

\smallskip\noindent
\emph{Sound hulls above the base.}
Repeat the finite sound-hull construction of
\cite[Lemma~2.3]{SchlutzenbergDefinability}. In a passive $1$-sound
target $H$ with $\rho_1^H\geq\gamma$, put $a$ into each generating set and include the desired
finite data among its prescribed parameters. The recursion adjusts
a finite decreasing ordinal parameter and a cardinal bound
$\gamma_i$, maintaining $\gamma\leq\gamma_i$. At a nonterminal
step it chooses a noncardinal new generator $\eta>\gamma$ while
putting the earlier solidity witnesses in the appropriate preceding
hull. The resulting collapse $W$ is $\rho$-sound, with
\[
 \rho=\gamma_i^{+W},\qquad \rho_1^W=\gamma_i;
\]
it has not at this stage been asserted $1$-sound.

Apply the stronger condensation just proved in the specialization
used by that lemma. Either $W$ is an initial segment of $H$, or the
remaining exceptional case presents $W$ as a type~I degree-zero
ultrapower with critical point $\gamma_i$. The exception contradicts
the choice of the new generator: that ultrapower is generated by
the old range and its normal-measure seed $\gamma_i$, with the
standard-parameter tail carried along. Since the old range already
contains $a$, the proposed new generator would be in the hull
generated by $a$, $\gamma_i+1$, and that tail.

Thus $W$ is an initial segment. Its description as the least relevant
projecting segment makes the next solidity witness available in the
next hull. Each nonterminal step strictly decreases the cardinal
bound. The bound remains at least $\gamma$, so the recursion
terminates with a sound collapse containing the prescribed objects
in its range. In particular it never crosses the base. This proves
soundness for these adjusted hulls, rather than for arbitrary hulls.

The simultaneous version is obtained by alternating these parameter
adjustments for two passive successor structures with the same pure
universe and the same initial segment $m$. Include $a,e$, the two
preceding structures, and a prescribed element $x$ in the common
parameters. At each adjustment retain the previously arranged
ordinal agreement and mutual containment. The construction in
\cite[Theorem~4.1, Subclaim~1.1]{SchlutzenbergDefinability}, with
trivial forcing, then gives sound collapses of the same ordinal
height whose uncollapse maps agree on ordinals and on $a$. The
enumeration $e$ ensures that the common pure-universe collapse
also fixes the whole base.

\smallskip\noindent
\emph{The internal regular stack.}
Let $\eta\geq\theta$ be regular in $C$. Compute inside $C$ the Jensen
stack of sound classical $a$-premice extending $C\restr\eta$,
projecting to $\eta$, and satisfying the condensation properties
above. This stack is exactly $C\restr\eta^{+C}$.

Here both the internal qualification and the equality are important.
For comparability, put the base pointwise into the common hull in
the proof of \cite[Fact~3.1]{SchlutzenbergDefinability}. Regularity
makes the hull's intersection with $\eta$ an ordinal. The two
collapse maps have the corresponding common critical point, and
their projecta collapse to it. Sound-source condensation places
the collapses in the same initial segment of the root, or in the
same exceptional ultrapower of that root, giving comparability.
For a candidate coded in $C$, compare with a sufficiently high
true sound projecting segment containing it. The argument of
\cite[Fact~3.2]{SchlutzenbergDefinability} puts the candidate on the
true sequence. Conversely, the acceptable-hierarchy calculation
gives cofinally many sound projecting levels below $\eta^{+C}$:
the bounded power sets are already fixed below $\eta$, and the
levels adding the relevant subsets of $\eta$ project to $\eta$.
Their union is the true successor-cardinal segment. Its pure
universe is $H_{\eta^+}^C$. We have made no assertion about a stack
containing every externally available mouse.

\smallskip\noindent
\emph{Uniqueness over the pure universe.}
Use the excellent-candidate definition in the direct proof of
\cite[Theorem~4.1]{SchlutzenbergDefinability}, restricted to the
fixed base $a$ and trivial forcing. A candidate extends $m$, has
regular height $\eta$, has the local condensation just used, and
has a successor-cardinal stack whose pure universe is
$H_{\eta^+}^C$. The true $C\restr\eta$ is a candidate by the
preceding paragraph.

If $R$ is another candidate, alternately take the least appropriate
projecting extensions on the two sides containing the preceding
structure on the other side. The resulting $\omega$-unions, and
then their passive successors, have the same pure universe. For
each $x$ in this universe, take the simultaneous sound hulls just
constructed. Condensation places both collapsed mice in the common
initial segment $m$. They have the same ordinal height, hence are
the same premouse. Their uncollapse maps agree on ordinals and on
$a$; relative Skolem generation makes the maps agree everywhere.
Varying $x$ identifies the two successor structures and hence
identifies $R$ with $C\restr\eta$.

Excellence is expressed using set-coded premice, their satisfaction
relations, and their bounded stacks. Thus this uniqueness defines
$C\restr\eta$ over the pure universe $C$ from $m,a,e$, uniformly
for all sufficiently large regular $\eta$. Taking their union
defines the extender sequence of $C$ from those set parameters.
No iteration strategy for a possibly spurious candidate is assumed:
its required condensation is part of the candidate definition, and
the adjusted-hull argument uses those properties directly.

The relative Jensen hierarchy, together with the fixed enumeration
of the base, now gives a global wellorder definable over $C$ from
$m,a,e$. Every element is therefore hereditarily ordinal-definable
from this tuple. Choice in $C$ permits coding the tuple and its
transitive closure by one set of ordinals $t\in C$, giving
$C=\HOD_{\{t\}}^C$. Vop\v enka forcing over $\HOD^C$, for the
bounded ordinal code $t$, gives a generic filter $h\in C$ with
\[
 \HOD^C[h]=\HOD_{\{t\}}^C=C.
\]
This is the coarse own-HOD observation used in
\cite[Introduction]{SchlutzenbergOrdinal}; no additional hypothesis
for its subsequent refined conclusions is used here.

The proof assumed $\ZFC$ only in the pure universe. All candidates,
local satisfaction relations, and bounded stacks were sets. Codes
for true initial segments belong to $C$ by its premouse hierarchy
and amenability. After extender definability has been proved,
formulas in the expanded premouse language can be translated into
pure formulas with the fixed parameters. Expanded-language
Separation and Replacement thus follow at the end; they were not
assumed to obtain the definition.
\end{proof}

\subsection{Completion and recoding invariance}
\label{wd-envelope-completion}

\begin{proof}[Proof of Theorem~\ref{wd-envelope}]
Run the construction of Section~\ref{wd-background} over the pure
set $a=\HYP(r)$. If it never stops, use its stabilized limit from
Lemma~\ref{wd-never-stops}. If it stops, choose the least qualifying
type~I segment on the determined sequence, using
Lemma~\ref{wd-first-overlap}, and use its top-measure envelope from
Lemma~\ref{wd-top-envelope}. The transition rule, stopping tests,
least-index choice, and unique linear iteration are all first-order
uniform from $a$. This defines $F(a)$ without any additional
choice parameter.

In both cases $C=F(a)$ is a countably iterable classical premouse
containing $a$, and its pure universe satisfies $\ZFC+\PCW$.
Since $r\in a\in C$, we have $L[r]\subseteq C$. The ordinal
$\omega_1^{\mathrm{CK},r}$ and the set
$L_{\omega_1^{\mathrm{CK},r}}[r]$ are countable in $L[r]$;
their enumerations therefore belong to $C$. Thus $a$ is countable
in $C$ as well as in the ambient universe. Proposition~\ref{wd-relative-own-hod}
applies and supplies the own-HOD set-forcing presentation.

Finally, if $r\equiv_T s$, every admissible set containing $r$
contains $s$, and conversely: the recursive computation is available
inside the admissible set. The least transitive admissible sets
containing the two reals are consequently identical. Thus
$\HYP(r)=\HYP(s)$, and the uniform pure-set definition gives the
stated equality of envelopes.
\end{proof}

\begin{remark}\label{wd-envelope-scope}
This construction retains fine structure. Its role is confined to
the tame background construction, the first-overlap fragment, and
the condensation needed for the own-HOD presentation. It requires
neither the iterability of an arbitrary nontame premouse nor a
strategy on the ambient universe. The enumeration used in the
own-HOD proof is internal evidence about an already determined
model; it is not part of the definition whose recoding invariance
is used in the consistency argument.
\end{remark}

\section{From external HOD to a Woodin symmetric ground}\label{wd-reverse-section}

We apply the relative inner-model theorem to a real coding a rank of
external HOD. The recoding invariance is needed inside a single
collapse extension. No absoluteness of the relative construction
between different universes is used.

\begin{proposition}\label{wd-reverse}
Suppose $U\models\ZFC+\PCW$, let $H=\Hseq^U$, and fix an ordinal
$\alpha$. In a set-generic extension of $U$ there are transitive
classes $D\sq N$ such that
\[
 D\models\ZFC+\PCW,\qquad
 N\models\ZF+\DC+\SVC,\qquad
 V_\alpha^N=V_\alpha^H.
\]
Moreover, $N$ has a set-symmetric presentation over $D$.
\end{proposition}
\begin{proof}
Choose $\lambda>\alpha+\omega$ with $\cf^U(\lambda)>\omega$, and put
$X=V_\lambda^H$. By Lemma~\ref{lem:HDC}, $H$ is closed under
$U$-countable sequences of its members. The cofinality condition
then gives
\begin{equation}\label{wd-X-closure}
 {}^\omega X\cap U\sq X.
\end{equation}
Indeed, the ranks of the countably many entries have a bound below
$\lambda$, and adding the finitely many levels needed to code the
sequence still leaves a rank below $\lambda$. The set $X$ is
ordinal-definable in $U$ from $\lambda$.

Force over $U$ with the finite partial functions from $\omega$ to
$X$. Let $e:\omega\twoheadrightarrow X$ be the generic enumeration.
Let $r$ code both relations
\[
 i\mathrel{E_r}j\ \longleftrightarrow\ e(i)=e(j),
 \qquad
 i\mathrel{R_r}j\ \longleftrightarrow\ e(i)\in e(j).
\]
Set forcing preserves the sufficiently high Woodins, so
$U[e]\models\ZFC+\PCW$ by \cite{HamkinsWoodin}. Apply
Theorem~\ref{wd-envelope} there to obtain
\[
 C=F(\HYP(r)),\qquad C\models\ZFC+\PCW,
 \qquad C=\HOD^C[h].
\]
Since $r\in C$, the extensional quotient of $(\omega,R_r)$ by
$E_r$ and its Mostowski collapse belong to $C$. Their collapse is
the actual transitive set $X$. Thus $X\in C$. Define
\[
 N=\HOD_{X\cup\{X\}}^C.
\]
Here definitions may use $X$, finitely many of its elements, and
ordinals. Relative hereditary definability gives $N\models\ZF$,
$X\in N$, and $\HOD^C\sq N$.

\smallskip\noindent
\emph{Descent to the original external HOD.}
Finite permutations of $\omega$ witness weak homogeneity of the
collapse. If $\pi$ is such a permutation, the real coding
$e\circ\pi$ is Turing equivalent to $r$. Consequently
\[
 \HYP(r_{e\circ\pi})=\HYP(r_e),
 \qquad F(\HYP(r_{e\circ\pi}))=F(\HYP(r_e)).
\]
Both equalities are evaluated in $U[e]$, which is also
$U[e\circ\pi]$. The class $C$ and the parameters $X$ and
$x\in X$ are therefore fixed by these recodings.

Induct on hereditary rank to show $N\sq H$. If $y\in N$ and all
members of $y$ have already descended to $U$, then $y$ is a subset
of a ground rank $V_\beta^U$. Choose a relative-HOD definition of
$y$ over $C$, evaluated at a sufficiently large set-sized rank.
The formula uses $X$, finitely many elements of $X$, and ordinals.
The preceding invariance and weak homogeneity make its membership
decisions independent of the collapse condition. The forcing
relation defines $y$ in $U$ from these same parameters. Since $X$
is OD in $U$ and its finitely many specified elements belong to
$H$, their ordinal-sequence definitions can be combined into a
single ordinal-sequence parameter. Hence $y$ is definable in $U$
from an allowed parameter. The induction hypothesis supplies the
hereditary requirement, proving $y\in H$.

Transitivity and $X\in N\sq H$ now give
\[
 V_\lambda^N=X=V_\lambda^H,
 \qquad V_\alpha^N=V_\alpha^H.
\]

\smallskip\noindent
\emph{Small violations of choice and dependent choice.}
The bounded evaluation map for relative-OD definitions gives
$\SVC(X)$ in $N$. To spell out its parameter bound, fix a nonempty
$Y\in N$ and a member $y_*\in Y$. Collection in $C$ bounds ranks
witnessing relative-OD definitions of all members of $Y$.
Formula codes, stage ordinals, and ordinal
parameters are absorbed into an ordinal $\eta$. Finite tuples of
elements of $X$ can be coded by elements of $X$. Thus an evaluation
map in $N$ gives a surjection
\[
f:\eta\times X\twoheadrightarrow Y.
\]
Send invalid codes to $y_*$. The graph is relatively OD in $C$
from $Y$, $y_*$, $X$, and the ordinal bounds, and all its hereditary
constituents belong to $N$. Thus $f\in N$. This does not choose a
wellordering of $X$.

Let $R\in N$ be a serial relation on a nonempty set $Y\in N$, and
fix $y_0\in Y$. Use such a surjection $f$. In $U$, having chosen
$y_n$, take the least $\xi_n<\eta$ such that
\[
 \exists x\in X\quad y_n\mathrel R f(\xi_n,x),
\]
choose one such $x_n$ using choice in $U$, and put
$y_{n+1}=f(\xi_n,x_n)$. All these operations take place in $U$,
since $R,Y,f\in N\sq U$. By \eqref{wd-X-closure}, the whole
sequence $\seq{x_n:n<\omega}$ belongs to $X$, and hence to $N$.
Inside $N$, reconstruct the $y_n$ by recursion using this sequence,
$y_0$, $f$, $R$, and the least-ordinal rule. Its defining
quantifiers range over the same sets in $N$ and $U$, so the
reconstruction gives exactly the chosen chain. This proves $\DC$
in $N$. No bound on $\eta$ by $\lambda$ is needed.

\smallskip\noindent
\emph{The Woodin ground.}
Set $D=\HOD^C$. Theorem~\ref{wd-envelope} gives a set forcing
$P\in D$ and a $D$-generic $h$ such that
\[
 D\sq N\sq C=D[h].
\]
Let $\tau=|P|^D$. Every sufficiently high Woodin of $C$, in
particular each above the small-forcing bound, is already Woodin
in $D$ by \cite{HamkinsWoodin}. Thus $D\models\ZFC+\PCW$.
Since $N\models\ZF+\SVC$, the symmetric-intermediate theorem
\cite[Corollary~5.10]{Usuba} implies that $N$ is a symmetric
extension of $D$. Its set-forcing presentation is supplied by
\cite[Proposition~4.4]{Usuba}. The presenting forcing need not be
the particular $P$ used above. This is sufficient for the claimed
existence of a set-symmetric presentation.
\end{proof}

\section{Woodin realization by set forcing}\label{sec:wd-realization}

Write $\PCW$ for the assertion that the Woodin cardinals form a
proper class.  For a rank-restricted conclusion, recognizing an entire
choice envelope is unnecessary.  We shall recognize enough of that
envelope to evaluate the capped-history predicate of
Theorem~\ref{thm:spectrum}.  This permits a construction using only set
forcing over the original Woodin ground.

\subsection{Keeping the original symmetric ground}

\begin{lemma}[The old-ground version of local transfer]
\label{wd-old-ground-transfer}
Suppose $D\models\ZFC+\PCW$, and $N$ is a set-symmetric extension
of $D$ satisfying $\ZF+\DC$.  Given an ordinal $\alpha$, there are
models and sets
\[
 D\subseteq N\subseteq M=D(A)\subseteq B=M[K]=D[b]
\]
such that
\begin{enumerate}
\item $V_\alpha^M=V_\alpha^N$ and $M\models\ZF+\DC+\SVC$;
\item $A\subseteq\mu^\omega$ for some ordinal $\mu$;
\item $B\models\ZFC+\PCW$, and $B/M$ is weakly homogeneous
and adds no ordinal $\omega$-sequences;
\item $B=D[b]$ is an ordinary set-forcing extension of this same $D$.
\end{enumerate}
There is no large-cardinal hypothesis on $N$ in this statement.
\end{lemma}

\begin{proof}
Fix a symmetric presentation $D\subseteq N\subseteq E_0=D[G]$,
with forcing $T\in D$.  By \cite[Lemma~4.1 and Proposition~5.4]{Usuba},
there is $z\in N$ with $N=D(z)$, and $N$ satisfies SVC.  Fix an
SVC witness $S\in N$.  The ground is already given; the measurable
capture argument in Proposition~\ref{prop:coretransfer} is not needed.

Choose an uncountable regular $D$-cardinal $\vartheta>|T|^D$ above
the presentation bounds.  Choose a sufficiently large ground beth
fixed point $\lambda>\alpha$ of $D$-cofinality $\vartheta$, with
$z,S\in V_\lambda^N$, and put $X=V_\lambda^N$.  The small forcing
$T$ preserves this cofinality.  The old cofinal sequence belongs to
$N$, and the full cover has no shorter one, so
$\cf^N(\lambda)=\vartheta>\omega$.  The transitive-base convention
gives $N=D(X)$, and extending each SVC surjection by a default value
gives $N\models\SVC(X)$.

A Woodin cardinal is a limit of measurable cardinals, so $D$ has
arbitrarily large measurable cardinals.  After bounding the ground
names for $X$ and the old collapse below, choose a $D$-measurable
$\kappa>\lambda+\omega$ above these bounds.  Fix a normal
$D$-measure $\UU$ on $\kappa$.  Let $\pp_\UU^D$ be the old
Prikry poset and let
\[
 \cc_0=\{k:k\text{ is an }N\text{-countable partial function }
                    \omega_1^N\longrightarrow X\}^{N}.
\]
Both the presentation and $\cc_0$ have size below $\kappa$ in
the full cover.  Take $(c,K_0)$ generic for the fixed product
$\pp_\UU^D\times\cc_0$ over $E_0$, and set
\[
 W=N[K_0][c]=N[c][K_0],\qquad E=E_0[K_0][c].
\]

Here the use of the old measure is essential.  In each small choice
cover $Y$ of $D$, the normal generated lift is
\[
 \UU_Y=\{a\in\Pow(\kappa)^Y:
                  \exists u\in\UU\ (u\subseteq a)\}.
\]
Every tail in $\UU_Y$ contains an old tail, so $\pp_\UU^D$ is
dense in the Prikry forcing recomputed from $\UU_Y$ in $Y$.
We do not require $\UU$ itself to measure all the new subsets of
$\kappa$ in $N$.

In $N$, DC makes $\cc_0$ countably closed.  Its generic union
enumerates $X$; SVC then wellorders every old set and every set of
names.  Hence $N[K_0]\models\ZFC$.  The generated measure in
$E_0[K_0]$ restricts to the normal measure defined by the same
ground-witness formula in $N[K_0]$.  Product genericity makes $c$
Prikry generic over this choice model, so $W\models\ZFC$.
These are exactly the hypotheses of Section~\ref{sec:ambient}.
In particular Lemma~\ref{lem:ambientfacts} gives
\[
 {}^\omega X\cap W={}^\omega X\cap N,
 \qquad V_\kappa^{N[c]}=V_\kappa^N,
\]
and gives an $N$-wellorderable cover for every $W$-countable family
of $N$-elements.  The countable closure is used before adjoining
$c$; closure of the old collapse after adjoining $c$ is not assumed.

The old initial-segment collapse projects onto the labeling forcing
of Section~\ref{sec:caps}.  Let $g$ be its projected generic.
The finite interpolation, paired lifting and literal comparison
lemmas give $Q=Q(c,g)$ and $F=F(c,g):Q\twoheadrightarrow X$.
Put $M=N(Q,F)$.  Lemma~\ref{lem:eliminate} and
Corollary~\ref{cor:rankpres} give $V_\lambda^M=X$;
Lemmas~\ref{lem:CD} and~\ref{lem:DCM} give DC in $M$.
These results use the old countable arrays and the facts just
verified, rather than a measure belonging to $N$.

For clarity, the SVC and seed conclusions can also be read directly
from the proof of Proposition~\ref{prop:seed}.  Bounded presentations
and the surjection onto $X$ give surjections
$\xi\times Q^{<\omega}\twoheadrightarrow Y$ for every nonempty
$Y\in M$, so $Q^{<\omega}$ witnesses SVC.  Code $Q$ and the relation
\[
 R(p,q)\quad\Longleftrightarrow\quad F(p)\in F(q)
\]
by differently tagged ordinal $\omega$-sequences.  Equality of
predecessor sets is equality of $F$-values.  The quotient relation's
unique Mostowski collapse recovers the literal $X$ and $F$.
Thus the resulting $A\subseteq\mu^\omega$ satisfies $M=D(A)$.
No measurable-tail conclusion about $M$ is needed here.

Apply Lemma~\ref{lem:envelope} to $M$, taking its fresh generic
$K$ over the full choice cover $E$ as well.  Then $B=M[K]$ is a
weakly homogeneous choice extension with no new ordinal
$\omega$-sequences.  Moreover
\[
 D\subseteq B\subseteq E[K].
\]
The outer model $E[K]$ is a set-forcing extension of $D$: the
posets defined in intermediate models have ground names.  Since
$D$ and $B$ satisfy ZFC, the ordinary intermediate-model theorem
gives a set forcing $P\in D$ and a $D$-generic $b$ with $B=D[b]$;
see \cite{Grigorieff}.  Small forcing preserves Woodinness
\cite{HamkinsWoodin}.  Every $D$-Woodin above $|P|^D$ therefore
survives in $B$, proving $B\models\PCW$.
\end{proof}

\subsection{One bounded homogeneous ordinal code}

\begin{lemma}[Bounded homogeneous coding]\label{wd-bounded-code}
Let $W\models\ZFC$ have a proper class of inaccessible cardinals,
let $d\subseteq\delta$ be a set of ordinals, and let $\beta$ be
an ordinal.  There is a weakly homogeneous set forcing, uniformly
defined from $d,\delta$ and an ordinal bound $b$, which preserves
$V_\beta^W$, adds no ordinal $\omega$-sequences, and makes $d$
ordinal definable.  No GCH hypothesis is required.
\end{lemma}

\begin{proof}
If $\delta=0$, use trivial forcing.  Assume $\delta>0$.
Choose $b>\beta,\delta,\omega$ with $|V_\beta^W|<b$.  Let
$\seq{\kappa_i:i<\delta}$ enumerate the first $\delta$
inaccessible cardinals strictly above $b$, in their ordinal order.
In particular $\delta<\kappa_0$.  All calculations until an
extension is specified take place in $W$.

At an inaccessible $\kappa=\kappa_i$, put
\[
 \theta=2^\kappa,\qquad \ell=\theta^+,
 \qquad C=\Col(\kappa^+,\theta),
 \qquad A=\Add(\kappa,\ell).
\]
Here $C$ consists of partial functions from $\kappa^+$ to
$\theta$ of size $<\kappa^+$, and $A$ consists of partial
binary functions on $\kappa\times\ell$ of size $<\kappa$.
Both are ordered by reverse inclusion.  The factor at $i$ is
\[
 R_i=\begin{cases}
 C,&i\notin d,\\
 C\times A,&i\in d.
 \end{cases}
\]
The Cohen factor is the ground poset in this displayed product.

Since $\theta^\kappa=(2^\kappa)^\kappa=\theta$, we have
$|C|=\theta$.  The collapse is $<\kappa^+$-closed, preserves
$\kappa^+$ and $\theta^+$, and makes $|\theta|=\kappa^+$.
It adds no subsets of $\kappa$.  Consequently, in $W[H_C]$,
\[
 2^\kappa=\kappa^+,\qquad \ell=\kappa^{++}.
\]
Moreover, $C$ adds no ordinal sequences of length $<\kappa$,
so the old $A$ is exactly the recomputed
$\Add(\kappa,\ell)^{W[H_C]}$.
In this intermediate model $\kappa^{<\kappa}=\kappa$.
Regularity of $\ell=\kappa^{++}$ and
$(\kappa^+)^\kappa=(2^\kappa)^\kappa=\kappa^+$ give
$\ell^\kappa=\ell$, by bounding the range of each function
$\kappa\to\ell$.  The Cohen chain condition and the count of
nice names therefore give $2^\kappa=\ell$ after $A$, while
preserving cardinals.  The two choices of $R_i$ thus force,
respectively, $2^\kappa=\kappa^+$ and
$2^\kappa=\kappa^{++}$.

For later size calculations, already in $W$ we have
$\ell^{<\kappa}=\ell$.  Indeed each short function into the
regular $\theta^+$ has bounded range and
$\theta^\kappa=\theta$.  Hence $|R_i|\leq\theta^+$.
Each $R_i$ is $<\kappa_i$-closed and weakly homogeneous.
Permutations of the collapse domain, and of the Cohen coordinates,
can move either of two condition domains off the other.

Use the full-support set product
\[
 R(d,\delta,b)=\prod_{i<\delta}R_i.
\]
It is $<\kappa_0$-closed and weakly homogeneous, using componentwise
unions and automorphisms.  Since $|V_\beta^W|<\kappa_0$,
induction on rank shows that it preserves $V_\beta^W$.
We must also verify that the inaccessible coding sites remain
recognizable and that different coordinates do not erase their bits.

Let $\nu\geq\kappa_0$ be any inaccessible of $W$.  There are
fewer than $\nu$ coordinates below $\nu$, and their number is
bounded by $|\delta|<\kappa_0$.  Their sites and factor sizes
are bounded below $\nu$ by regularity and the strong-limit
property.  For a common size bound $\zeta<\nu$, their product
has size at most $\zeta^{|\delta|}<\nu$.  The remaining old
product is $<\nu$-closed.  Easton's lemma
\cite[Fact~5.10]{CummingsForcing} makes that fixed tail
$<\nu$-distributive over the small lower product.  Small forcing
preserves the inaccessibility of $\nu$, and the tail preserves
its regularity and all smaller power sets.  Thus $\nu$ remains
inaccessible.  We use distributivity of the old tail here;
closure of that tail in the intermediate extension is unnecessary.

Forcing cannot create a strongly inaccessible ordinal.  An old
noncardinality or singularity witness persists.  If an old regular
cardinal $\nu$ was not strong limit, an old injection
$\nu\to\Pow(\zeta)$ for some $\zeta<\nu$ persists as well.
Therefore the first $\delta$ inaccessibles above $b$ are the
same ordinals in the extension.

Fix $i$.  Force first with $R_i$.  The product of lower coordinates
has size $<\kappa_i$, and its old enumeration remains an enumeration
after $R_i$.  It preserves the relevant successor cardinals and
the calculated value of $2^{\kappa_i}$: names for subsets of
$\kappa_i$ are coded by subsets of the product of its condition
set with $\kappa_i$, so there are at most the current
$2^{\kappa_i}$ such names.  The old subsets give the opposite
inequality.  These factors commute.  If a tail remains, put
$\nu=\kappa_{i+1}$.  The initial product through $i$ has size
$<\nu$ in $W$, whereas the old remaining product is
$<\nu$-closed.  Easton's lemma makes the tail
$<\nu$-distributive over that initial product.  It changes
neither $\Pow(\kappa_i)$ nor the successor cardinals used in
the bit test, all of which are below $\nu$.

Consequently the final extension satisfies
\begin{equation}\label{eq:wd-bit-code}
 i\in d\quad\Longleftrightarrow\quad
 2^{\kappa_i}=\kappa_i^{++}\qquad(i<\delta),
\end{equation}
with $2^{\kappa_i}=\kappa_i^+$ at the other sites.  Enumerating
the inaccessibles above $b$ and applying this test defines $d$
from the ordinals $b,\delta$.

Finally, the entire forcing recipe uses only $d,\delta,b$.
It chooses no auxiliary wellorder or enumeration of names.
If $W$ is an earlier generic extension and $d$ is an old parameter,
the name given by this recipe is invariant, up to forced equality,
under every earlier automorphism fixing that parameter and the
ordinals.  This is the form of uniformity needed for homogeneous
composition below.
\end{proof}

\subsection{Bounding all capped histories}

For a fixed schedule as in Theorem~\ref{thm:spectrum}, write
$\operatorname{Hist}_B(e,h,k)$ for the assertion that $h$ is
$Q_e$-generic over $B$ and $k$ is generic over $B[h]$ for the
recomputed cap $\Add(\rho,1)^{B[h]}$.  All generic filters in
this predicate are actual sets in the universe in which it is
evaluated.  The next lemma bounds the ground data needed to test
the predicate, including the dense sets for the recomputed cap.

\begin{lemma}[Bounded history transfer]\label{wd-bounded-history}
Fix $B\models\ZFC$, the universal schedule, the candidate
forcings $Q_e$ for $e\in(2^\tau)^B$, and the cap $\rho$
from Theorem~\ref{thm:spectrum}.  There is an ordinal $\chi$
such that every transitive inner model $C\subseteq B$ of ZFC
with $V_\chi^C=V_\chi^B$ computes the same schedule and candidate
forcings and satisfies
\[
 \operatorname{Hist}_C(e,h,k)
 \quad\Longleftrightarrow\quad
 \operatorname{Hist}_B(e,h,k)
\]
for all candidates and all possible outer filters $h,k$.
The bound can be increased arbitrarily.
\end{lemma}

\begin{proof}
For each $e\in(2^\tau)^B$, put $\bb_e=\RO^B(Q_e)$.
Let $T_e$ be the full bounded system of total Boolean names for
cap conditions used after \eqref{eq:PF}.  For every $B$-generic
$h\subseteq Q_e$,
\begin{equation}\label{eq:wd-cap-names}
 \{t^h:t\in T_e\}=\Add(\rho,1)^{B[h]}.
\end{equation}
Indeed, localize any name for a cap condition, mix with the empty
cap condition off that Boolean cone, and normalize to the bounded
Boolean-name codes.  This normalization takes place in the choice
model $B$ and yields a set $T_e$ independent of $h$.

Let $\mathcal N_e$ consist of all Boolean names for subsets of
the ground set $T_e$.  A name has one Boolean value in $\bb_e$
for each possible member of $T_e$, so the codes are contained in
$\Pow^B(T_e\times\bb_e)$, together with the fixed check-name
coding.  Choose $\chi$ so large that its rank cut contains, for
every candidate $e$, the whole sets
\[
 Q_e,\quad\Pow^B(Q_e),\quad\bb_e,\quad T_e,
 \quad\Pow^B(T_e\times\bb_e),\quad\mathcal N_e,
\]
their elements, and all the lower universal products and their
power sets used to form the schedule and Boolean completions.
Include the wellordering and cardinal-bound witnesses from the
schedule recursion, and take $\chi>\rho+\omega$.
There is one such bound because the candidate family and schedule
have set size.  Agreement above $\rho+\omega$ includes all
bounded witnesses to cardinality, cofinality and the strong-limit
tests below $\rho$; thus no earlier inaccessible is introduced
in the schedule computed in $C$.  The complete Boolean algebras
and the indicated name systems are computed identically.

Every $B$-dense subset of $Q_e$ belongs to $C$, and every
$C$-dense subset is a $B$-dense subset of the same poset.
Hence $C$-genericity and $B$-genericity of $h$ are equivalent.
For such an $h$ we have $C[h]\subseteq B[h]$.  Absolute evaluation
of the names in $T_e$, together with \eqref{eq:wd-cap-names},
gives equality of the two recomputed cap posets.  In the convention
$\Add(\rho,1)=2^{<\rho}$, membership is absolute for a sequence
already present in the smaller model.

In fact the full power sets of these cap posets agree.  Given
$Z\subseteq\Add(\rho,1)^{B[h]}$ in $B[h]$, form
\[
 Y=\{t\in T_e:t^h\in Z\}.
\]
Some member of $\mathcal N_e$ evaluates to $Y$.  That name belongs
to $C$, so $Y\in C[h]$, and evaluating its members reconstructs
$Z$ inside $C[h]$.  The opposite inclusion follows from
$C[h]\subseteq B[h]$.  Therefore the two models have exactly
the same dense subsets of the common cap poset.  Genericity of
any outer filter $k$ is equivalent over them, proving the lemma.
\end{proof}

\begin{lemma}[A sufficient ground fragment]\label{wd-ground-fragment}
Suppose $D\subseteq M=D(A)\subseteq B=D[b]$, where $D,B$
satisfy ZFC and $b$ is generic for a set forcing $P\in D$.
Fix ordinals $\alpha,\chi,\eta$, with $P,b\in V_\eta^B$.
There is $\theta$ such that
\begin{enumerate}
\item every element of $V_\chi^B$ is the value of a $P$-name
in $V_\theta^D$;
\item $V_\alpha^M\subseteq L(V_\theta^D,A)$.
\end{enumerate}
Here and below $L$ with set arguments denotes transitive-base
constructibility, including all members of its arguments.
\end{lemma}

\begin{proof}
The bounded-rank name lemma supplies the first bound.  One can
obtain it by recursion in $D$, using at successor stages all
Boolean names whose possible members are in the preceding
name set, and taking unions at limits.  Their evaluations cover
the corresponding ranks in every generic extension.  Equivalently,
in $B$ choose a ground name for each element of the set
$V_\chi^B$ and bound their ranks.  The ground $D$ is definable
in its set extension from a set parameter; the chosen collection
need not itself belong to $D$ for its ordinal rank bound to do so.

For the second bound, use the convention and bounded-witness
argument of Lemma~\ref{lem:presentation}.  Each
$y\in V_\alpha^M$ belongs to a set-sized construction level
over $\TC(\{V_\xi^D,A\})$ for some $\xi$.  Collection in
$B$ bounds the ground ranks $\xi$ needed for the set
$V_\alpha^M$.  Increase $\theta$ past this bound and the name
bound.  The inner model $L(V_\theta^D,A)$ contains every one
of the indicated levels: recursion through the definable-subset
operation on set structures is absolute.  This proves the second
inclusion without claiming that an arbitrary directed union of
inner models satisfies ZF.
\end{proof}

\subsection{Realizing the protected rank}

\begin{theorem}[Woodin rank realization]\label{wd-set-realization}
Let $D\models\ZFC+\PCW$, let $N$ be a set-symmetric extension
of $D$ satisfying $\ZF+\DC$, and let $\alpha$ be an ordinal.
In a common outer universe there is an ordinary set-forcing
extension $U$ of $D$ such that
\[
 U\models\ZFC+\PCW,
 \qquad V_\alpha^{\Hseq^U}=V_\alpha^N.
\]
In particular, a rank sentence true in such a symmetric extension
is true in the corresponding rank of external
$\HOD_{\Ord^\omega}$ in a model of $\ZFC+\PCW$.
\end{theorem}

\begin{proof}
Apply Lemma~\ref{wd-old-ground-transfer}.  We obtain
\[
 D\subseteq N\subseteq M=D(A)\subseteq B=M[K]=D[b],
 \qquad V_\alpha^M=V_\alpha^N,
\]
with $A\subseteq\mu^\omega$ and the stated homogeneous
sequence-preserving envelope.  Fix its forcing presentation
$P\in D$ and choose $\eta$ with $A,K,P,b\in V_\eta^B$.

Choose an infinite $\tau$ large enough to code
$(\TC(\{A\}),\in,A)$, and let $F_A\subseteq2^\tau$ be
the family of all its pointed well-founded extensional
presentations, with domain, root and unused-position data as in
Corollary~\ref{cor:ODcoding}.  This family is defined from
$A,\tau$; no presentation is selected.  Apply
Theorem~\ref{thm:spectrum} over $B$, placing the first site
above a regular bound greater than $|V_\eta^B|$.  With its
universal schedule and cap $\rho$, use the actual forcing
\[
 S=P_{F_A}=
 \prod_{e\in F_A}^{\mathrm{full}}
       \bigl(Q_e*\dot{\Add}(\rho,1)^{B[\dot h_e]}\bigr),
 \qquad U_0=B[G_S].
\]
The terminal forcing is recomputed in its own row extension.
The extension retains the evaluated row and cap generics; the
larger raw termspace generics are not adjoined.  Theorem~\ref{thm:spectrum}
gives the spectrum $F_A$, weak homogeneity and preservation of
$V_\eta^B$.  Since $U_0$ is a set extension of $D$, it still
has a proper class of Woodins, and hence of inaccessibles.

Next choose $\chi>\eta+\omega$ satisfying
Lemma~\ref{wd-bounded-history}, and choose $\theta$ by
Lemma~\ref{wd-ground-fragment}.  In $D$, code the transitive set
$V_\theta^D$ by a set of ordinals $d\subseteq\delta$.
Include domain and root information in its well-founded
extensional presentation.  The code $d$ belongs to $D$, hence
to $M$, regardless of how these ordinal bounds were selected.

In $U_0$ choose a rank bound $\zeta$ above $\eta$ and all
possible capped-history witnesses for this schedule.  Such a
bound depends only on the set family of lower posets and on
$\rho$: a lower filter is a subset of some $Q_e$, and a cap
filter is a subset of $2^{<\rho}$.  It bounds the ranks of all
potential witnesses, not just of the generics originally planted
by $S$.  Enlarge it to include the candidate codes.
Apply Lemma~\ref{wd-bounded-code} over $U_0$ to $d$, preserving
$V_\zeta^{U_0}$, and call the resulting extension $U$.

We now define the decoder inside $U$.  The set $d$ is OD, and
$V_\eta^U=V_\eta^B$.  Thus the class
\begin{equation}\label{eq:wd-envelope-cut}
 C=L(d,V_\eta^U)
\end{equation}
is definable from ordinal parameters in $U$.  Its bases are
sets of $B$, so the absolute transitive-base construction gives
$C\subseteq B$.  Decoding $d$ puts $V_\theta^D$ in $C$;
the second base contains $P,b$.  Evaluation of the names in
Lemma~\ref{wd-ground-fragment} therefore gives
\[
 V_\chi^C=V_\chi^B.
\]
A $B$-wellordering of $V_\eta^B$ has rank below $\chi$ after
the stipulated increase of $\chi$, and hence belongs to $C$.
Together with the ordinal-set code $d$, it wellorders the
transitive base of \eqref{eq:wd-envelope-cut}.  Consequently
$C\models\ZFC$.

Lemma~\ref{wd-bounded-history} identifies the history predicate
relative to $C$ with the predicate relative to $B$.  The last
forcing preserved every possible history witness and changed
neither $C$ nor its ground-name and dense-set families.  Thus
in $U$ the spectrum of histories relative to the definable class
$C$ is exactly $F_A$.  Taking the pointed Mostowski collapse
of any member recovers $A$ and makes $A$ OD in $U$.
We have transferred a bounded predicate; we have not reapplied
Theorem~\ref{thm:spectrum} to $C$, which need not have a proper
class of inaccessible cardinals.

Put $H=\Hseq^U$.  The forcing from $M$ to $U$ is weakly
homogeneous.  The spectrum name is uniformly defined from the
old $A$ and ordinals, and the final code name is uniformly
defined from the old $d$ and ordinals.  Earlier automorphisms
therefore send each name to a forced equal name and lift to
the iteration.  The local Boolean-mixing argument in the proof
of Proposition~\ref{prop:secure-realize} combines these
automorphisms: a locally forced automorphism is given the
identity value off its Boolean cone.  This uses no global
maximum principle over the choiceless $M$ and introduces no
wellorder from the choice envelope as a parameter.

The first stage adds no ordinal $\omega$-sequences, and both
later stages are highly distributive.  Lemma~\ref{lem:homdesc}
gives $H\subseteq M$.  Conversely, the ordinal code
$d\in\HOD^U$ and its unique Mostowski collapse put
$V_\theta^D$, with all its hereditary elements, in $H$.
Also $A\in H$: it is OD and each of its members is an allowed
ordinal $\omega$-sequence.  Since $H\models\ZF$ by
Lemma~\ref{lem:HDC}, it contains $L(V_\theta^D,A)$.  Hence
\[
 V_\alpha^M\subseteq L(V_\theta^D,A)
       \subseteq H\subseteq M,
\]
which proves the required rank equality.

The definability of $C$ places none of the extra choice
wellorderings of $B$ in $H$.  In particular $V_\eta^U$ is
used as an OD set defining a class, not as a hereditarily OD
parameter.  This is the distinction of Remark~\ref{rem:ODnotHOD}.
The object coded hereditarily is the ground fragment
$V_\theta^D$, not the envelope rank $V_\chi^B$.

Finally, $U/D$ is a composition of set forcings.  Every sufficiently
high ground Woodin survives by small-forcing preservation
\cite{HamkinsWoodin}.  Thus $U\models\ZFC+\PCW$.
The bounds were chosen in the order
\[
 \eta,\quad S\text{ and }\rho,\quad\chi,\quad
 \theta\text{ and }d,\quad\zeta,\quad
 \text{the last coding block}.
\]
There is no demand that $S$ preserve $V_\theta^B$: the ground
fragment is recovered from $d$.  All preparation and coding in
this proof are set forcing, and no conclusion that $N$ or $M$
has Woodin cardinals is used.
\end{proof}

\section{GCH realization as a symmetric ground}\label{wd-exact-section}

We now strengthen the consistency conclusion by using the special ground
produced in Proposition~\ref{wd-reverse}. The additional hypotheses on
that ground allow a bounded GCH-preserving code to identify all its sets
inside the final HOD. We then recover the sequence-generated model
exactly and exhibit the ambient universe as its set-forcing extension.

\subsection{The normalized choice ground}

The reverse construction supplies more structure than is needed for
Theorem~\ref{wd-set-realization}.  We use that structure to obtain
global GCH and to identify the entire external HOD.

\begin{lemma}[Properties of the normalized ground]
\label{wd-normalized-ground}
Let $D\subseteq N$ be the models constructed in
Proposition~\ref{wd-reverse}, and let $C$ be its canonical choice
envelope, so that
\[
 D=\HOD^C,\qquad D\subseteq N\subseteq C=D[h].
\]
Then $D$ satisfies eventual GCH, and there is a set of ordinals
$t\in D$ such that
\[
                         D=\HOD_{\{t\}}^D.
\]
Every set-forcing extension of $D$ satisfies eventual GCH.
\end{lemma}

\begin{proof}
The model $C$ supplied by Theorem~\ref{wd-envelope} is an acceptable
relative pure-extender premouse over a bounded self-wellordered base.
In either branch of that construction, the usual acceptability
calculation applies above the size and height of the base.  Subsets
of an infinite cardinal $\kappa$ occur below $J_{\kappa^+}$, and
the standard surjections from the base times $\kappa$ bound the
earlier levels by $\kappa$.  Thus $C$ satisfies GCH above a fixed
bound; compare \cite[Section~2.2]{SteelOutline}.

Fix a forcing presentation $C=D[h]$ by $T\in D$.  Above $|T|^D$,
the relevant cardinals are preserved.  If GCH failed in $D$ at an
arbitrarily large cardinal $\kappa$, its old subsets of $\kappa$
would still contradict GCH in $C$.  Hence $D$ satisfies eventual
GCH.  Conversely, for any fixed set forcing $P\in D$ and every
sufficiently large cardinal $\kappa$, names for subsets of $\kappa$
can be coded by subsets of $\kappa\times P$.  When GCH holds in
$D$ there, this bounds their number by $(\kappa^+)^D$, which the
forcing preserves.  The old subsets give the opposite inequality.
This proves eventual GCH in every set extension of $D$.

Replace $T$ by an ordinal-coded presentation in $D$, and let $t$
be one ordinal-set code for its domain and order.  If $x\in D$,
then $x$ is ordinal definable in $C$.  There are a formula
$\varphi$, a finite tuple of ordinals $\vec\xi$, and $p\in h$
such that
\[
 p\forces_T \text{``$\check x$ is the unique $y$ satisfying
                         $\varphi(y,\check{\vec\xi})$''}.
\]
Inside $D$, the set $x$ is therefore the unique $y$ for which
$p\forces_T\varphi(\check y,\check{\vec\xi})$.  The forcing
relation is definable in $D$ from $T$, and $p$ is represented by
an ordinal.  Thus $x$ is definable over $D$ from $t$ and ordinals.
The same argument applies to every member of its transitive
closure, proving $D=\HOD_{\{t\}}^D$.
\end{proof}

\subsection{Bounded coding while preserving GCH}
\label{wd3-bounded-coding-section}

The continuum code of Lemma~\ref{wd-bounded-code} can be replaced,
over a GCH model, by a code in the $\diamondsuit^*$ pattern.  We
need a bounded set product whose definition uses only the set being
coded and ordinal parameters.  We also need its actual closure,
rather than only distributivity, for the ground-recognition argument.

\begin{lemma}[Bounded GCH-preserving coding]\label{wd-gch-bounded-code}
Let $W\models\ZFC+\GCH$ have a proper class of inaccessible
cardinals.  Suppose $d\sq\delta$, and let $\beta,\chi$ be ordinals.
For every sufficiently large ordinal $b$ there is a set forcing
$R(d,\delta,b)$, uniformly defined from these three parameters,
which is locally homogeneous and $<\chi$-closed.  It preserves
all cardinals, GCH, the class of inaccessible cardinals, and
$V_\beta^W$, adds no ordinal $\omega$-sequences, and makes $d$
ordinal definable from $b,\delta$.
\end{lemma}

\begin{proof}
If $\delta=0$, use trivial forcing.  Otherwise take
\[
 b>\delta,\beta,\chi,\omega,|V_\beta^W|,
\]
and let $\seq{\kappa_i:i<\delta}$ enumerate the first $\delta$
inaccessible cardinals above $b$.  Put
\[
 \lambda_i=\kappa_i^{+++}.
\]
Until an extension is specified, all posets and cardinal calculations
in this proof are computed in $W$.  In particular $\delta<\kappa_0$.

We first specify the two possible factors at an infinite successor
cardinal $\lambda=\theta^+$.  For the positive factor $P_\lambda$,
use pairs $(X,\xi)$ with
\[
 \varnothing\ne X\sq{}^\lambda2,
 \qquad |X|\leq\theta,
 \qquad \xi<\lambda,
\]
and a separate largest condition.  Declare $(Y,\zeta)\leq(X,\xi)$
if $Y\supseteq X$, $\zeta\geq\xi$, and
\[
 \forall f\in Y\ \exists g\in X\quad
 f\restr\xi=g\restr\xi.
\]
The nonempty-branch convention is part of this presentation.
An empty branch set would give a degenerate component under the
displayed order and is excluded.

This is the positive coding forcing of
\cite[Section~2]{BrookeTaylorCoding}.  It forces
$\diamondsuit^+_\lambda$, and hence $\diamondsuit^*_\lambda$.
The relevant feature of its proof is that the generic tree captures
every old binary $\lambda$-sequence up to a bounded modification.
Indeed, below any nonempty condition, alter the initial part of a
given sequence to match a represented node and then add that branch.
Nice names for new binary $\lambda$-sequences use only $\lambda$
many conditions and branches, so they have old binary
$\lambda$-sequence codes.  The tree recovers the generic filter.
Together with a code for the old $\Pow(\theta)$, these facts give
the hypothesis of the positive coding theorem in
\cite[Section~2]{BrookeTaylorCoding}.  That argument uses only
the nonempty component specified here.

The negative factor is $Q_\lambda=\Add(\lambda,\lambda^+)$,
which forces $\neg\diamondsuit^*_\lambda$; see
\cite[Appendix, Lemma~18]{CFH}.  Both factors are
$<\lambda$-closed, have size $\lambda^+$, and satisfy the
$\lambda^+$-chain condition under GCH.  For $P_\lambda$, unions
of increasing branch sets and suprema of heights give the closure.
At each fixed height $\xi<\lambda$ there are at most $\lambda$
possible sets of at most $\theta$ binary $\xi$-prefixes.
Conditions with the same prefix set are compatible by taking the
union of their branch sets.  This proves its chain condition.
The size bound follows from
$(2^\lambda)^\theta=2^\lambda=\lambda^+$.
The usual Cohen calculation gives the same bounds for $Q_\lambda$.
For either factor, closure, the chain condition, and nice-name
counting preserve all cardinals and GCH.

Define
\[
 R_i=
 \begin{cases}
 P_{\lambda_i},&i\in d,\\
 Q_{\lambda_i},&i\notin d,
 \end{cases}
 \qquad
 R(d,\delta,b)=\prod_{i<\delta}R_i,
\]
using full support and the old factors.  The product is
$<\lambda_0$-closed by coordinatewise lower bounds in $W$.
Since $\lambda_0>b>\chi,|V_\beta^W|$, it is $<\chi$-closed,
adds no ordinal $\omega$-sequences, and preserves $V_\beta^W$.
For the rank assertion, a first new set below that rank would be
a new subset of an old set of size $<\lambda_0$, contrary to
closure and Choice in $W$.

\smallskip\noindent
\emph{Local homogeneity.}
Given two nontrivial conditions in $P_\lambda$, extend both
distinguished branch sets to have size $\theta$.  Choose a common
height $\gamma<\lambda$ above their heights which separates
every pair of distinct branches in either set.  Match their
$\theta$ many nodes at height $\gamma$ bijectively, together
with the unique distinguished branch through each node.
Within each matched pair of binary cones, replace the prefix
and translate the tail by coordinatewise addition modulo two
so that the distinguished branches correspond.  The resulting
map preserves equality of initial segments at every height
at least $\gamma$.  Applied to branch sets, it is an isomorphism
between the two forcing cones below the strengthened conditions.
Thus $P_\lambda$ is locally homogeneous.  The Cohen factor
$Q_\lambda$ is weakly homogeneous.  Choice in $W$ combines
the component cone isomorphisms into a cone isomorphism for the
full product.  This proves the required local homogeneity.

\smallskip\noindent
\emph{Factorization estimates.}
For each $i<\delta$, the product below $i$ has size
$<\kappa_i$.  Its number of coordinates is less than
$\kappa_i$, and regularity bounds their sizes below
$\kappa_i$.  The strong-limit property then bounds the full
product below $\kappa_i$.  Consequently the product through
$i$ has size at most $\lambda_i^+$ and is
$\lambda_i^+$-cc.  To verify the latter claim, an antichain
of size $\lambda_i^+$ could be thinned to have the same
lower-product coordinate, contradicting the chain condition
of $R_i$.

If a tail remains after $i$, it is
$<\lambda_{i+1}$-closed in $W$.  The prefix has size
$<\lambda_{i+1}$.  Easton's lemma
\cite[Fact~5.10]{CummingsForcing} makes this fixed old tail
$<\lambda_{i+1}$-distributive over the prefix extension.
Thus it adds no subsets to the cardinals at or immediately
above $\lambda_i$.  This use of distributivity does not require
the old tail to remain closed in the intermediate model.

\smallskip\noindent
\emph{Cardinals and GCH.}
At an active site $\lambda_i$, the preceding size and chain
condition bounds give at most
\[
 (\lambda_i^+)^{\lambda_i}=\lambda_i^+
\]
nice prefix names for subsets of $\lambda_i$.  At its successor
and at larger cardinals above that prefix's size, a subset name
is coded by a subset of the product of the domain cardinal
with the forcing.  Ground GCH gives the corresponding successor
bound.  Upper-tail distributivity transfers these calculations
to the full extension.  At cardinals below the active site,
one may instead force with $R_i$ first and then with the small
lower product.  These factors commute, the first preserves
GCH, and the second preserves the continuum function above its
size by the same name count.  This also handles the gaps
between a completed prefix and the next active site.

We give the limit calculation separately.  Suppose
\[
 \mu=\sup_{i<j}\lambda_i,
 \qquad 0<j\leq\delta,
 \qquad j\text{ is a limit ordinal}.
\]
The underlying inaccessibles are cofinal in $\mu$, so $\mu$
is strong limit and
\[
 \varepsilon=\cf^W(\mu)\leq |j|\leq|\delta|<\kappa_0.
\]
Ground GCH gives
\begin{equation}\label{wd3-limit-prefix-bound}
 \left|\prod_{i<j}R_i\right|
 \leq\mu^{|j|}=\mu^+.
\end{equation}
If $j<\delta$, then $\kappa_j>\mu^+$, and the remaining
tail is distributive beyond $\mu^+$ after this prefix.

The earlier factorization calculations preserve all cardinals
below $\mu$, so $\mu$ remains a cardinal.  Its old short
cofinal sequence shows that it remains singular.  If the old
$\mu^+$ ceased to be regular, its new cofinality would be
a regular cardinal $\nu<\mu$.  Choose $i<j$ sufficiently
large that $\nu<\lambda_{i+1}$ and the prefix through $i$
still has size $<\mu$.  This prefix preserves the regularity
of the old $\mu^+$, while its remaining old tail is
$<\lambda_{i+1}$-distributive by Easton's lemma.  The tail
cannot add the purported cofinal $\nu$-sequence.  Hence
$\mu^+$ is preserved.  The prefix size bound
\eqref{wd3-limit-prefix-bound} preserves the higher cardinals
and gives GCH at $\mu^+$ and above, up to the next active site.

To count subsets of $\mu$, fix in $W$ a cofinal sequence
$\seq{\lambda_{i_\xi}:\xi<\varepsilon}$ of active cardinals
below $\mu$.  For each $\xi$, the isolated-site nice-name
calculation supplies a fixed ground set $T_\xi$ of names of
size at most $\lambda_{i_\xi}^+$.  Every subset of
$\lambda_{i_\xi}$ in the full extension is the value of a
name in $T_\xi$, because the upper tail adds no such subsets.
Wellorder these sets of names in $W$.  For each $x\sq\mu$
in the extension, select the least index of a name evaluating
to $x\cap\lambda_{i_\xi}$, for each $\xi<\varepsilon$.
This is an $\varepsilon$-sequence of ordinals below $\mu$.

The whole product is $<\lambda_0$-closed and
$\varepsilon<\kappa_0<\lambda_0$, so this entire sequence
of indices already belongs to $W$.  There are at most
\[
 (\mu^\varepsilon)^W=\mu^+
\]
such ground sequences.  Evaluating the named restrictions
and taking their union gives all subsets of $\mu$ in the
extension.  Since $\mu^+$ has been preserved, this proves
$2^\mu=\mu^+$.  In particular no $\mu^+$-chain-condition
claim about the limit product is needed.

An induction through the active sites and their limits now
proves cardinal preservation and GCH at every cardinal.
Away from a limit of active sites, the sites below a given
cardinal are bounded and the prefix-size calculation applies.
Above the size of the entire set product it applies directly.

\smallskip\noindent
\emph{Preservation and recognition of the code.}
Let $\nu\geq\kappa_0$ be any inaccessible of $W$.
Fewer than $\nu$ coordinates lie below $\nu$, their sizes
are bounded below $\nu$, and their product has size $<\nu$.
The remaining old tail is $<\nu$-closed.  Small forcing
preserves the inaccessibility of $\nu$, and Easton's lemma
makes the tail $<\nu$-distributive over that small extension.
It preserves regularity and all smaller power sets.  Inaccessibles
below $\kappa_0$ are preserved by closure.  Conversely, old
cofinality witnesses, noncardinality witnesses, and injections
witnessing a failure of strong limit persist.  Thus the class
of inaccessible cardinals is unchanged.

Fix $i<\delta$ and first force with $R_i$.  The old lower
product has size $<\kappa_i<\lambda_i$ and therefore
preserves the truth value of $\diamondsuit^*_{\lambda_i}$
by \cite[Appendix, Lemma~19]{CFH}.  The upper tail adds no
subsets of $\lambda_i$.  Under GCH below $\lambda_i$,
each candidate $\diamondsuit^*$ sequence and its club witnesses
can be coded by subsets of $\lambda_i$.  Hence the upper
tail preserves that truth value as well.  We have proved
\begin{equation}\label{wd3-diamond-code}
 i\in d\quad\Longleftrightarrow\quad
 \diamondsuit^*_{\lambda_i}\qquad(i<\delta)
\end{equation}
in the final extension.  Its first $\delta$ inaccessibles
above $b$, and their triple successors, are the same ordinals
as in $W$.  Equation~\eqref{wd3-diamond-code} defines $d$
from $b,\delta$.

Finally, the formula for the forcing chooses no enumeration
or wellorder of a ground power set.  The choices of matches
and wellorders in the proof establish homogeneity and counting;
they are not parameters in the forcing definition.  If $W$
is an earlier generic extension and $d$ is old, this uniform
definition gives an invariant name under any earlier cone
isomorphism fixing the old parameters and ordinals.
\end{proof}

\subsection{Local homogeneity and hereditary descent}
\label{wd3-local-homogeneity-section}

For composition, we understand local homogeneity on separative
quotients, equivalently on regular-open cones.  Thus two arbitrary
conditions have stronger conditions whose forcing cones are
isomorphic.  Transporting names gives isomorphisms in this sense;
it need not biject redundant syntactic iteration names.

\begin{lemma}[Local homogeneous descent]\label{wd-local-descent}
Let $M\models\ZF$ and suppose $U=M[G]\models\ZFC$ is a
locally homogeneous set-forcing extension with
\[
 (\Ord^\omega)^U=(\Ord^\omega)^M.
\]
Then $\Hseq^U\sq M$.

Moreover, local homogeneity is preserved by finite iterations
whose next forcing is forced locally homogeneous and is uniformly
defined from fixed old parameters and ordinals.  For this assertion
the defining names are taken invariant, up to forced equality,
under the preceding cone isomorphisms.  The composition argument
requires no maximum principle in $M$.
\end{lemma}

\begin{proof}
An isomorphism of forcing cones transports check names to check
names.  If two conditions forced opposite truth values for a
sentence with only ground parameters, local homogeneity would
give stronger isomorphic cones with the same opposite decisions,
a contradiction.  Thus the largest condition decides every
such sentence.  This is the local-homogeneity argument of
\cite[Appendix, Lemma~16]{CFH}.

The rank induction of Lemma~\ref{lem:homdesc} now applies.
Suppose $x\in\Hseq^U$ and every member of $x$ belongs to $M$.
Choose $\xi$ with $x\sq V_\xi^M$.  A definition of $x$ uses
an ordinal $\omega$-sequence already in $M$ and ordinal
parameters.  The top condition forces the uniqueness assertion
and uniformly decides membership of each ground
$y\in V_\xi^M$ in its unique value.  Separation using the
forcing relation in $M$ recovers $x$.  This proves
$\Hseq^U\sq M$.

For composition, consider two conditions in a two-step iteration.
First strengthen their first coordinates to isomorphic cones
using local homogeneity of the first forcing.  Transport one
second-coordinate name to the other cone.  Uniformity of the
definition identifies the two second-forcing names there.
After strengthening the common first-coordinate condition,
choose names for stronger second-coordinate conditions and
an isomorphism between their cones.  Such witnesses exist
locally by the forcing existential clause, since the second
forcing is forced locally homogeneous.  No name for witnesses
below all first-coordinate conditions is required.

The resulting named isomorphism induces an isomorphism of the
separative iteration cones over that common first-coordinate
cone.  Compose it with the transported first-coordinate
isomorphism.  This proves local homogeneity of the two-step
iteration, and finite induction proves the stated extension.
All these cone maps fix check names for old parameters and
ordinals, so the same reasoning applies at the next uniformly
defined stage.
\end{proof}

In particular, the weakly homogeneous fresh envelope of
Lemma~\ref{lem:gch-envelope}, the locally homogeneous forcing
of Lemma~\ref{wd-gch-bounded-code}, and the uniformly defined
spectrum forcing can be composed over the original choiceless
model.  Their definitions use the old sequence generator,
the old ordinal code, and ordinal parameters.  Closure at the
two choice-model stages preserves the allowed ordinal
$\omega$-sequences, so Lemma~\ref{wd-local-descent} applies
to the combined extension.

\subsection{Closure of the evaluated spectrum}

The ground-recognition step below needs closure of the actual
forcing, in addition to the distributivity already established
for the spectrum.

\begin{lemma}[Closure of the evaluated spectrum]
\label{wd-spectrum-closure}
In Theorem~\ref{thm:spectrum}, let $\gamma_0$ be the first coding
cardinal.  The actual evaluated forcing $P_F$ of \eqref{eq:PF}
is $<\gamma_0$-closed in $B$.
\end{lemma}

\begin{proof}
Each lower row $Q_e$ is $<\gamma_0$-closed.  Consider a descending
chain
\[
       \seq{(q_i,\dot u_i):i<\zeta},\qquad \zeta<\gamma_0,
\]
in the actual two-step row
$Q_e*\dot{\Add}(\rho,1)^{B[\dot h_e]}$.  Take a lower bound
$q$ for the $q_i$.  Below $q$, the cap names form a descending
chain of conditions of the recomputed Cohen forcing.  Since
$\gamma_0<\rho$ and $Q_e$ preserves regularity of $\rho$, their
forced union is a cap condition and a lower bound for the chain.
Mix its name below $q$ with the empty cap off $q$, and normalize
the resulting total name to the bounded Boolean-name system
$T_e$ used after \eqref{eq:PF}.  If $\dot u$ is this name, then
$(q,\dot u)$ is a lower bound in the actual row.

Given a descending chain in the full product $P_F$, use Choice
in $B$ to select these lower bounds across its set of rows.
Their product is a lower bound in $P_F$.  The construction uses
names for evaluated cap conditions; it does not adjoin generics
for the larger ground termspaces.
\end{proof}

\subsection{Exact realization and ground recognition}

\begin{theorem}[Exact GCH realization over the normalized ground]
\label{wd-exact-gch-realization}
Let $D\subseteq N$ be supplied by Proposition~\ref{wd-reverse},
and fix an ordinal $\alpha$.  In a common outer universe there
are $M$ and $U$ such that
\[
 \begin{gathered}
 U\models\ZFC+\GCH+\PCW,\qquad
 M=\Hseq^U\models\ZF+\DC+\SVC,\\
 V_\alpha^M=V_\alpha^N.
 \end{gathered}
\]
Moreover, $U$ is an ordinary set-forcing extension both of $D$
and of $M$.  In particular, $\Hseq^U$ is a symmetric ground of $U$.
\end{theorem}

\begin{proof}
By Lemma~\ref{wd-old-ground-transfer}, there are
\[
 D\subseteq N\subseteq M=D(A)\subseteq B=M[K]=D[b]
\]
with $A\subseteq\mu^\omega$, $M\models\ZF+\DC+\SVC$,
$V_\alpha^M=V_\alpha^N$, and $B/M$ weakly homogeneous with no
new ordinal $\omega$-sequences.  The model $B$ satisfies
$\ZFC+\PCW$.  By Lemma~\ref{wd-normalized-ground}, it also
satisfies eventual GCH.

Apply Lemma~\ref{lem:gch-envelope}.  Its measurable-cardinal
hypothesis follows from $\PCW$.  We obtain a choice envelope
$B'$ satisfying global GCH, still weakly homogeneous over $M$
and with the same ordinal $\omega$-sequences.  It is a set
extension of $D$, so a proper class of ground Woodins survives.

We may arrange that $B'/D$ is a genuinely nontrivial forcing
extension.  If necessary, append $\Add(\omega_1,1)^{B'}$ before
fixing the presentation of $B'/D$.  Over the GCH model $B'$ this
forcing is countably closed, homogeneous, and GCH preserving.
Its literal countable conditions and the countably supported
permutations witnessing homogeneity belong to $M$, because
$M$ and $B'$ have the same ordinal $\omega$-sequences.  The
product presentation therefore preserves weak homogeneity over
$M$, and the additional forcing adds no such sequences.  Include
this extension in the notation $B'$.

Fix a presentation $B'=D[b']$ by $P\in D$, and put
\[
 \kappa=|P|^D,\qquad \delta=(\kappa^+)^D,
 \qquad s=({}^{<\delta}2)^D.
\]
Take $t\in D$ from Lemma~\ref{wd-normalized-ground}.  In $D$,
choose one ordinal-set code $z$ for $(s,t)$, including the
well-founded extensional relation and the distinguished points
needed for its unique Mostowski collapse.  This fixes $\delta$
and $z$ before any further coding.

Apply Lemma~\ref{wd-gch-bounded-code} over $B'$ to $z$, placing
its sites high enough that the coding forcing is
$<\delta^+$-closed and preserves $\delta^+$.  Let the result be
$B''=B'[J]$.  It satisfies GCH and has the same ordinal
$\omega$-sequences as $M$.  The set $z$ is ordinal definable
in $B''$.  Its uniquely decoded components $s,t$ are therefore
ordinal definable there as well.  The forcing is uniformly
defined from the old $z$ and ordinal parameters and is locally
homogeneous.

Choose $\eta$ large enough to contain all generators and forcing
data for $B''/D$, and all bounded data of the preceding code.
In particular, it bounds the coding sites, their relevant power
sets, and the candidate diamond-star sequences and club witnesses.
Thus preservation of $V_\eta^{B''}$ preserves the whole code,
including its decoding locations.

Choose an infinite ordinal $\tau$ large enough to present
$(\TC(\{A\}),\in,A)$, and let $F_A\subseteq2^\tau$ be the
family of all its pointed well-founded extensional presentations
as in Corollary~\ref{cor:ODcoding}.  Apply
Theorem~\ref{thm:spectrum} over the actual choice model $B''$,
placing its first cardinal $\gamma_0$ above $\delta^+$ and a
regular bound greater than $|V_\eta^{B''}|$.  Form the extension
\[
                         U=B''[G],\qquad G\subseteq P_{F_A}.
\]
The forcing preserves $V_\eta^{B''}$, so the code for $z$ and
its components $s,t$ remains ordinal definable in $U$.
Lemma~\ref{lem:spectrum-gch} gives $U\models\GCH$.
By Lemma~\ref{wd-spectrum-closure}, this actual final forcing
is $<\gamma_0$-closed.

Consequently the complete forcing from $D$ to $U$ has the fixed
nontrivial stem $P$, of $D$-size $\kappa$, followed by a
$<\delta^+$-closed tail.  The tail preserves $(\delta^+)^D$;
the small stem preserves it as well.  The closure-point lemma
gives the $\delta$-approximation and $\delta$-cover properties
for $D\subseteq U$, where $\delta=(\kappa^+)^D$ is regular.
The ground-definition theorem now defines $D$ inside $U$ from
$s=({}^{<\delta}2)^D$ and ordinal parameters; see
\cite[Theorem~6 and Lemma~8]{FuchsHamkinsReitz}.  This uses the
fixed approximation bound $\delta$, not a bound exceeding the
size of the complete coding forcing.

Since $s$ is ordinal definable in $U$, the class $D$ is ordinal
definable there.  Since $t$ is also ordinal definable and
$D=\HOD_{\{t\}}^D$, relativizing the internal definitions to
this class shows that every member of $D$, and each of its
hereditary members, is ordinal definable in $U$.  Hence
\[
                            D\subseteq\HOD^U.
\]
This conclusion uses the parameter $t$; definability of the
class $D$ by itself would not suffice.

The full choice envelope is now recognizable inside $U$ as
\[
                            B''=D(V_\eta^U).
\]
Indeed $V_\eta^U=V_\eta^{B''}$ contains the generators for
$B''/D$, whereas the generated model on the right is constructed
inside $B''$.  Thus $B''$ is an ordinal-definable class in $U$.
The fixed-base spectrum of Theorem~\ref{thm:spectrum}, over this
very class $B''$, recovers $F_A$.  The common pointed collapse
of its members is $A$, so $A$ is ordinal definable in $U$.
Every member of $A$ is an ordinal $\omega$-sequence.  It follows
that $A\in\Hseq^U$.

In the other direction, the entire forcing $U/M$ is locally
homogeneous.  Start with its weakly homogeneous choice-envelope
factor and apply the composition assertion of
Lemma~\ref{wd-local-descent} to the coding and spectrum factors.
Their defining parameters $z,A$ are old elements of $M$, and
their other parameters are ordinals.  The local cone maps fix
these parameters; no selected wellordering of the presentation
codes is part of a forcing definition.  None of the factors adds
ordinal $\omega$-sequences, so the descent conclusion of that
lemma gives
\[
                              \Hseq^U\subseteq M.
\]
Conversely, $D\subseteq\HOD^U\subseteq\Hseq^U$ and
$A\in\Hseq^U$.  Minimality of the generated model yields
\[
                       M=D(A)\subseteq\Hseq^U.
\]
Therefore $\Hseq^U=M$, with the required protected rank.

The construction exhibits $U$ as an ordinary set-forcing
extension of $M$.  Since $U$ satisfies Choice, $M$ is also a
symmetric ground of $U$ by \cite[Corollary~5.9]{Usuba}.
Every forcing used above is set forcing over an intermediate
set extension of $D$, so their composition is a set forcing
over $D$.  All sufficiently high $D$-Woodins remain Woodin
in $U$ by small-forcing preservation \cite{HamkinsWoodin}.
This proves $U\models\ZFC+\GCH+\PCW$ and all the conclusions.
\end{proof}

\section{The Woodin consistency implications}\label{wd-consistency-section}

\begin{proof}[Proof of Theorem~\ref{wd-main}]
First suppose $U\models W_1(\sigma)$, and choose $\alpha$ with
$V_\alpha^{\Hseq^U}\models\sigma$.
Proposition~\ref{wd-reverse} supplies a model $D\models\ZFC+\PCW$
and a set-symmetric extension $N\models\ZF+\DC$ of $D$ with
\[
 V_\alpha^N=V_\alpha^{\Hseq^U}.
\]
Thus $N\models R_\sigma$, giving the required interpretation of
$W_2(\sigma)$.

Conversely, start with a $\ZFC+\PCW$ ground $D$ and a set-symmetric
extension $N\models\ZF+\DC+R_\sigma$. Choose $\alpha$ witnessing
$R_\sigma$ in $N$. Theorem~\ref{wd-set-realization} gives a
set-forcing extension $U$ of $D$ such that
\[
 U\models\ZFC+\PCW,
 \qquad V_\alpha^{\Hseq^U}=V_\alpha^N.
\]
It follows that $U\models W_1(\sigma)$.

To pass from $W_1$ to $W_3$, use the particular $D,N$ supplied
by Proposition~\ref{wd-reverse} from the original $W_1$ model.
Lemma~\ref{wd-normalized-ground} gives eventual GCH in $D$ and
a set of ordinals $t\in D$ with $D=\HOD_{\{t\}}^D$.
Theorem~\ref{wd-exact-gch-realization} yields a set-forcing
extension $U^*$ of $D$ and a model $M$ such that
\[
 U^*\models\ZFC+\GCH+\PCW,\qquad
 \Hseq^{U^*}=M,\qquad V_\alpha^M=V_\alpha^N,
\]
and $U^*$ is an ordinary set-forcing extension of $M$.
Thus $U^*\models W_3(\sigma)$. The implication from $W_3$ to
$W_1$ follows by omitting the two additional assertions.
Together with the first two implications, this proves the three
consistency equivalences. The passage through the normalized ground
is part of the construction; it makes no GCH assumption about the
original ground of an arbitrary $W_2$ model.

These are ordinary syntactic consistency implications. Fix $\sigma$
before taking either interpretation. The relative construction
$F(a)$ is a uniform first-order class definition. Its assertions
of $\ZFC$ in the resulting class are read as the corresponding
axiom scheme. Every iteration history used for a set-length
argument, every relative-HOD definition, and every occurrence of
satisfaction is coded in a set-sized structure. The argument
requires no truth predicate for the whole universe or an external
class iteration strategy.

For $W_2$, use a set symmetric system and a condition forcing
$\DC+R_\sigma$ in the hereditary-symmetric interpretation. The
symmetric forcing theorem gives the interpretation of $\ZF$.
Conversely, if a particular generic interpretation satisfies
$\DC+R_\sigma$, its truth lemma supplies a condition forcing this
fixed conjunction. There is no requirement that the largest
condition force it. For $W_3$, the construction supplies an actual
set-forcing presentation over the definable class $\Hseq$.
The ground and symmetric-ground assertions have the first-order
formulations of \cite{Usuba}; the choice ground $D$ is recovered by
\cite[Theorem~6]{FuchsHamkinsReitz}. All additional coding stages
are set forcings. The usual set-forcing and definable-inner-model
interpretations therefore give all three consistency implications.
Equivalently, the constructions can be carried out internally in
countable models supplied by completeness. External transitivity
of those models is not an additional hypothesis.
\end{proof}

Theorem~\ref{wd-main} places the Woodins in the choice ground of
$W_2$ and in the ambient universes of $W_1$ and $W_3$.
The symmetric model satisfies dependent choice and has a rank
satisfying $\sigma$. The general realization theorem preserves
that rank. The realization over the normalized ground identifies
the whole model $M=D(A)$ with external HOD and makes it a ground
of a $\ZFC+\GCH+\PCW$ universe. Neither conclusion requires
Woodins inside $M$. The original collapse real and its canonical
envelope need not belong to the final universe.

\appendix
\section{Two bookkeeping details}\label{app:bookkeeping}

\subsection{Names, parameters, and nonstandard models}
All definability arguments use bounded-stage codes. To say that a set belongs to $D(A)$, specify a set-sized stage over a rank of $D$ and the transitive closure of $A$. A relative-HOD definition can likewise be evaluated at a sufficiently large rank level, with that ordinal included in the code. No satisfaction predicate for the entire universe is required.

The forcing comparisons concern fixed names and formulas. Auxiliary collapses produce common generic realizations only to refute incompatible decisions. The Prikry-only names in Lemma~\ref{lem:eliminate} are constructed in the original model $N$. These facts allow the relative-consistency argument to be formalized over arbitrary countable models, without assuming transitivity.

For class preparation, use the conservative class-theoretic expansion for progressively closed coding. Each set needed in the proof appears in a set-sized initial segment. The corresponding forcing and reflection argument handles any specified finite list of set-theoretic axioms and formulas. The preparation is the high-start $\diamondsuit^*$ version of
\cite[Appendix, Lemma~20]{CFH}. Its closure and lifting checks are
given in Lemma~\ref{lem:secure}. The later GCH collapse and
capped-history coding are both set forcing.

\subsection{Uniformity of the high coding}
The decoder uses three ordinal parameters: the presentation length $\tau$, an initial protection bound, and the envelope rank $\eta$. The site recursion takes its size bounds from all of $2^\tau$, so the candidate forcing $Q_e$ and cap cardinal $\rho$ are defined before the spectrum is recovered. The final predicate asserts that $e$ admits the two generic filters over the class $B=D(V_\eta)$. This is first-order because the posets and their dense subsets belong to the specified definable inner models.

Homogeneity requires the family of all presentation codes. Its common pointed collapse makes $A$ OD without making any particular wellordering of $\TC(\{A\})$ OD. The final HOD can therefore remain choiceless.


\section{The obstruction without measurables}\label{app:no-meas}

The measurable hypothesis cannot simply be omitted, even from the
comparison between the weakest external theory and the theory with
dependent choice. Put
\[
 A_L\equiv
 \Pow(\omega)=\Pow^L(\omega)\ \wedge\ \omega_2=\omega_2^L.
\]

\begin{lemma}\label{lem:constructible-sequences}
In any model of $\ZF+A_L$, every countable ordinal sequence
belongs to $L$. Consequently
\[
 \ZF+A_L+\KW\vdash\mathsf{AC}.
\]
If $U\models\ZFC$ and $\Hseq^U\models A_L$, then
$\Hseq^U=\HOD^U$ and this inner model satisfies Choice.
\end{lemma}
\begin{proof}
Fix an ordinal sequence $s$ in the given universe $W$, and apply
Jensen covering in the choice inner model $L[s]$; see
\cite[Theorem~1.1]{MitchellCovering}. This use of covering does not
assume Choice in $W$. The models $L$ and $L[s]$ have the same reals,
hence the same $\omega_1$, and $\omega_2^L$ remains a cardinal in
$L[s]$ because it remains a cardinal in $W$. Also $0^\#$ does not
exist in $L[s]$, since it would be a nonconstructible real.

Covering gives a constructible set $y$ containing
$\ran(s)\cup\omega_1$ whose cardinality in $L[s]$ is $\omega_1$.
Its cardinality in $L$ is at most $\omega_1$. Otherwise an
$L$-injection of $\omega_2^L$ into $y$ would collapse that cardinal
in $L[s]$. Fix an $L$-injection $e:y\to\omega_1$.
In $L[s]$, the sequence $e\circ s$ is bounded in $\omega_1$.
An $L$-enumeration of a countable bound codes this sequence by a
real. That real is constructible, so $s\in L$.

Thus each whole sequence space $\mu^\omega$ is its constructible
counterpart and is wellorderable. Under $\KW$, least preimages
wellorder every nonempty set.
For the last assertion, $\Hseq^U$ contains every ordinal
$\omega$-sequence of $U$. Applying the first assertion inside
$\Hseq^U$ makes every allowed sequence parameter constructible.
Every constructible set is ordinal-definable in $U$, using its
index in a canonical constructible wellordering. Replacing the
sequence parameters by these ordinal definitions gives
$\Hseq^U=\HOD^U$.
\end{proof}

\begin{proposition}\label{prop:no-meas-rank}
There is a first-order sentence $\sigma_0$ such that
\[
 \ZFC\vdash
 \Hseq\models\neg\exists\theta\,(V_\theta\models\sigma_0),
\]
whereas, relative to $\Con(\ZFC)$,
\[
 \Con\bigl(\ZF+\DC+
       \exists\theta\,(V_\theta\models\sigma_0)\bigr).
\]
The countermodel can also satisfy
$A_L+\neg\mathrm{DC}_{\omega_1}$.
\end{proposition}
\begin{proof}
Over $L$, force with the countable partial functions
\[
 P=\operatorname{Fn}_{<\omega_1}
       (\omega_1\times\omega_1,2),
\]
ordered by reverse inclusion. Permute the first coordinate, and
use the normal filter generated by pointwise stabilizers of
countable sets of coordinates. Let $N$ be the hereditarily
symmetric model. The forcing is countably closed and, by CH in
$L$, has size $\omega_1$. It therefore adds no reals and preserves
all cardinals. The same reals, $\omega_1$, and $\omega_2$ occur in
$L$, $N$, and the full extension. The countably complete symmetry
filter and countable closure give $N\models\DC$ by
\cite[Lemma~3.1]{KaragilaDC}.

Let $g_i$ be the generic subset of $\omega_1$ at coordinate $i$.
The set $X=\{g_i:i<\omega_1\}$ belongs to $N$. Every
ordinal-indexed map into $X$ in $N$ has range contained in its
countable family of supported generics. Indeed, if a supported name for $f$
satisfied $f(\xi)=g_i$ outside its support, transpose $i$ with a
coordinate absent from both the support and a condition deciding
this equality. The condition and its image are compatible and
force two distinct values at $\xi$. This is impossible.
The generics on each countable support have a countable enumeration in $N$.
Hence $X$ is not wellorderable, and neither is $\Pow(\omega_1)$.
The tree of injections from countable ordinals into $X$ is closed
under increasing countable sequences and has no maximal node.
It has no $\omega_1$-branch, since such a branch would give an
injection of $\omega_1$ into $X$. Thus $N$ also fails transfinite
$\mathrm{DC}_{\omega_1}$, in the tree formulation of
\cite[Theorem~2.1]{KaragilaDC}.

Choose $\sigma_0$ to assert that there is no greatest ordinal;
that $\lambda=\omega_1$ and $\mu=\omega_2$ exist, with a limit
ordinal $\nu>\mu$; that every real is constructible;
that $\lambda=\omega_1^L$ and $\mu=\omega_2^L$; and that
$\Pow(\lambda)$ is not wellorderable. These are combined into one
first-order sentence, with $\lambda,\mu,\nu$ existentially
quantified.

Suppose $V_\theta^W\models\sigma_0$, where $W\models\ZF$.
The first clause makes $\theta$ a limit, and the guard gives
$\theta>\mu+\omega$. All functions between ordinals at most
$\mu$, and all subsets and binary relations on
$\Pow(\lambda)$, belong to this rank whenever they belong to
$W$. Thus its $\omega_1$ and $\omega_2$ are those of $W$, and
its nonwellorderability assertion remains true in $W$.
The local constructible hierarchy agrees with the actual one at
every level below $\theta$. To verify this, code finite
assignments by finite function graphs: the satisfaction relation
for a set structure and a hierarchy sequence through a fixed
level require only a fixed finite increase in rank. All such
witnesses remain below the limit $\theta$.
In particular, every real of $W$ is constructible.

It remains to check the second constructible cardinal.
Equality of reals gives $\lambda=\omega_1^L$ globally, and
inner-model cardinal comparison gives
$\gamma=\omega_2^L\leq\mu=\omega_2^W$.
If $\gamma<\mu$, then $\gamma$ is also a cardinal of the rank's
local $L$: a local constructible bijection collapsing it would
be a global constructible bijection. This puts an $L$-cardinal
strictly between the asserted first and second uncountable
$L$-cardinals, a contradiction. Therefore $W\models A_L$.

If $W=\Hseq^U$ for a choice universe $U$,
Lemma~\ref{lem:constructible-sequences} gives Choice in $W$,
contradicting the retained nonwellorderability assertion.
On the other hand, the symmetric model $N$ satisfies
$\sigma_0$ globally, so reflection gives a rank satisfying it.
This is a formal relative-consistency construction; no
transitive-model assumption is required.
\end{proof}

The same sentence gives a rank-conservativity obstruction:
$\ZF+\KW$ proves that no rank satisfies $\sigma_0$, while
$\ZF+\DC$ does not, assuming consistency. This proves that CM
cannot be deleted from the stated theorem. It does not identify
the weakest sufficient large-cardinal hypothesis.


\section{A Ramsey obstruction}\label{app:ramsey}

Let $\mathsf{CR}$ assert that the cardinals $\rho$ satisfying
$\rho\longrightarrow(\rho)^{<\omega}_2$ are unbounded in the ordinals,
and put $T_{\mathrm R}=\ZF+\DC+\mathsf{CR}$. In this section
$\mathbb R=\Pow(\omega)$. The three sorts of arithmetic are
$\omega$, $\mathbb R$, and $\Pow(\mathbb R)$. A $\Pi^2_3$ sentence
has three alternating blocks of third-sort quantifiers, beginning
universally, followed by a matrix with only number and real
quantifiers. Finite tuples of third-sort variables may be combined
by fixed recursive tags.

\begin{theorem}[Ramsey obstruction]\label{thm:ramsey-obstruction}
There is a parameter-free $\Pi^2_3$ sentence $\Phi$ such that
\[
 T_{\mathrm R}+\KW\vdash\Phi,
 \qquad
 \Con(\ZFC+\mathsf{CR})
 \ \Longrightarrow\ \Con(T_{\mathrm R}+\neg\Phi).
\]
Consequently, relative to $\Con(\ZFC+\mathsf{CR})$, adding $\KW$
is not conservative over $T_{\mathrm R}$ for third-order
arithmetic. The same example gives a structural $\Pi_2$ assertion
separating these theories.
\end{theorem}

We use the classical Dodd--Jensen core model. In a choiceless
universe $W$, set $K=(K^{\mathrm{DJ}})^{\HOD^W}$. The semantic
form of the separating implication is
\begin{equation}\label{eq:ramsey-semantic}
 \bigl[\neg\exists0^\dagger\ \wedge\
       \mathbb R=\mathbb R^K\ \wedge\
       \omega_2=\omega_2^K\bigr]
 \ \longrightarrow\ \mathsf{Fodor}(\omega_1).
\end{equation}
We first prove this implication with $\KW$, then give its
third-order formulation and a countermodel.

\begin{proof}
\emph{The covering implication.}
Under $\mathsf{CR}$, absence of $0^\dagger$ excludes an inner
model with a measurable. Indeed, from such an inner model obtain
its canonical $L[U]$, with measurable $\kappa$. Its wellordering
gives an ordinal code $A\subseteq\alpha$ for the transitive
closure of $U$, where $\alpha>\kappa$, $A\in L[U]$, and
$L[A]=L[U]$. A Ramsey above $\alpha+\omega$ supplies indiscernibles
for $L[A]$. Shifting above $\alpha$ gives a nontrivial
self-embedding of $L[U]$ with critical point above $\kappa$,
which implies $0^\dagger$. Here the relativized indiscernibles
argument is applied to canonical structures with definable
Skolem functions; it is available in ZF
\cite[Propositions 7--8]{ApterKoepke}
\cite[p.~262]{MitchellRamsey}.
The last implication is the standard nonrigidity
characterization of $0^\dagger$ \cite{CompleatDagger}.
The possibly uncountable parameter sharp is not itself being
identified with a real.

Assume the antecedent of \eqref{eq:ramsey-semantic}, and fix
$s\in(\Ord^\omega)^W$. Code its graph by a set of ordinals
$\bar s$. Apter--Koepke's adjoining construction gives
\[
 C=(\HOD^W)[\bar s]\models\ZFC,\qquad
 K\subseteq C\subseteq W,\qquad K^{\mathrm{DJ},C}=K
\]
with $s\in C$ \cite[Proposition 9]{ApterKoepke}.
The endpoint agreements give common $\omega_1$ and $\omega_2$
in these three models. Apply strong Dodd--Jensen covering
inside $C$ to $\operatorname{ran}(s)\cup\omega_1$
\cite[Theorem 1.2]{MitchellCovering}. It has a cover
$y\in K$ with $|y|^C=\omega_1$. Correct $\omega_2$ implies
$|y|^K\leq\omega_1$. For a $K$-surjection $e:\omega_1\to y$,
the sequence of least $e$-indices of the values of $s$ is bounded
in $\omega_1$ in $C$. A $K$-enumeration of a countable bound
codes that sequence by a real. The real belongs to $K$, so
$s\in K$.

Thus $(\xi^\omega)^W=(\xi^\omega)^K$ for every ordinal $\xi$.
Each of these spaces has a $K$-wellordering. Under $\KW$,
least preimages under a sequence-space surjection wellorder
every nonempty set. Hence AC, and therefore Fodor's lemma,
follows.

\emph{The third-order formula.}
Use the classical one-measure lower-part presentation of $K$
\cite[Section 3]{BenNeriaLuecke}. Countable mousehood has a
projective definition, and the corresponding countable-hull
tests use only real codes
\cite[Lemma 3.32 and p.~92]{MitchellCovering}.
For $A\subseteq\mathbb R$, a certificate for
$\operatorname{KSet}(A)$ consists of a wellordered real domain,
a well-founded extensional premouse relation on it, its measure
predicate and satisfaction relation, actual-real labels, and
a distinguished lower-part node whose elements are exactly $A$.
Require every real-coded countable elementary hull to collapse
to an iterable classical premouse. Also allow a constructible
level certificate.

The domain wellorder makes the entire labelled certificate
ordinal-codable. In the choice model obtained by adjoining
that code to HOD, the countable-hull copying criterion gives
iterability, and Proposition 9 of \cite{ApterKoepke} identifies
the core with $K$. This proves soundness. For completeness
under $\mathbb R=\mathbb R^K$, choose the witness inside $K$,
with both $\mathbb R^K$ and $A$ in its lower part. A canonical
hull containing every real and these two nodes has size at
most $\omega_1$ and its collapse fixes $A$.
The small-mouse argument in the proof of
\cite[Lemma 3.1]{BenNeriaLuecke}, applied inside $K$ at
$\omega_2^K$, also gives this witness directly.
CH in $K$ supplies a real-domain code. Equality of reals
preserves its countable-hull tests. Consequently
\[
 \mathbb R=\mathbb R^K
 \quad\Longrightarrow\quad
 \bigl(\operatorname{KSet}(A)\longleftrightarrow A\in K\bigr).
\]

Under DC, a relation on a real domain is well-founded exactly
when it has no real-coded descending omega-sequence. All the
certificate conditions therefore have a second-order matrix
after one existential third-sort quantifier:
$\operatorname{KSet}(A)\equiv\exists P\,k(A,P)$.
Let $\operatorname{CopyK}$ say that every wellorder on a subset
of the reals has an isomorphic real-domain wellorder with such
a core certificate. Bundling the copy, certificate and
isomorphism gives
\[
 \operatorname{CopyK}\equiv\forall E\,\exists P\,c(E,P),
\]
where $c$ is second-order; malformed inputs permit a dummy
certificate. On $\mathbb R=\mathbb R^K$, this is equivalent
to $\omega_2=\omega_2^K$. For if the cardinals agree, every
real-domain order type is below $\omega_2^K$ and has a core
copy. If $\omega_2^K<\omega_2^W$, an ambient real-domain
wellorder of type $\omega_2^K$ has no such copy. The old
CH bijection excludes the reverse cardinal inequality.

Write $B$ for the second-order guard
$\neg\exists0^\dagger\wedge\forall x\in\mathbb R\,(x\in K)$.
Here $0^\dagger$ uses its usual real-coded two-block
indiscernible theory and countable well-foundedness tests.
Let $U$ code a wellorder $w$ of all reals with countable
proper initial segments, a set $S$ of its points, and a
regressive function on $S$. Such a $w$ has type $\omega_1$.
Fodor's assertion says either that a club avoids $S$, or
that one fiber meets every club. An existential third-sort
witness $V$ stores a tag and either that first club or the
single real naming the fiber. Thus its real-domain version
is $\forall U\,\exists V\,\forall D\,q(U,V,D)$, where $D$
tests clubs and $q$ is second-order. Invalid $U$ is accepted
automatically. No family of clubs is selected.

The required $\Phi$ is the following normal form:
\begin{equation}\label{eq:ramsey-pi23}
 \forall U\,\exists(E,V)\,\forall(P,D)\,
 \bigl[\neg B\ \vee\ \neg c(E,P)\ \vee\ q(U,V,D)\bigr].
\end{equation}
All prefix variables have the third sort. If
$\operatorname{CopyK}$ fails, a single $E$ with no certificate
makes the matrix true. If it holds, a certificate for the
selected $E$ reduces the matrix to the Fodor clause.
Thus \eqref{eq:ramsey-pi23} is equivalent, over $\ZF+\DC$,
to \eqref{eq:ramsey-semantic}. This proves the advertised
$\Pi^2_3$ upper bound, without claiming minimality.

\emph{The countermodel.}
Start with $\ZFC+\mathsf{CR}$ and pass to its classical core
$D$. Mitchell's Ramsey preservation theorem gives
$D\models\ZFC+\GCH+\mathsf{CR}$ \cite[Theorem 1]{MitchellRamsey}.
Also $K^D=D$ and $D$ has no $0^\dagger$.
Apply Karagila's local symmetric construction at
$\kappa=\mu=\omega_1^D$ \cite[Section 4]{KaragilaFodor}.
First add $g:\kappa\to\kappa$ by bounded conditions, and set
$S_\beta=\{\xi<\kappa:g(\xi)\geq\beta\}$.
Then use the countable-support product of the club-shooting
orders on the $S_\beta$, with the normal filter fixing old
countable sets of coordinates. Let $N$ be the symmetric model.

The combined forcing has a countably closed dense
presentation of size $\omega_1$. A dense condition synchronizes
the club maxima and chooses the value of $g$ there above every
active index. Countable unions have lower bounds of the same
form, and CH bounds the number of codes. The reals and
cardinals are unchanged, and the countably complete support
filter gives DC in $N$. Every ordinal-coded object of $N$
belongs to a bounded head $D[G_\alpha]$, and every such head
is contained in $N$.

Put $T_*=\kappa\setminus\Delta_{\beta<\kappa}S_\beta$.
Then $f=g\restriction T_*$ is regressive and each fiber of
value $\eta$ misses the retained club through $S_{\eta+1}$.
The set $T_*$ is stationary in $N$. Indeed, a retained club
belongs to a head $D[G_\alpha]$. Fusion in that head produces
a limit point $\delta>\alpha+1$ of the club at which one may
set $g(\delta)=\alpha$. Appending $\delta$ to every active
coordinate below $\alpha$ is legal, while
$\delta\notin S_{\alpha+1}$. This is stationarity in $N$;
the full choice extension retains an additional diagonal club.

For any old Ramsey $\rho>\omega_1$, a coloring in $N$ of
$[\rho]^{<\omega}$ belongs to one bounded head.
That head is a choice extension by a forcing of size
$\omega_1<\rho$. Small-forcing preservation supplies a
homogeneous set there \cite[Theorem 4.1]{GitmanJohnstone},
so that set also belongs to $N$. Hence $N\models\mathsf{CR}$.
Classical mouse absoluteness and set-forcing invariance
identify $K^N=D$; unchanged reals preserve the
$0^\dagger$ predicate. Thus the antecedent of
\eqref{eq:ramsey-semantic} holds in $N$, while its conclusion
fails. The old CH bijection transports this counterexample
to the third-order formula. These forcing and inner-model
constructions give the stated consistency implication.
\end{proof}

For the structural conclusion, use a finite signature with
number and real sorts partitioning its domain, the arithmetic
operations on the number sort, and a membership relation
between the two sorts. A full second-order sentence asserts the Peano
axioms with second-order induction on the first sort and
requires every subset of that sort to be coded by exactly
one element of the second. Such a structure is isomorphic
to the canonical structure on $\omega\sqcup\Pow(\omega)$.
Translate third-sort variables in $\Phi$ into second-order
subsets of its real sort. The universal assertion that every
such frame satisfies this translation is equivalent to
$\Phi$ in ZF. It has the structural $\Pi_2$ form of
Definition~\ref{def:structural-pi2}. The $\Pi^2_3$ bound concerns
third-order arithmetic, whereas this $\Pi_2$ classification
concerns the L\'evy hierarchy.

In particular, the Ramsey replacement of
Theorem~\ref{thm:main} fails even with Foundation retained in
the starting theory, and hence also with Foundation omitted.
This argument does not assert that $\KW$ alone implies
Fodor's lemma, nor does it supply a Ramsey replacement
counterexample for the separate external rank-equiconsistency
theorem without a further rank translation.



\section{A strong-cardinal separation for the Chang model}\label{app:chang}

We give the separation used in the introduction to explain the
choice of $\HOD_{\Ord^\omega}$ in Theorem~\ref{thm:external-main}.
Throughout this appendix, $C^U=L(\Ord^\omega)^U$ is computed in an
ambient universe $U\models\ZFC$.

Define
\[
 E_1^C(\sigma)=\ZFC+\CM+
 [C\models\exists\kappa\,(V_\kappa\models\sigma)].
\]
Let $\CS$ assert that strong cardinals are unbounded, with
strongness given by its usual first-order extender formulation, and put
\[
 B_{\rm str}=\ZFC+\CS+
 \text{``there is an inaccessible limit of strong cardinals.''}
\]
The following single first-order sentence will separate the theories:
\begin{equation}\label{eq:chang-strong-sentence}
 \begin{aligned}
 \sigma_{\rm str}\equiv\exists\lambda\,\exists\nu\,[&
  \lambda,\nu\in\Ord\ \wedge\ \lambda<\nu\\
  &{}\wedge\ \nu\text{ is a limit ordinal}\\
  &{}\wedge\ V_\lambda\models\ZFC+\CS].
 \end{aligned}
\end{equation}
Here the rank and its satisfaction relation are expressed by set
witnesses, including satisfaction of the recursive set of all ZFC
axioms. The ordinal $\nu$ is a height guard: in a witnessing outer
rank $V_\kappa$, it ensures $\lambda+\omega\leq\nu<\kappa$.

\begin{theorem}[Chang rank separation]\label{thm:chang-rank-separation}
If $U\models\ZFC$, no full rank $V_\lambda^{C^U}$ satisfies
$\ZFC+\CS$. In particular, $E_1^C(\sigma_{\rm str})$ is
inconsistent, whereas
\[
 \Con(B_{\rm str})\ \Longrightarrow\ \Con(E_3(\sigma_{\rm str})).
\]
\end{theorem}

Thus, relative to $B_{\rm str}$, substituting the Chang model for
$\Hseq$ destroys even the equiconsistency of theories (1) and (3).
The prohibition on Chang ranks requires no ambient measurable
cardinals. Strong cardinals enter the consistency bound for the
separating instance of theory (3). For a tautological $\sigma$, the
two Chang analogues remain consistent relative to $\ZFC+\CM$.

\subsection{The imported mouse construction}

We use Mitchell's Chang-sharp construction. The relevant results are \cite{MitchellChang}, Theorem 1.5, Sections 1.3 and 3, Definition 4.1, and Proposition 1.4.

The construction uses a set mouse over all the reals with a sufficiently long last extender. Section 3 reduces the hypothesis to set-sized data, and Definition 4.1 specifies the additional approximation system. The relevant mice use the linear iteration theory below a strong cardinal. Iterating the final extender gives a continuous class $I=\{\kappa_\alpha:\alpha\in\Ord\}$. Definition 3.9 gives uniformly interpreted term codes from the mouse. These terms represent every element of Chang. Proposition 1.4 extends appropriate order-preserving maps of $I$ to elementary embeddings of Chang.

The large cardinal existence input is that $\ZFC+\CS$ supplies such a mouse over its entire real set, with the required iterability. Gappo and Sargsyan explicitly identify this upper bound in the introduction to \cite{GappoSargsyan}. We call the full set data specified by Mitchell a \emph{Mitchell datum}. This abbreviation includes the approximation system; a bare premouse with a long extender is not sufficient.

All structural requirements on a fixed datum are first-order assertions about set structures, their satisfaction relations, functions, and ordinals. Their interpretation is absolute when the data and the relevant countable sequences are shared. The remaining nonlocal requirement is iterability. The next section addresses that requirement rather than assuming it is absolute.

\subsection{Countable hull iterability transfers from a Chang rank}

\begin{lemma}\label{lem:chang-rank-closure}
Suppose $W=V_\lambda^{C^U}\models\ZFC$. Then $W$ and $U$ have the same reals and the same $\omega_1$. If $a\in W$, every $U$-countable sequence of elements of $a$ belongs to $W$.
\end{lemma}
\begin{proof}
The Chang model is closed under countable sequences formed in $U$. In particular it contains every real and every countable ordinal sequence of $U$.

If $a\in W$ and $s:\omega\to a$ belongs to $U$, then $s\in C^U$. Its rank is bounded by $\max\{\operatorname{rank}(a),\omega\}$ plus a fixed finite ordinal. This is below $\lambda$, since $W\models\ZFC$. Hence $s\in W$.

Every real has bounded finite rank above $\omega$, so the real sets agree. If an ordinal in $W$ is countable in $U$, its enumeration belongs to $W$ by the preceding argument. Applying this to $\omega_1^W$, if necessary, proves equality of $\omega_1$.
\end{proof}

\begin{lemma}\label{lem:chang-iterability}
Let $W$ be as above. Suppose $M\in W$ is a set premouse in the linear iteration setting of \cite{MitchellChang}. If $W$ regards every countable elementary hull of $M$ as iterable through all countable linear iterations, then the same assertion holds in $U$. Moreover $M$ has no ill-founded set-length linear iteration in $U$.
\end{lemma}
\begin{proof}
Include the extender predicates and the final extender in the structure being considered. Let $X\prec M$ be any elementary substructure countable in $U$. An enumeration of $X$ is a countable sequence from the fixed set underlying $M$, so Lemma~\ref{lem:chang-rank-closure} puts $X$ and an enumeration in $W$. Set satisfaction is absolute, so $W$ recognizes its elementarity. The two universes form the same transitive collapse $\bar M$.

A countable premouse, a countable putative linear iteration, and a descending sequence witnessing ill-foundedness can be coded by reals. The relevant code relations are statements of second-order arithmetic. Since $W$ and $U$ have exactly the same reals, they agree on these relations and on the assertion that every countable legal linear iteration of $\bar M$ is well-founded. Thus every countable hull remains countably iterable in $U$.

For completeness, suppose there were a bad set-length linear iteration of $M$ in $U$. Retain the iteration through a first bad model, together with a descending-sequence witness for its formal ultrapower or direct-limit relation. In $U$, take a countable elementary hull of a sufficiently large set structure containing these data and the definitions of the iteration operations. Its transitive collapse gives a countable bad linear iteration of the collapse of a countable elementary substructure of $M$.

The collapse preserves the iteration rules. Formal ultrapowers use functions from the relevant source structure, and elementarity supplies representatives of their quotient elements in the hull. The same reasoning applies to the direct-limit construction. The retained descending sequence witnesses that the last relation is still ill-founded. This contradicts the preceding paragraph.

This argument concerns linear iteration. It does not assert preservation or existence of a strategy for arbitrary uncountable branching iteration trees.
\end{proof}

\subsection{Why the datum cannot belong to Chang}

\begin{proposition}\label{prop:chang-forbidden-datum}
A Mitchell datum over all the reals of $U$, with the requisite external linear iterability, cannot belong to $C^U$.
\end{proposition}
\begin{proof}
Suppose the datum belongs to $C=C^U$. Write $(M,E)$ for its mouse and final extender. The canonical linear iteration by $E$ can be computed inside $C$ at every set-length stage. Its computation agrees with that in $U$: the source structures and their functions are the same, formal ultrapowers and direct limits are the same, and well-founded collapses are unique. Transfinite recursion in $C$ supplies the set-length iteration systems. Thus the class
\[
 I=\{\kappa_\alpha:\alpha\in\Ord\}
\]
and the interpretation of Mitchell's term codes are definable over $C$ from set parameters in the datum.

This last definability assertion can be read directly from Definition 3.9 of \cite{MitchellChang}. Its ingredients are the canonical iteration maps, countable sequences of term codes, composition, and satisfaction in a set-sized Chang construction stage. In particular, the apparently class-length term $i_{\Ord}(f)(a)$ is evaluated at the first set stage $\alpha$ for which all entries of the finite ordinal tuple $a$ are below $\kappa_\alpha$. The value $i_\alpha(f)(a)$ is below $\kappa_\alpha$ and is fixed by every subsequent iteration map. Thus evaluation is uniformly first-order definable from the set data. No truth predicate for the whole Chang model is required. The auxiliary approximation system and generic objects used in the proof of indiscernibility do not enter the interpretation of these terms.

Define an increasing map on ordinal indices by
\[
 f(\alpha)=
 \begin{cases}
 \alpha,&\alpha<\omega_1,\\
 \omega_1+\alpha,&\alpha\geq\omega_1,
 \end{cases}
 \qquad s(\kappa_\alpha)=\kappa_{f(\alpha)}.
\]
The map preserves successors and limit-index status. It is continuous at limit indices of countable cofinality. Its only initial discontinuity is at $\omega_1$, whose cofinality is uncountable in both universes. Therefore $s$ satisfies the hypotheses of Mitchell's Proposition 1.4.

Using the term representation of Chang, define
\[
 j(\tau(B))=\tau(s[B]).
\]
The equality and formula preservation furnished by the sharp make this definition independent of the chosen representation and make $j:C\to C$ elementary. The relation defining $j$ quantifies over set term codes and countable suitable sets $B$; all its ingredients are definable over $C$ from the datum. In particular $j$ is a parameter-definable class of $C$. It is nontrivial because
\[
 j(\kappa_{\omega_1})=\kappa_{\omega_1\cdot2}
 >\kappa_{\omega_1}.
\]
Suzuki's theorem \cite{Suzuki}, which requires ZF rather than Choice, excludes such a parameter-definable nontrivial elementary self-embedding of $C$. This is a contradiction.
\end{proof}

\subsection{The separation}

\begin{proof}[Proof of Theorem~\ref{thm:chang-rank-separation}]
Suppose $W=V_\lambda^{C^U}\models\ZFC+\CS$. Inside $W$, the imported mouse construction produces a Mitchell datum over $\mathbb R^W$. Lemma~\ref{lem:chang-rank-closure} gives $\mathbb R^W=\mathbb R^U$ and the same countable sequences from each fixed set in the datum. The bijection witnessing $|M|=|\mathbb R|$ remains a bijection externally. All elementarity and internal cardinal computations in the fixed structures remain correct. The height of $M$ still has cofinality $\omega_1$: its cofinal $\omega_1$-sequence remains cofinal, while any new countable cofinal sequence would belong to $W$ by bounded closure. Thus the structural requirements hold externally. Lemma~\ref{lem:chang-iterability} transfers the required linear iterability. The datum belongs to $W\subseteq C^U$, contradicting Proposition~\ref{prop:chang-forbidden-datum}.

If a rank $V_\kappa^{C^U}$ satisfies $\sigma_{\rm str}$, the height guard puts $\lambda+\omega$ below $\kappa$. Induction on the witnessed cumulative hierarchy identifies its nested rank with the actual $V_\lambda^{C^U}$. Every subset required at an earlier hierarchy stage is present in the outer rank. Finite assignments and the full set satisfaction relation also lie below $\lambda+\omega$, so satisfaction agrees. This gives precisely the impossible $W$ above, proving inconsistency of $E_1^C(\sigma_{\rm str})$.

For the consistency direction, work in a model of $B_{\rm str}$, and let $\delta$ be an inaccessible limit of ambient strong cardinals. Then $V_\delta\models\ZFC$. Every ambient strong $\rho<\delta$ is strong in $V_\delta$: for each target rank $V_\eta$ below $\delta$, choose the usual bounded extender $E$ derived from an ambient strongness embedding. Inaccessibility of $\delta$ puts its support, the extender, and the required power sets below $\delta$.

To check the target-rank containment, represent each $x\in V_\eta$ in the ambient extender ultrapower as $[a,f]_E$. Replace $f$ by the function that keeps its values in $V_{\eta+1}$ and uses $\varnothing$ elsewhere. It represents the same $x$, since the ultrapower image of $\eta+1$ exceeds $\eta$. The new function has rank below $\delta$. Hence the natural factor map from the ultrapower of $V_\delta$ has every element of $V_\eta$ in its range. Its transitive collapse fixes these elements by induction on rank. Thus the ultrapower of $V_\delta$ contains $V_\eta$, as required. This proves $V_\delta\models\CS$.

The ambient universe satisfies $\CM$, since strong cardinals are measurable and are unbounded. It satisfies $\ZFC$, hence $\ZF+\DC$. The rank $V_{\delta+\omega\cdot2}$ satisfies $\sigma_{\rm str}$, using $\lambda=\delta$ and $\nu=\delta+\omega$. This realizes $E_3(\sigma_{\rm str})$.
\end{proof}

The obstruction is a set-sized iterable datum that a full Chang rank satisfying $\ZFC+\CS$ would contain. The height guard in $\sigma_{\rm str}$ ensures that the nested rank and its satisfaction relation have their intended meanings.

\subsection{The anti-inner-model question}

The separation above leaves open whether restricting the available
inner models restores the comparison. Use the standard Mitchell-order
convention, so a measurable cardinal has order at least one, and let
\[
 A_2\quad\equiv\quad
 \text{``there is no inner model of $\ZFC$ with a measurable
 $\nu$ satisfying $o(\nu)\geq2$.''}
\]
The Mitchell order is computed in the indicated inner model.
Equivalently, the excluded configuration consists there of two
normal measures $U_0,U_1$ on the same $\nu$ with
$U_0\in\operatorname{Ult}(M,U_1)$. Here $M$ is the inner model,
and the ultrapower uses its functions. We use the semantic
inner-model formulation of $A_2$; any formal version of the
following question must fix a corresponding mouse or scheme
formulation on both sides.

\begin{question}\label{q:chang-anti-inner}
Can an anti-inner-model hypothesis restore the equiconsistency of
theories (1) and (3) after the Chang substitution? In particular,
for every fixed first-order sentence $\sigma$, is
\[
 \Con\bigl(E_1^C(\sigma)+A_2\bigr)
 \quad\longleftrightarrow\quad
 \Con\bigl(E_3(\sigma)+A_2\bigr)?
\]
\end{question}

In this question $A_2$ is appended to both ambient theories. The
left-hand witnessing rank is still computed inside Chang, and
the right-hand rank is computed in the $\ZF+\DC+\CM$ universe.
The restriction permits order-one measurable cardinals and does
not exclude measurable limits of measurable cardinals. No
equiconsistency under this restriction is asserted here. Whether
the Chang separation itself can be obtained from ordinary
measurable-cardinal strength also remains open.


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\end{document}