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\hypersetup{pdftitle={Ordinal-definable families in Cohen, random, and collapse extensions, version \paperversion},pdfauthor={Elliot Glazer}}
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\title{Ordinal-definable families in Cohen, random, and collapse extensions}
\author{Elliot Glazer}
\date{Version \paperversion, September 28, 2026}
\begin{document}
\maketitle
\begin{abstract}
We study ordinal-definable families of sets of arbitrary rank in
Cohen, random, and collapse extensions. Over $L$, countable OD families
have OD enumerations after adding one Cohen or one random real.
A single generalized Cohen subset gives the corresponding sharp theorem
at every regular uncountable cardinal. When $\kappa$ is singular strong
limit and has uncountable cofinality, adding $\kappa$ Cohen reals makes
every member of a short-parameter definable family of size at most
$\kappa$ definable from a short parameter. Adding $\kappa$ random reals
gives a single short-parameter definable enumeration for every such
family of size strictly below $\kappa$. The random bound is sharp.
The proofs use coordinate amalgamation and small-index arguments. We also obtain arbitrary-rank descent for countable families
after collapsing any infinite cardinal, with applications to choice
in the relative Feferman--Levy model.
In the full Solovay collapse extension of an arbitrary ZFC ground,
every family definable from a real and ordinals that represents fewer
than $2^{\mathfrak c}$ classes modulo null sets consists of measurable
sets; the corresponding category assertion also holds.

In the extensions of $L$ by $\omega_1$ Cohen or random reals, no
model of $I\Delta_0$ of size at most $\aleph_1$, definable from
ordinals and a real, has full binary-coded standard system.
$\Pi^1_1\text{-}\mathsf{CA}_0$ proves a finite-fragment Borel obstruction
to a full binary standard system for $I\Delta_0$. For
regular uncountable $\kappa$, the generalized Cohen extension by
$\Add(\kappa,\Lambda)^L$, $\Lambda>\kappa$, has no ambiently
$\kappa^+$-saturated $I\Delta_0$ model in
$\operatorname{HOD}_{H_{\kappa^+}}$, regardless of its size.
Consequently every infinite regular $\kappa$ of $L$ admits a
cofinality-preserving GCH extension with no
$\mathrm{OD}_{\mathcal P(\kappa)}$
saturated arithmetic presentation of size $\kappa^+$. ZFC also proves that every singular strong limit $\kappa$ admits
an OD $\kappa^+$-saturated elementary extension of $\mathbb N$ of
size $2^\kappa$, hence of size $\kappa^+$ under GCH.
\end{abstract}

{\small\tableofcontents}
\newpage
\section{Introduction}

An ordinal-definable set need not have ordinal-definable elements.
Kanovei and Lyubetsky proved that countable OD sets of reals in several
one-real extensions lie in the ground model, and asked about families
of sets of reals \cite{KL}. At higher rank, the forcing matters.
A Sacks extension
of $L$ can contain an OD pair of non-OD sets of reals
\cite{EnayatKanovei}. We prove arbitrary-rank small-family theorems
for Cohen, random, and collapse forcing, and apply them to
arithmetic and choice in inner models of ZF.

Definability and cardinality are computed in the ambient extension.
The notation $\OD(a)$ permits the parameter $a$ in addition to
ordinals, and $\ODR$ permits one real. For ordinary Cohen and random
forcing, $\ODlt{\kappa}$ permits fewer than $\kappa$ reals or,
equivalently, one subset of an ordinal of cardinality less than $\kappa$.
For generalized Cohen forcing, $\mathrm{OD}_{H_{\kappa^+}}$
permits one parameter of hereditary cardinality at most $\kappa$;
$\mathrm{OD}_{\mathcal P(\kappa)}$ permits one subset of $\kappa$.
Members of the families may have arbitrary rank unless stated otherwise.

\begin{remark}[Ground models]\label{rem:ground-convention}
Forcing lemmas are stated over arbitrary ZFC grounds, with their
cardinal-arithmetic assumptions explicit. Every definability theorem
stated over $L$ remains valid over an original ZFC ground $W$ satisfying
$\mathsf{GCH}+\mathsf{GA}+(V=\operatorname{HOD})$.
These are the only features of $L$ used. Here GA is the Ground Axiom.
By \cref{lem:stable-ground-codes}, these assumptions give definitions
of the ground and its objects that remain valid in further set-forcing
extensions. Intermediate grounds need not satisfy
GA or $V=\operatorname{HOD}$. Statements over $L[a]$ similarly permit
such a ground with a fixed ground parameter $a$. Several theorems
explicitly weaken or dispense with GCH.
\end{remark}

\subsection{Small definable families}

If $c$ is Cohen over $L$, every countable $\OD(a)$ family in $L[c]$
has an $\OD(a)$ bijective enumeration, even for a new set of ordinals
$a\in L[c]$ (\cref{thm:one-cohen}). This also gives uniformization
of countable sections with real first coordinates. One random real
gives the corresponding enumeration theorem for ground parameters
(\cref{thm:one-random}). For regular uncountable $\kappa$ in $L$,
one Cohen subset of $\kappa$ gives OD enumerations of OD families
of size at most $\kappa$
(\cref{thm:single-cohen-od}).

For every infinite $\mu$, a $\Coll(\omega,\mu)$-extension of $L[a]$
makes every countable $\OD(a)$ family pointwise $\OD(a)$.
More precisely, it has an $\OD(a)$ bijection from a ground cardinal
$\nu\leq\mu$ (\cref{cor:countable-collapse-od}). There is no rank
or cofinality restriction. The indexing ordinal may become countable;
an OD enumeration indexed by $\omega$ need not exist.

The full L\'evy collapse gives a different small-family theorem.
Let $W\models\mathsf{ZFC}$, let $\kappa$ be inaccessible in $W$,
and let $G\subseteq\Coll(\omega,{<}\kappa)^W$ be generic.
In $W[G]$, every $\ODR$ family representing fewer than
$2^{\mathfrak c}$ classes modulo null sets consists of measurable
sets; modulo meager sets, every member has the Baire property
(\cref{thm:friedman-family}). In particular, an $\ODR$ class
modulo either ideal has a Borel representative
(\cref{cor:friedman-class}). Friedman obtained these conclusions
in a further extension and asked whether the single-class assertion
already holds in the original Solovay extension \cite{FriedmanNull}.
Our theorem gives both conclusions there, over an arbitrary ground
with an inaccessible cardinal.

After adding $\kappa$ ordinary Cohen reals to $L$, for any
uncountable $\kappa$,
\[
 A\in\ODlt{\kappa},\quad |A|\leq\kappa
 \quad\Longrightarrow\quad A\subseteq\ODlt{\kappa}
 \qquad
 (\cref{thm:regular-cohen,cor:scf-countable-od,thm:ssf-cohen-endpoint}).
\]
At singular $\kappa$, families of size less than $\kappa$ have one
$\ODlt{\kappa}$ bijective enumeration, without any cardinal-arithmetic
hypothesis (\cref{thm:scf-all-singular}). At countable cofinality,
OD families of size at most $\kappa$ have OD enumerations
(\cref{cor:scf-countable-od}). At uncountable cofinality,
the assertion for individual members requires only strong limitness
in place of GCH.

For random forcing of uncountable Maharam type $\kappa$, the bounds
differ. At singular countable cofinality, OD families of size at most
$\kappa$ have OD enumerations (\cref{thm:singular-random-od}).
At uncountable cofinality, every $\ODlt{\kappa}$ family of size
strictly below $\kappa$ has an $\ODlt{\kappa}$ enumeration if
$\kappa$ is singular strong limit, or regular with
$\kappa^{<\kappa}=\kappa$
(\cref{thm:ssf-random-enumeration,cor:regular-random-enumeration}).
An OD family of size $(\kappa^{\aleph_0})^L$ with no
$\ODlt{\kappa}$ member proves sharpness over $L$:
its size is $\kappa^+$ at countable cofinality and $\kappa$
at uncountable cofinality (\cref{thm:higher-random}).

For regular uncountable $\kappa$, in the extension by
$\Add(\kappa,\kappa^+)^L$, every
$\mathrm{OD}_{H_{\kappa^+}}$ family of size at most $\kappa^+$
is pointwise $\mathrm{OD}_{H_{\kappa^+}}$.
The uniform ultrafilters on $\kappa$ form a counterexample of
size $\kappa^{++}$; this sharpness argument works over every
ZFC+GCH ground. More general cardinal hypotheses for the positive
result are given in \cref{sec:generalized-small-families}.

\subsection{Arithmetic and inner models}

In the extension of $L$ by $\omega_1$ Cohen or random reals,
no $\ODR$ model of $I\Delta_0$ of size at most $\aleph_1$
has full binary standard system
(\cref{thm:arithmetic-negative,thm:random-arithmetic-negative}).
For $I\Delta_0$, binary standard systems use residues modulo the
standard numerals $2^{n+1}$; total exponentiation is not assumed.
The forcing-theoretic proofs pass to canonical exponential cuts
(\cref{subsec:bounded-arithmetic-cuts}); EFA is a tool inside those
proofs, while the metatheory is ZF. A direct Borel argument applies
to a fixed finite subtheory
$F_{\mathrm{Bor}}\subseteq I\Delta_0$ such that
$\Pi^1_1\text{-}\mathsf{CA}_0$ proves that no Borel quotient
$F_{\mathrm{Bor}}$-model realizes all traces for one fixed bounded
binary-digit formula
(\cref{thm:borel-finite-fragment}). Because $F_{\mathrm{Bor}}$ is
fixed and finite, this is one sentence of second-order arithmetic;
it uses no finite axiomatizability assumption for $I\Delta_0$.

Let $\kappa$ be regular uncountable and
$V=L[G]$, where $G\subseteq\Add(\kappa,\Lambda)^L$ is generic
and $\Lambda>\kappa$ is any ground cardinal. Then
$S^*=\operatorname{HOD}_{H_{\kappa^+}}^V$ is a ZF inner model,
closed under ambient sequences of length at most $\kappa$,
containing $L(\mathcal P(\kappa)^V)$, and containing no uniform
ultrafilter on $\kappa$. It has no ambiently
$\kappa^+$-saturated $I\Delta_0$ model, of any size
(\cref{thm:generalized-pa-obstruction}). Every completion of
$I\Delta_0$ nevertheless has a $\kappa$-saturated model of size
$\kappa$ in the smaller inner model. At $\Lambda=\kappa^+$,
the small-family theorem rules out definable presentations of
$\kappa^+$-saturated models of size at most $\kappa^+$.

Consequently, for every infinite regular $\kappa$ of $L$ there
is a cardinal- and cofinality-preserving GCH extension in which
saturated arithmetic models of size $\kappa^+$ exist but have
no $\mathrm{OD}_{\mathcal P(\kappa)}$ copies
(\cref{cor:uniform-definable-saturation-obstruction}).
ZFC proves that every singular strong
limit $\kappa$ admits an OD $\kappa^+$-saturated elementary
extension of $\mathbb N$ of size $2^\kappa$
(\cref{thm:definable-saturation}). Under GCH its size is
$\kappa^+$, so the regularity restriction is exact.

The collapse theorem also applies to the relative Feferman--Levy
model $M_0=W_0(\mathcal R,\mathcal A)$, where $\mathcal A$ lists
the finite-stage real sets and $\mathcal R$ is their union.
It equals the corresponding ambient hereditary-definability model
(\cref{thm:fl-relative-identification}). Its reals are exactly
$\mathcal R$, a countable union of countable sets internally;
it satisfies $\mathsf{AC}^{\mathrm{fin}}_\omega$, and every
infinite set in it has a countably infinite subset.

\subsection{Proof structure}

To prove the small-family theorems, we first find minimal nonempty
invariant subfamilies and then show that each has one member. For one
Cohen real, this uses Kuratowski--Ulam and generic orbit comparison
\cite{SWW,CohenNotes}. For generalized Cohen forcing, we conjugate
small automorphism groups into piecewise translations. For random
forcing, we also compare equivalent probability measures. At singular
countable cofinality, commuting coordinate automorphisms give invariant
enumerations. At higher cofinality, we preserve the maps already
constructed and use a fixed chain of proper factors to eliminate
constant comparisons. At singular strong limits, the same small-index
construction on the forcing coordinates works for Cohen and random
forcing. The collapse proof compares three factors and uses
variable-depth prefix codes.
For the full L\'evy collapse, relative homogeneity lets us extend
isomorphisms onto whole bounded initial algebras. Random or
Cohen witnesses then give a binary tree of automorphisms whose
branches produce $2^{\mathfrak c}$ inequivalent members of any
definable family containing a member without the relevant regularity
property.

Definition-code lemmas transfer the arithmetic problems to hereditary
inner models. Category or probability traces and uniformization then
produce small coinitial families above arithmetic cuts, contradicting
full standard systems or saturation. The positive saturation theorem
uses all codes for small ultrafilter constructions in iterated
finite-support ultrapowers. It does not require a definable choice
of one such code.
For Borel quotient presentations, analytic separation makes the
remainder decoder Borel, while the Harrington--Marker--Shelah
representation and a coded lexicographic argument supply the required
countable coinitial family.


\section{Preliminaries}\label{sec:names}

All Boolean algebras used for forcing are complete, and their embeddings
preserve arbitrary joins. Computations involving names,
automorphisms, and Boolean products take place in the ground model $W$.
Write $\|\varphi\|_\B$ for a Boolean truth value and
$\B\restr b=\{a\in\B:a\leq b\}$, with unit $b$.
For Boolean-valued forcing and the intermediate model theorem see
\cite{Jech}; our category conventions are those of \cite{Kechris}.

\begin{definition}
A $\B$-name $\tau$ is \emph{invariant} if
$\one\Vdash g\tau=\tau$ for every $g\in\Aut^W(\B)$.
More generally, it is fixed by a subgroup if these equalities hold for
that subgroup. Invariance always means forced equality, not literal
equality of names.
\end{definition}

Two elementary observations will be used repeatedly. First, if $g$ is
the identity on $\B\restr b$, induction on names gives
$b\Vdash g\tau=\tau$ for every name $\tau$. Indeed
$b\wedge g(a)=b\wedge a$ for every coefficient $a$, and the induction
applies to the names attached to those coefficients. Second, if an
invariant name is forced finite or countable and the full automorphism
group has fixed algebra $\{0,1\}$, its cardinality is decided by $1$.
The Boolean values of its possible cardinalities are fixed elements.

We use the same-extension theorem of Vop\v enka--H\'ajek and Grigorieff:
if $G,H$ are $W$-generic filters on a complete Boolean algebra $\B\in W$
and $W[G]=W[H]$, then $H=gG$ for some $g\in\Aut^W(\B)$
\cite[Theorem~3.5.1]{Grigorieff}; see also
\cite[Theorem~2.16]{Smythe}. Recall the covariance identity
$(g\tau)^{gG}=\tau^G$.

\begin{lemma}[Stable ground codes]\label{lem:stable-ground-codes}
Suppose $W\models\mathsf{ZFC}+\mathsf{GA}+(V=\operatorname{HOD})$.
In every set-forcing extension $V$ of $W$, the class $W$ is
parameter-free definable and every member of $W$ is OD.
If $W\subseteq W[a]\subseteq V$ is an intermediate model generated
by a set of ordinals $a$, then $W[a]$ and a set-like well-order of
its universe are definable in $V$ from $a$ and ordinals.
\end{lemma}
\begin{proof}
Usuba's downward-directed grounds theorem \cite{Usuba} implies that
$W$ is the mantle of $V$. Indeed, any other ground of $V$ has a
common ground below it and $W$; the intermediate model theorem
makes this a ground of $W$, and GA makes it equal to $W$.
The mantle has a uniform parameter-free definition. Relativizing
the canonical HOD well-order to this class gives ordinal codes for
all members of $W$, including forcing notions and names.

For the intermediate model, fix a $W$-name for $a$ and use the
complete subalgebra generated by its Boolean membership values.
Its generic trace is recoverable from $a$ and the ground name;
the intermediate model theorem identifies its extension with $W[a]$.
Order its names by the ground order and give each value its least
name code. This defines the asserted well-order from $a$ and ordinals.
The same argument applies in every further set-forcing extension.
No Ground Axiom or HOD assumption on $W[a]$ is required.
\end{proof}

\begin{lemma}[Invariant names give ordinal definitions]\label{lem:invariant-od}
Let $W\models\mathsf{ZFC}$, let $\B\in W$ be a complete Boolean
algebra, and let $V=W[G]$. Suppose that $a\in W$ is a set of
ordinals, $W$ is definable in $V$ from $a$ and ordinals, and
$\B,\tau\in W$ are $\OD(a)$ in $V$.
If $\tau$ is invariant, then $\tau^G$ is $\OD(a)$ in $V$.
\end{lemma}
\begin{proof}
In $V$, use the given definitions to form the set of values
\[
 \{\tau^H:H\subseteq\B\text{ is }W\text{-generic and }W[H]=V\}.
\]
This definition is first-order: genericity quantifies over dense sets
in the definable class $W$, and the last condition says that every
set is the value of a ground-model name under $H$. The displayed
collection is a set, by Separation on $\mathcal P(\B)$ followed by
Replacement. The same-extension theorem gives $H=gG$, and invariance gives
$\tau^H=(g\tau)^{gG}=\tau^G$. Thus its unique member is $\OD(a)$.
\end{proof}

The hypotheses hold for $W=L[a]$ by its canonical well-order.
They also hold for the intermediate grounds of
\cref{lem:stable-ground-codes}, with the indicated parameter.

We obtain invariant names from definitions as follows. Suppose the full automorphism group of $\B$ has fixed algebra
$\{0,1\}$, and in $W[G]$ a formula with fixed ground parameters and ordinals
defines a unique nonempty countable set. The Boolean value of this
assertion is fixed and belongs to $G$, so it is $1$. Choose a name
forced to satisfy the definition. Uniqueness makes that name invariant.
The same argument applies to a finite family or to any other property
expressed using fixed parameters.

\begin{lemma}[Quotient invariance]\label{lem:hc-quotient-invariance}
Let $C\subseteq B$ be complete in a ground $W$, let $F$ be
$B$-generic, and put $U=W[F\cap C]$. If a ground $B$-name
$\tau$ is invariant under the pointwise stabilizer
$G_C=\{g\in\Aut^W(B):\,g\restr C=\id\}$, its quotient interpretation over
$U$ is invariant under all automorphisms of the quotient completion
computed in $U$.
\end{lemma}
\begin{proof}
Use the two-step presentation of $B$ as $C$ followed by its quotient,
and let $\dot Q$ name that quotient completion. Given a quotient
automorphism $\pi\in U$, choose a $C$-name and a condition in
$F\cap C$ forcing it to be an automorphism. Mix the name with the
identity outside that condition, so
$1_C\Vdash\dot\pi\in\Aut(\dot Q)$ and its actual value is $\pi$.
On the dense iteration presentation, the map
\[
 (d,\dot q)\longmapsto(d,\dot\pi(\dot q))
\]
is an automorphism, with inverse defined by $\dot\pi^{-1}$.
Complete it and identify the iteration with $B$. This gives a ground
$g\in G_C$ whose quotient action is $\pi$. The equality
$1_B\Vdash g\tau=\tau$ therefore yields the corresponding forced
equality in the quotient. This covers every automorphism in $U$;
it does not assume that an old completion remains complete there.
\end{proof}

\section{Countable families in a Cohen extension}\label{sec:cohen}

\begin{theorem}[Invariant enumeration for Cohen forcing]\label{thm:cohen-invariant}
In any ground model $W$ of ZFC, let $\B=\RO(2^\omega)^W$. Every
invariant name forced to be a nonempty countable set has an invariant
bijective enumeration by some $I\in(\omega\setminus\{0\})\cup\{\omega\}$.
The ranks of its members are unrestricted.
\end{theorem}

The proof has two steps. We first find minimal nonempty invariant
subfamilies. We then show that each such subfamily has exactly one
member. The first step uses Kuratowski--Ulam; the second uses the
generic orbit-equivalence theorem of Sullivan--Weiss--Wright.

\subsection{Category and orbit comparison}

\begin{lemma}\label{lem:countable-fixed-algebra}
If a group $H$ acts on a complete Boolean algebra $R$ with a countable
order-dense subset, then $R^H$ has a countable order-dense subset.
Moreover $R^{H_0}=R^H$ for some countable subgroup $H_0\leq H$.
\end{lemma}
\begin{proof}
Let $(u_n)_{n<\omega}$ be order-dense in $R^+$ and put
$v_n=\bigvee_{h\in H}h(u_n)$. If $0<d\in R^H$ and $u_n\leq d$,
then $v_n\leq d$, so the $v_n$ are order-dense in $R^H$. The algebra
$R$ is ccc. For each $n$ choose countably many $h$ whose images of
$u_n$ have join $v_n$, and let $H_0$ be the subgroup they generate.
If $d\in R^{H_0}$, then
$d=\bigvee\{v_n:u_n\leq d\}$, so $d\in R^H$.
\end{proof}

\begin{lemma}[A countable collection of membership patterns]\label{lem:ku-patterns}
Let $I$ be nonempty and countable, let $r_i:2^\omega\to2^\omega$ be
Borel, and put $S(x)=\{r_i(x):i\in I\}$. If
$S(x\mathbin\oplus a)=S(x)$ on a comeager set for each $a\in2^\omega$,
then $S(x)$ has a fixed countable value on a comeager set.
\end{lemma}
\begin{proof}
The relation $E(x,y)$ asserting $S(x)=S(y)$ is Borel. By
Kuratowski--Ulam, $E(x,x\oplus a)$ holds on a comeager subset of the
product. The map $(a,x)\mapsto(x,x\oplus a)$ is a homeomorphism,
so $E$ itself is comeager. Another application of Kuratowski--Ulam
gives $x_*$ such that $S(x)=S(x_*)$ for comeager many $x$.
\end{proof}

An isomorphism between principal ideals is \emph{piecewise given by}
a group $H$ of automorphisms if it agrees with members of $H$ on the
pieces of a countable Boolean partition of its domain. Its inverse
has the same property, and compositions do too: refine the first
partition by inverse images of the second.

\begin{lemma}[Generic orbit comparison]\label{lem:sww-comparison}
Let $R,S$ be atomless complete Boolean algebras with countable
order-dense subsets. Let $H\leq\Aut(R)$ and $K\leq\Aut(S)$ be
countable, with $R^H=\{0,1\}$ and $S^K=\{0,1\}$. There is an
isomorphism $\alpha:R\to S$ such that every $\alpha h\alpha^{-1}$
is piecewise given by $K$.
\end{lemma}
\begin{proof}
Choose countable invariant order-dense Boolean subalgebras. Their
Stone spaces are Cantor spaces. The induced actions are by
homeomorphisms and are generically ergodic: an invariant Borel set
has a fixed class in the category algebra, hence is meager or
comeager. Sullivan--Weiss--Wright \cite[Theorem~1.8]{SWW} gives an
orbit equivalence through a homeomorphism between invariant dense
$G_\delta$ subspaces. To obtain this two-space formulation, compare
each action to the canonical finite-bit-flip action and intersect
the two invariant comeager subspaces there.

Direct image induces a complete Boolean isomorphism $\alpha$.
Enumerate $K=(k_n)_{n<\omega}$. For each $h\in H$, partition the
target subspace according to the least $n$ for which its conjugate
under the orbit equivalence agrees pointwise with $k_n$. These sets
are Borel. Their nonzero category classes give the required partition.
\end{proof}

\subsection{The Boolean algebra of subfamilies}

We first work with Boolean algebras alone. Put
$G=\Aut(\B)$ and $C=\B^I$, the complete Boolean \emph{direct product}.
This is a lottery of copies of $\B$, not forcing with independent
Cohen generics. Write $\delta(b)=(b)_{i\in I}$ and let $e_i$ be the
unit of its $i$th slice. For $z\in C$ put
\[
 s(z)=\bigvee_{i\in I}z_i,\qquad
 \iota_z(b)=\delta(b)\wedge z\quad(b\leq s(z)).
\]
The map $\iota_z:\B\restr s(z)\to C\restr z$ is a complete
embedding. It is onto exactly when the coordinates of $z$ are
pairwise disjoint; in that case its inverse is $s$. For the converse,
if $z\wedge e_i=\iota_z(b_i)$, then $z_i\leq b_i$ and
$b_i\wedge z_j=0$ for $j\ne i$.

Suppose $\Theta:G\to\Aut(C)$ is an action satisfying
\begin{align}
 \Theta_g\delta(b)&=\delta(gb),\label{eq:cohen-diagonal}\\
 g\restr(\B\restr b)=\id
 &\ \Longrightarrow\ 
 \Theta_g\restr(C\restr\delta(b))=\id.\label{eq:cohen-locality}
\end{align}
\begin{lemma}[Full quotient actions]\label{lem:full-quotient-action}
Let $A\subseteq B$ be complete Boolean algebras in a ZFC ground,
and let $I$ be any ground set. After forcing with $A$, a full local
diagonal-equivariant action on $B^I$ induces such an action on
$Q^I$, where $Q$ is the quotient completion computed in the
intermediate extension. It includes every automorphism of that
completion. For a product probability algebra and a coordinate
factor $A$, the assertion also holds for the measure-preserving groups.
\end{lemma}
\begin{proof}
Use $B=A*\dot Q$. Every $A$-name for a $Q$-value has a
representative in $B$; coordinatewise the same holds for $Q^I$
and $B^I$. For $b,c\in B$ their relative equality has $A$-value
\[
 \bigvee\{a\in A:a\wedge b=a\wedge c\};
\]
for tuples take the meet over $I$. Diagonal equivariance makes
the lifted maps respect this equality.

The lifting in \cref{lem:hc-quotient-invariance} represents every
named quotient automorphism by a ground automorphism fixing $A$,
mixing with the identity where necessary. If two operator names
agree below $a\in A$, their lifts agree on $B\restr a$; locality
makes their cover actions agree below $\delta(a)$. Thus these
actions define one full relative group action. The same argument
on a relative principal algebra proves its locality, and diagonal
equivariance descends directly. For product probability algebras,
a conditionally measure-preserving quotient map has a globally
measure-preserving lift by integration. No bound on $I$ or
restriction to ground-model quotient operators was used.
\end{proof}

The least diagonal element above $z$ is $\delta(s(z))$. Therefore
\[
 s(\Theta_gz)=g(s(z)),\qquad
 \Theta_g\iota_z(b)=\iota_{\Theta_gz}(gb).
\]
In particular, the action preserves the property of having disjoint
coordinates. Put $D=C^{\Theta[G]}$. Every nonzero $d\in D$ has
$s(d)=1$, since $\B^G=\{0,1\}$. To see the latter assertion, finite
bit flips can carry a basic cylinder below any $0<b<1$ to a cylinder
below its complement.

\begin{proposition}\label{prop:cohen-fixed-atomic}
The algebra $D$ is atomic.
\end{proposition}
\begin{proof}
By \cref{lem:countable-fixed-algebra}, choose an order-dense sequence
$(v_n)$ in $D^+$. Represent $v_{n,i}$ by a Borel set $V_{n,i}$ and
define $r_i(x)(n)=1$ exactly when $x\in V_{n,i}$.
For the translation $t_a(x)=x\oplus a$ write
$m^a_{ij}=(\Theta_{t_a}e_i)_j$. Each row partitions $1_\B$ by the
disjoint-coordinate observation, and each column partitions $1_\B$
because the $e_i$ partition $1_C$. Since $v_n$ is fixed,
\[
 m^a_{ij}\wedge t_a(v_{n,i})=m^a_{ij}\wedge v_{n,j}.
\]
Taking all $n$ shows that $\{r_i(x\oplus a):i\in I\}$ equals
$\{r_i(x):i\in I\}$ on a comeager set. By \cref{lem:ku-patterns}
there is a countable set $S_*$ of patterns such that this collection
equals $S_*$ for comeager many $x$.

For $t\in S_*$ let $a_t=\bigwedge_n v_n^{t(n)}$, where
$v_n^1=v_n$ and $v_n^0=\neg_Cv_n$. Coordinatewise this is the
category class of $\{x:r_i(x)=t\}$. Hence the $a_t$ are disjoint
and have join $1_C$. If $0<z\leq a_t$ belongs to $D$, choose
$0<v_n\leq z$. The element $a_t$ decides $v_n$, so it must lie
below $v_n$. Thus $z=a_t$, and every nonzero $a_t$ is an atom.
\end{proof}

Atomicity uses only diagonal equivariance. We next use locality
to show that each minimal invariant subfamily is a singleton.

\begin{lemma}\label{lem:cohen-selector-factor}
If $0<d\in D$ and $\alpha:\B\to C\restr d$ is an isomorphism,
there are $q\leq d$, whose coordinates partition $1_\B$, and an
isomorphism $U:C\restr q\to C\restr d$, piecewise given by
$\Theta[G]$, such that $\alpha=U\iota_q$.
\end{lemma}
\begin{proof}
Split each nonzero $\alpha^{-1}(d\wedge e_i)$ into two positive
pieces and enumerate the resulting partition as $(p_n)$. Let $i_n$
be the slice containing $\alpha(p_n)$. The map
$b\mapsto\alpha(b)_{i_n}$ is an isomorphism between principal
ideals of $\B$. Both units are proper and nonzero, because of the
splitting. Both complementary ideals are Cohen algebras, so the
map extends to $g_n\in G$. Thus, for $b\leq p_n$,
\[
 \alpha(b)=e_{i_n}\wedge\delta(g_nb).
\]
Put $u_n=\Theta_{g_n^{-1}}\alpha(p_n)$ and $q=\bigvee_nu_n$.
Each $u_n$ has disjoint coordinates and support $p_n$. Consequently
the coordinates of $q$ partition $1_\B$. Also $q\leq d$, since
$d$ is fixed. The formula
$U(z)=\bigvee_n\Theta_{g_n}(z\wedge u_n)$ gives the required
isomorphism. The displayed identity shows $U\iota_q=\alpha$
on each $p_n$, hence everywhere.
\end{proof}

\begin{theorem}\label{thm:cohen-fixed-singletons}
Every atom of $D$ has coordinates partitioning $1_\B$. There are
exactly $|I|$ atoms, and their matrix has every row and every column
a partition of $1_\B$.
\end{theorem}
\begin{proof}
Choose a countable $\Gamma\leq G$ with $C^{\Theta[\Gamma]}=D$.
The diagonal embedding shows $\B^\Gamma=\{0,1\}$. For an atom
$d\in D$, the action on $C\restr d$ also has trivial fixed algebra.
Both algebras are atomless and countably order-dense. Apply
\cref{lem:sww-comparison} to obtain $\alpha:\B\to C\restr d$
such that $\alpha\gamma\alpha^{-1}$ is piecewise given by
$\Theta[\Gamma]$. By \cref{lem:cohen-selector-factor}, write
$\alpha=U\iota_q$. Then
\[
 V_\gamma=U^{-1}\alpha\gamma\alpha^{-1}U
           =\iota_q\gamma\iota_q^{-1}
\]
is piecewise given by $\Theta[G]$.

On a piece $u=\iota_q(b)$ where $V_\gamma$ agrees with $\Theta_g$,
take $a\leq b$. Taking supports in
$\Theta_g\iota_q(a)=\iota_q(\gamma a)$ gives $g(a)=\gamma(a)$.
Locality implies that $\Theta_g$ and $\Theta_\gamma$ agree below
$\delta(b)$, hence below $u$. On all the pieces together,
$V_\gamma=\Theta_\gamma\restr(C\restr q)$, so
$\Theta_\gamma(q)=q$. Thus $q\in D$; since $0<q\leq d$ and
$d$ is an atom, $q=d$.

There are countably many atoms because $C$ is ccc. Their join is $1_C$,
so every column partitions $1_\B$, and the preceding argument gives
the row partitions. These partitions imply the number of rows is
$|I|$. If there are $n$ columns, $n+1$ rows are impossible: expand
the meet of their row joins, and every term has two disjoint entries
in one column. Transposing the matrix gives the reverse inequality
when the number of rows is finite. The countably infinite case follows
as well.
\end{proof}

\subsection{Applying the calculation to names}

\begin{proof}[Proof of \cref{thm:cohen-invariant}]
Choose names $(\tau_i)_{i\in I}$ forced to enumerate the given
family $\dot A$ bijectively. Set
\[
 m^g_{ij}=\|g\tau_i=\tau_j\|_\B,
 \qquad (\Theta_gz)_j=\bigvee_i\bigl(g(z_i)\wedge m^g_{ij}\bigr).
\]
The matrix has partition rows and columns, and
\[
 m^{gh}_{ik}=\bigvee_j\bigl(g(m^h_{ij})\wedge m^g_{jk}\bigr).
\]
Thus $\Theta$ is an action by complete automorphisms and satisfies
\eqref{eq:cohen-diagonal}. The rank induction on names from
\cref{sec:names} gives \eqref{eq:cohen-locality}.

For a name $\sigma$ forced to belong to $\dot A$, the element
$q_\sigma=(\|\sigma=\tau_i\|)_i$ has coordinates partitioning $1$,
and $\Theta_gq_\sigma=q_{g\sigma}$. Enumerate the atoms of $D$
as $(d_k)_{k\in I}$. Mix the names $\tau_i$ along the $k$th row:
$d_{k,i}\Vdash\sigma_k=\tau_i$. Then $q_{\sigma_k}=d_k$.
Since this atom is fixed, $\sigma_k$ is invariant. Disjointness
of the columns gives pairwise distinctness of the $\sigma_k$, and
the column joins show that every member of the family occurs. The enumeration name is
itself invariant.
\end{proof}

In this calculation, $z=(z_i)\in\B^I$ codes the subfamily in which $\tau_i$ is present with Boolean value $z_i$.
An atom of $D$ is a minimal nonempty invariant subfamily. Its
disjoint coordinates say that exactly one member is selected.

\begin{corollary}\label{cor:cohen-ground-parameter}
If $a$ is a set of ordinals and $c$ is Cohen over $L[a]$, every
countable $\OD(a)$ set in $L[a,c]$ has an $\OD(a)$ bijective
enumeration by a finite ordinal or by $\omega$.
\end{corollary}
\begin{proof}
More generally, use any ground with the definability and ordinal
codes of \cref{lem:invariant-od}; these are supplied by
\cref{lem:stable-ground-codes} in the grounds of the introduction.
The definition gives an invariant name, to which we apply
\cref{thm:cohen-invariant,lem:invariant-od}. The empty enumeration
handles the empty set.
\end{proof}

\begin{lemma}\label{lem:cohen-intermediate}
Let $W\models\mathsf{ZFC}$, let $c$ be Cohen over $W$, and let
$a\in W[c]$ be a set of ordinals. Then either $W[a]=W[c]$ or
$W[c]$ is a Cohen-real extension of $W[a]$.
\end{lemma}
\begin{proof}
By the intermediate model theorem, $W[a]=W[H]$ for the generic
trace on a complete subalgebra $\B_0$ of the Cohen algebra in $W$.
Let $\pi:\B\to\B_0$ take an element to its least upper bound in
$\B_0$, and fix a countable order-dense $P\subseteq\B^+$ in $W$.
The quotient is represented by
$Q=\{b\in\B^+:\pi(b)\in H\}$. For $b\in Q$,
\[
 \pi(b)=\bigvee\{\pi(p):p\in P,\ p\leq b\}.
\]
Genericity for this ground-model join supplies $p\leq b$ in
$P\cap Q$. Thus the quotient completion is countably order-dense.
If its generic meets an atom it adds nothing. Otherwise the generic
lies below the complement of the join of its atoms, a Cohen algebra.
See also \cite[Theorem~4]{KLSigma}.
\end{proof}

\begin{theorem}\label{thm:one-cohen}
In $L[c]$, for $c$ Cohen over $L$ and every set of ordinals
$a\in L[c]$, every countable $\OD(a)$ set has an $\OD(a)$
bijective enumeration. In particular all its members are $\OD(a)$.
\end{theorem}
\begin{proof}
Work more generally with $V=W[c]$, where $W$ satisfies the hypotheses
of \cref{lem:stable-ground-codes}. Put $N=W[a]$.
If $N=V$, its $\OD(a)$ well-order chooses an enumeration.
Otherwise $V$ is a Cohen extension of $N$ by
\cref{lem:cohen-intermediate}. The definition gives an invariant name
for the family. Apply \cref{thm:cohen-invariant} to obtain an invariant
enumeration.
The ordinal-and-$a$ codes in $N$, followed by
\cref{lem:invariant-od}, make that enumeration $\OD(a)$ in $V$.
The empty set has its empty enumeration.
\end{proof}

\begin{corollary}[Uniformization]\label{cor:cohen-uniformization}
In $L[c]$, let $a$ be a set of ordinals and let $R\in\OD(a)$
be a set of ordered pairs with real first coordinates and countable
vertical sections. There is an $\OD(a)$ sequence $(f_n)_{n<\omega}$
of functions on $\operatorname{dom}(R)$ with
$R_x=\{f_n(x):n<\omega\}$. In particular $R$ has an $\OD(a)$
uniformization.
\end{corollary}
\begin{proof}
Each nonempty $R_x$ is $\OD(a,x)$, and $(a,x)$ is coded by a set
of ordinals. Choose the least $\OD(a,x)$ surjection
$e_x:\omega\to R_x$ in the canonical definable well-order of
$\OD(a,x)$, repeating entries for finite sections. This selection
is uniform in $x$. Replacement gives $f_n(x)=e_x(n)$.
\end{proof}

\begin{remark}
For families of reals, the invariant enumeration has an invariant
real code. Each bit has Boolean value $0$ or $1$, so the code belongs
to the ground. This recovers the rank-one Cohen theorem. At arbitrary
rank, invariant values need not lie in the ground; the conclusion is
ordinal definability, supplied by \cref{lem:invariant-od}.
\end{remark}

\section{One generalized Cohen subset}\label{sec:single-cohen}

Throughout this section, $\kappa$ is regular uncountable and
$\kappa^{<\kappa}=\kappa$ in the ZFC ground $W$. Fix a set $\Omega$
of cardinality $\kappa$, and write
\[
 P=\operatorname{Fn}_{<\kappa}(\Omega,2),\qquad B=\RO(P),
 \qquad G=\Aut^W(B).
\]
We identify each condition with its nonzero Boolean value, which we call
\emph{basic}. All constructions in this section take place in $W$.
The group $\Gamma$ consists of translations $\tau_u(x)=x\mathbin\triangle u$
for $u\in[\Omega]^{<\kappa}$, and $\Sigma=\Sym(\Omega)$ acts by
permuting coordinates. The group $\Gamma$ has cardinality $\kappa$
and acts weakly homogeneously on $B$. Write $[\Gamma]$ for the
group of automorphisms which agree with members of $\Gamma$ on
the pieces of a Boolean partition of $1$. Such partitions have
size at most $\kappa$, since $B$ has order density $\kappa$.

\begin{theorem}[Invariant enumeration]\label{thm:single-cohen-invariant}
Every invariant $B$-name forced to be a nonempty set of cardinality
at most $\kappa$ has an invariant bijective enumeration by a ground
cardinal $\nu\leq\kappa$. The ranks of its members are unrestricted.
\end{theorem}

\begin{theorem}\label{thm:single-cohen-od}
Let $\kappa$ be regular uncountable in $L$ and let
$c\subseteq\kappa$ be $\Add(\kappa,1)^L$-generic. In $L[c]$, every
OD family of cardinality at most $\kappa$ has an OD bijective
enumeration by an ordinal at most $\kappa$. In particular, every
member of the family is OD.
\end{theorem}

We prove the invariant-name theorem first. For
$0<|I|\leq\kappa$, put $C=B^I$. Use the diagonal embedding $\delta$,
support $s$, and maps $\iota_z$ from \cref{sec:cohen}. A
\emph{selector} is $q\in C$ whose coordinates partition $1_B$;
equivalently, $s(q)=1$ and $\iota_q:B\to C\restr q$ is an
isomorphism. A \emph{frame} is a partition of $1_C$ into selectors.
Suppose $\Theta:G\to\Aut(C)$ satisfies diagonal equivariance and
locality, as in \eqref{eq:cohen-diagonal} and
\eqref{eq:cohen-locality}.
For a subgroup $H\leq G$, write $C^H$ for its fixed algebra under
$\Theta$. We will prove that the atoms of $C^G$ form a frame.

Coordinate permutations and short translations first give a frame in
$C^\Gamma$. We then compare small groups of Boolean automorphisms
to show that all of $G$ fixes this frame.

\subsection{Atomicity of the fixed algebra}

The following permutation-kernel calculation will also be used at
singular width.

\begin{lemma}\label{lem:boolean-commuting-bound}
If a Boolean algebra $D$ has order density at most an infinite
cardinal $\lambda$, then $\Aut(D)$ has at most $\lambda$
pairwise elementwise-commuting nonabelian subgroups.
\end{lemma}
\begin{proof}
Fix an order-dense $P\subseteq D^+$ of size at most $\lambda$.
For noncommuting $f,g$, choose $d>0$ with
$gf(d)\wedge fg(d)=0$, then choose successively
\[
 b\in P,\ b\leq g(d),\qquad
 p\in P,\ p\leq d\wedge g^{-1}(b),\qquad
 a\in P,\ a\leq f(p),\qquad q\in P,\ q\leq g(a).
\]
Thus $g(p)\leq b$, $a\leq f(p)$, $q\leq g(a)$, and
$f(b)\wedge q=0$. Assign such a quadruple to a noncommuting
pair in each subgroup. If $(f,g)$ and $(f',g')$ receive the
same quadruple, then $q\leq g'f(p)$ whereas
$q\wedge fg'(p)=0$, so $f$ and $g'$ do not commute.
There are at most $\lambda$ quadruples.
\end{proof}

\begin{lemma}[Branch permutation kernel]\label{lem:branch-permutation-kernel}
\label{lem:single-cohen-small-kernel}\label{lem:scf-permutation-kernel}
Let $G$ have at most $\lambda$ pairwise elementwise-commuting
nonabelian subgroups, where $\lambda$ is infinite.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $\operatorname{cf}(\kappa)=\omega<\kappa$ and
$\lambda\leq\kappa$, every homomorphism $\Sym(\kappa)\to G$
killing finitary permutations kills countably supported permutations.
\item If $\lambda<2^{\aleph_0}$, every homomorphism
$\Sym(\omega)\to G$ killing finitary permutations is trivial.
\item If $\kappa$ is regular uncountable,
$\kappa^{<\kappa}=\kappa$, and $\lambda\leq\kappa$, every
homomorphism $\Sym(\kappa)\to G$ killing permutations of
support less than $\kappa$ is trivial.
\end{enumerate}
These conclusions apply to $G=\Aut(D)$ whenever $D$ has order
density at most $\lambda$, and to $G=\Sym(J)$ when $|J|\leq\lambda$.
\end{lemma}
\begin{proof}
For the three cases use the trees $\kappa^{<\omega}$, $2^{<\omega}$,
and $2^{<\kappa}$, respectively. Identify the coordinate set with five
copies of the tree together with a reserve of the same cardinality.
For each branch, repeat the same copy of $A_5$ on the five copies of
every node. Distinct branch copies have commuting images:
their commutators are supported on their common nodes, finitely
many in (i)--(ii) and fewer than $\kappa$ in (iii).
The number of branches exceeds $\lambda$, by K\"onig's theorem
in (i) and Cantor's theorem in the other cases. Simplicity of
$A_5$ and the hypothesis on $G$ therefore put one branch copy
in the kernel.

Its double transposition is an involution with $\theta$ pairs
and as many fixed points as the coordinate set, where
$\theta=\omega$ in (i)--(ii) and $\theta=\kappa$ in (iii).
The normal closure of any such involution contains every
involution with at most $\theta$ pairs. For one with exactly
$\theta$ pairs, divide the pairs into two equally large parts;
both restrictions are conjugate to the given involution.
For one with fewer pairs, multiply by a disjoint conjugate
on its fixed region, leaving a reserve of fixed points; the
product is another conjugate. Finally, every permutation is a
product of two involutions, by reflections on its finite cyclic
and bilateral infinite orbits. This proves all three assertions.
For the final assertion use \cref{lem:boolean-commuting-bound},
taking $D=\mathcal P(J)$ in the permutation case.
\end{proof}


\begin{lemma}\label{lem:single-cohen-fixed-atomic}
Let $\kappa$ be regular uncountable with
$\kappa^{<\kappa}=\kappa$, let $B=\RO(\Add(\kappa,1))$, and let
$C=B^I$, where $0<|I|\leq\kappa$. Let $\Gamma$ be the translations
with support less than $\kappa$ and let $\Sigma=\Sym(\kappa)$
act by coordinate permutations. Suppose $\Theta$ is a local
diagonal-equivariant action of $\Gamma\rtimes\Sigma$ on $C$.
Then $C^\Gamma$ is atomic with at most $\kappa$ atoms, and
$\Sigma$ fixes it pointwise.
\end{lemma}
\begin{proof}
Identify the coordinate set $\Omega$ with $\kappa$.
Write $\delta(b)=(b)_{i\in I}$ and
$s(c)=\bigvee_{i\in I}c_i$. The scalar $\Gamma$ action is weakly
homogeneous, since a bounded translation makes any two basic conditions
compatible. Thus every nonzero $\Gamma$-fixed element of $C$
has support $1$.

Put $D=C^\Gamma$. Choose a dense sequence
$(u_\alpha:\alpha<\kappa)$ in $C^+$ and put
\[
 v_\alpha=\bigvee_{\gamma\in\Gamma}\Theta_\gamma(u_\alpha).
\]
These elements are order-dense in $D$, since $0<u_\alpha\leq d\in D$
implies $0<v_\alpha\leq d$. Hence $D$ has order density at most
$\kappa$. The group $\Sigma$ normalizes $\Gamma$, and therefore
acts on $D$. If $\sigma\in\Sigma$ has short support, it is the
identity on the positive cone assigning zero to its support.
For $d\in D$, locality makes
$d\mathbin{\triangle}\Theta_\sigma(d)$ vanish on this cone.
That difference is $\Gamma$-fixed, so its support cannot be $1$
unless the difference is zero. Every such $\sigma$ therefore
fixes $D$ pointwise. Lemma~\ref{lem:single-cohen-small-kernel}
makes the entire $\Sigma$ action on $D$ trivial.

To prove atomicity, we use only the fact that $\Sigma$ fixes $D$
pointwise. Set
$X=2^\kappa$ with its bounded topology. Here a
$\kappa$-meager set is a union of at most $\kappa$ nowhere-dense
sets, and $\kappa$-Borel means generated from open sets by
complements and unions of at most $\kappa$ sets. These sets
have the $\kappa$-Baire property. Their quotient modulo
$\kappa$-meager sets is $B$, respecting joins and meets of at
most $\kappa$ elements.

Represent the $i$th coordinate of $v_\alpha$ by its regular-open
set $V_{\alpha i}\subseteq X$, and define
\[
 r_i(x)(\alpha)=1\ \Longleftrightarrow\ x\in V_{\alpha i},
 \qquad S(x)=\{r_i(x):i\in I\}.
 \tag{**}
\]
For each fixed $\sigma\in\Sigma$, diagonal equivariance means
that $\Theta_\sigma$ acts by a Boolean-valued permutation of
the sheets over the scalar action of $\sigma$. Its rows and
columns partition $1$. Since all $v_\alpha$ are fixed, these
partitions identify the patterns in (**) at $x$ and $\sigma x$
off a $\kappa$-meager set. Only $\kappa$ partition and coordinate
identities are involved for this fixed $\sigma$. Consequently
\[
 S(\sigma x)=S(x)\quad\hbox{for $\kappa$-comeager many }x.
 \tag{***}
\]
This argument does not assert uniform regularity of the coefficients
as a function of $\sigma$.

We show that (***), for all $\sigma$, makes $S$ constant on
a $\kappa$-comeager set. Give $\Sigma$ its topology of short
partial-bijection neighborhoods. This space, $X$, and their
finite products have bases of size $\kappa$ and are
$\kappa$-Baire. A recursion of length $\kappa$ meets
$\kappa$ dense open requirements. For a permutation, include
requirements that put every coordinate in its domain and range.
The union at each proper stage still has size less than $\kappa$.

The Kuratowski--Ulam argument applies with these bases.
For a closed nowhere-dense subset of a product and each basic
open set in the second factor, the first coordinates admitting
a smaller rectangle disjoint from it form a dense open set.
Intersecting these $\kappa$ sets shows that its sections are
nowhere dense off a $\kappa$-meager set. Taking $\kappa$ unions
gives the assertion for meager sets. In particular, if every section of
a relation with the
$\kappa$-Baire property is comeager, then the relation is comeager.
Otherwise its complement is comeager on a nonempty rectangle,
contradicting the section assertion.

Let $X_{\rm bal}$ consist of the points having $\kappa$ zeros
and $\kappa$ ones. It is dense and $\kappa$-comeager. The map
\[
 \Phi:\Sigma\times X_{\rm bal}\longrightarrow
                 X_{\rm bal}\times X_{\rm bal},
 \qquad \Phi(\sigma,x)=(x,\sigma x)
\]
is a continuous open surjection. In fact the image of
$[t]\times([p]\cap X_{\rm bal})$, for a short partial bijection
$t$, consists exactly of the balanced pairs $(x,y)$ such that
\[
 x\supseteq p,\qquad y(t(i))=x(i)\quad(i\in\operatorname{dom}t).
\]
This set is open. Match the remaining zero and one coordinates
to extend $t$ to the required permutation. This proves both the
description of the image and surjectivity.

The relation $E(x,y)\iff S(x)=S(y)$ is $\kappa$-Borel:
each inclusion has the form
$\forall i\in I\,\exists j\in I\,\forall\alpha<\kappa$
of equality tests from (**). Independently of the lift
coefficients, define $R(\sigma,x)=E(x,\sigma x)$.
It is $\kappa$-Borel by continuity of the coordinate action.
By (***) and the section argument, $R$ is comeager.
On $\Sigma\times X_{\rm bal}$ it is $\Phi^{-1}(E)$.
An open continuous surjection pulls back meager sets to
meager sets. Thus, if the complement of $E$ were comeager
on a nonempty open set, its inverse image would contradict
comeagerness of $R$. Hence $E$ is comeager on
$X_{\rm bal}^2$, and therefore on $X^2$. Applying the section
argument once more, choose $x_*$ with comeager $E$-section.
Then $S(x)=S_*:=S(x_*)$ on a comeager set, and $|S_*|\leq\kappa$.

For $t\in S_*$ put
\[
 a_t=\bigwedge_{\alpha<\kappa}v_\alpha^{\,t(\alpha)},
 \qquad v^1=v,\quad v^0=\neg v.
\]
These elements belong to $D$. Their coordinate representatives
are $\{x:r_i(x)=t\}$ modulo $\kappa$-meager sets, so they are
disjoint and join to $1$. Every nonzero $a_t$ is an atom:
if $0<d\leq a_t$ in $D$, choose $0<v_\alpha\leq d$.
Since $a_t$ decides $v_\alpha$, this decision must be positive. Hence
$a_t\leq v_\alpha\leq d\leq a_t$. This proves atomicity and
the stated bound.
\end{proof}

\subsection{The initial frame}\label{subsec:single-cohen-initial-frame}

We retain the notation $P,B,C,\delta,s,\Gamma,\Sigma,\Theta$.
A \emph{partial selector} has pairwise disjoint
coordinates and is a \emph{selector} if its scalar support is $1$.
Below a partial selector $u$, the support map is a complete isomorphism
\[
 s:C\restr u\longrightarrow B\restr s(u),
 \qquad s^{-1}(b)=\delta(b)\wedge u.
\]
A \emph{frame} is a partition of $1_C$ into selectors. We will show that
the atoms supplied by \cref{lem:single-cohen-fixed-atomic} form a
frame, also when $|I|=\kappa$.

We use the topology on $\Sigma=\Sym(\Omega)$ generated by restrictions
of permutations to fewer than $\kappa$ coordinates. Inverse restrictions
give the same topology. Its basis has size $\kappa$, and intersections
of fewer than $\kappa$ open sets are open. We use the
$\kappa$-Borel and $\kappa$-meager conventions of the preceding
proof; in particular $\kappa$-Borel sets have the $\kappa$-Baire
property.

\begin{lemma}[Uniform permutation factorization]
\label{lem:single-cohen-four-conjugates}
Allow also $\kappa=\omega$, with the usual Borel structure and
finite-restriction topology. Fix an involution
$r\in\Sigma$ having $\kappa$ transposed pairs and
$\kappa$ fixed points. There are $\kappa$-Borel maps
$a_m:\Sigma\to\Sigma$, $m<4$, such that
\begin{equation}
 \sigma=\prod_{m<4}a_m(\sigma)r a_m(\sigma)^{-1},
 \label{eq:single-cohen-four-conjugates}
\end{equation}
with composition from right to left.
\end{lemma}
\begin{proof}
Identify $\Omega$ with $\kappa$. On each $\sigma$-orbit, choose its
least element $o$ and write the orbit as $\sigma^n(o)$, with indices
in $\mathbb Z/d\mathbb Z$ for a finite orbit and in $\mathbb Z$
otherwise. The two reflections
\[
 \beta_\sigma(\sigma^n(o))=\sigma^{-n}(o),
 \qquad
 \alpha_\sigma(\sigma^n(o))=\sigma^{1-n}(o)
\]
are involutions with $\sigma=\alpha_\sigma\beta_\sigma$.
For uncountable $\kappa$, both depend continuously on $\sigma$:
the value at a coordinate is determined by its countable orbit,
and fewer than $\kappa$ such orbits still have size less than
$\kappa$. For $\kappa=\omega$ they are Borel, since orbit
membership and the choice of its least element use only countably
many iterate tests.

Call an involution \emph{balanced} if it has $\kappa$ pairs and
$\kappa$ fixed points. We next factor every involution $\tau$
into two balanced involutions, uniformly $\kappa$-Borel. Put
\[
 L(\tau)=\{\xi<\kappa:\xi<\tau(\xi)\},
 \qquad F(\tau)=\{\xi<\kappa:\tau(\xi)=\xi\}.
\]
The condition $|L(\tau)|=\kappa$ is $\kappa$-Borel, since it says
that $L(\tau)$ is unbounded (infinite when $\kappa=\omega$).
When this holds, enumerate $L(\tau)$
in increasing order and divide its index set into two fixed
$\kappa$-sized parts. Restrict $\tau$ to the pairs in each part,
fixing all other points. These restrictions are balanced, have
disjoint supports, and multiply to $\tau$.

In the other case $|F(\tau)|=\kappa$. Using its increasing
enumeration, divide $F(\tau)$ into three indexed sets of size
$\kappa$, pair the first two sets, and leave the third fixed.
Let $v$ exchange those pairs and fix the moved points of $\tau$.
Then $v$ and $\tau v$ are balanced and $(\tau v)v=\tau$.
These constructions are continuous on their respective domains.
Indeed, the $\eta$-th member of either increasing enumeration,
and the rank of a specified member, are determined by a restriction
of $\tau$ to a bounded initial segment. Thus both factor maps
are $\kappa$-Borel.

Finally, for a balanced involution $\eta$, match in increasing
order the smaller pair members of $r$ with those of $\eta$, their
partners with their partners, and the fixed points with the fixed
points. The resulting permutation $c(\eta)$ satisfies
$c(\eta)r c(\eta)^{-1}=\eta$. This choice is continuous on the
balanced subspace, by the same enumeration argument. Apply the
two-factor construction to $\alpha_\sigma$ and $\beta_\sigma$,
then apply $c$ to the resulting four balanced involutions.
This gives \eqref{eq:single-cohen-four-conjugates}.
For uncountable $\kappa$, the coordinatewise continuity assertions
give continuity in the stated topology by combining fewer than
$\kappa$ output requirements into one short input restriction.
At $\kappa=\omega$, all the choices just described are ordinary
Borel; continuity of the orbit reflections is not needed.
\end{proof}

\begin{lemma}[Borel permutation lifts]\label{lem:cohen-borel-lift}
Let $\kappa=\omega$, or let $\kappa$ be regular uncountable with
$\kappa^{<\kappa}=\kappa$. Put
$\Omega=\kappa$, $P=\operatorname{Fn}_{<\kappa}(\Omega,2)$, $B=\RO(P)$,
$\Gamma=[\kappa]^{<\kappa}$ acting by bit translations, and
$\Sigma=\Sym(\kappa)$. Suppose a local diagonal action of
$\Gamma\rtimes\Sigma$ on $B^J$, $0<|J|\leq\kappa$, is
$\Gamma$-ergodic. Assume also that every set action of $\Sigma$
on at most $\kappa$ points which kills short-support permutations
is trivial. Then the actual map $\sigma\mapsto\Theta_\sigma$
is $\kappa$-Borel in Boolean-matrix coordinates. For
$\kappa=\omega$, ``$\kappa$-Borel'' means ordinary Borel.
\end{lemma}
\begin{proof}
Write $e_j$ for the coordinate selectors and $\delta,s$ for the
diagonal and support maps. The four-conjugates factorization above
applies in both cases.

\emph{The fiber centralizer.}
Let $D$ be the group of complete automorphisms of $B^J$ that fix
the scalar diagonal pointwise and commute with every
$\Theta_{\tau_t}$, $t\in[\Omega]^{<\kappa}$. For $d,e\in D$,
the part of the cover where their Boolean permutations of the
fiber labels agree is $\Gamma$-invariant. It is therefore $0$
unless $d=e$. Consequently, for any selector $q$, the selectors
$d(q)$, $d\in D$, are pairwise disjoint. Since $B^J$ has order
density at most $\kappa$, this proves
\[
 |D|\leq\kappa.
\]
The same argument shows that two members of $D$ which agree on any
nonzero part of the cover agree everywhere.

Conjugation by $\Theta_\sigma$ gives an ordinary homomorphism
$\Sigma\to\Sym(D)$. If $\sigma$ is supported on a short set $E$,
it is the identity on the entire principal scalar algebra below
the basic condition assigning zero on $E$. Locality makes its lift
the identity there as well. Every member of $D$ preserves this
principal cover algebra, so conjugation by $\Theta_\sigma$ agrees
with the identity on $D$, by the preceding agreement argument.
The assumed set-action property makes the whole homomorphism
trivial. Thus
\[
 \Theta_\sigma d=d\Theta_\sigma
 \qquad(\sigma\in\Sigma,\ d\in D).
\]

Call $L$ a \emph{scalar-$\sigma$ intertwiner} if it covers $\sigma$
on the diagonal and
$L\Theta_{\tau_t}L^{-1}=\Theta_{\tau_{\sigma[t]}}$ for every $t$.
Every such map has the form $d\Theta_\sigma$ for a unique $d\in D$.
Hence, for every $\rho\in\Sigma$,
\begin{equation}
 L\Theta_\rho L^{-1}=\Theta_{\sigma\rho\sigma^{-1}}.
 \label{eq:single-cohen-intertwiner-conjugation}
\end{equation}
Thus the conjugation does not depend on the choice of intertwiner.
We do not assume that $D$ is commutative.

\emph{Closed charts for intertwiners.}
Fix $i\in J$. A seed is a pair $(p,j)$ with $p\in P$ and $j\in J$.
For this seed put
\[
 e=\delta(p)\wedge e_i,\qquad
 u_t=\Theta_{\tau_t}(e),\qquad
 v_t(\sigma)=
 \Theta_{\tau_{\sigma[t]}}\bigl(\delta(\sigma p)\wedge e_j\bigr).
\]
The two families $(u_t)_t$ and $(v_t(\sigma))_t$ consist of partial
selectors and each has join $1$, by ergodicity. Their supports
satisfy $s(v_t(\sigma))=\sigma s(u_t)$. Declare the seed valid at
$\sigma$ if
\begin{equation}
 \sigma s(u_t\wedge u_z)=s(v_t(\sigma)\wedge v_z(\sigma))
 \qquad(t,z\in[\Omega]^{<\kappa}).
 \label{eq:single-cohen-seed-overlaps}
\end{equation}
On the validity set there is a unique scalar-$\sigma$ intertwiner
taking $e$ to $\delta(\sigma p)\wedge e_j$, namely
\begin{equation}
 L(w)=\bigvee_t
 \bigl(\delta(\sigma s(w\wedge u_t))\wedge v_t(\sigma)\bigr).
 \label{eq:single-cohen-seed-map}
\end{equation}
To verify this, identify each principal algebra below $u_t$ or
$v_t(\sigma)$ with its scalar support. Equation
\eqref{eq:single-cohen-seed-overlaps} says precisely that the
canonical scalar-$\sigma$ maps on these principal algebras agree
on overlaps, and that their inverses agree on overlaps. As both
families cover $1$, they paste to a complete isomorphism given
by \eqref{eq:single-cohen-seed-map}. Translating the index $t$
proves the intertwining identity. Because the families cover $1$, the
specified seed determines the intertwiner uniquely.

Each validity set is closed. For fixed $t,z$, the right side of
\eqref{eq:single-cohen-seed-overlaps} is locally constant: it
depends only on $\sigma$ on $t\cup z\cup\operatorname{dom}(p)$.
For fixed $b\in B$ and basic $q$, the condition $q\leq\sigma b$
is clopen, since it is equivalent to $\sigma^{-1}q\leq b$ and
$\sigma^{-1}q$ is locally constant. Equality in
\eqref{eq:single-cohen-seed-overlaps} is therefore a closed
condition, tested by all basic $q$. There are at most $\kappa$
seeds, and their validity sets cover $\Sigma$. To see this, the actual
lift $\Theta_\sigma$ takes some positive portion of $e_i$ into an $e_j$.
A basic restriction of its source then supplies a valid seed.
A valid seed need not be realized by the actual lift.

\emph{Recovering the actual action.}
Well-order the seeds and choose the first valid one for each $\sigma$.
The choice pieces are $\kappa$-Borel. Formula
\eqref{eq:single-cohen-seed-map} gives a $\kappa$-Borel choice
$\sigma\mapsto L_\sigma$ in Boolean-matrix coordinates.
For the coding, represent $b\in B$ by
$\{p\in P:p\leq b\}$ and an automorphism of the cover lying over
a coordinate permutation by its scalar map and the coordinates
of the images of the $e_j$.
The maps $v_t(\sigma)$ are locally constant, and applying $\sigma$
to a fixed Boolean element is coordinatewise continuous by the
preceding inclusion test. Boolean meets and complements are
$\kappa$-Borel on these codes. Joins of at most $\kappa$ elements
are $\kappa$-Borel as well, using
\begin{equation}
 p\leq\bigvee_\xi b_\xi
 \quad\Longleftrightarrow\quad
 (\forall q\in P,\ q\leq p)(\exists r\in P,\ r\leq q)
                   (\exists\xi)\ r\leq b_\xi.
 \label{eq:single-cohen-code-join}
\end{equation}
All quantifiers here range over sets of size at most $\kappa$.
The joint scalar action is also $\kappa$-Borel, since
$\sigma^{-1}p$ has at most $\kappa$ possible, locally specified
values. Matrix composition uses $\kappa$ joins of meets and this
scalar action; inversion uses the scalar inverse and transposed
coefficients. They are consequently $\kappa$-Borel, and so is the
choice of intertwiners. No regularity of $\Theta$ has been assumed.

Fix the actual lift $\Theta_r$ of the involution in
\cref{lem:single-cohen-four-conjugates}. Equation
\eqref{eq:single-cohen-intertwiner-conjugation} and the factorization
there give
\begin{equation}
 \Theta_\sigma
   =\prod_{m<4}
       L_{a_m(\sigma)}\Theta_r L_{a_m(\sigma)}^{-1}.
 \label{eq:single-cohen-actual-borel}
\end{equation}
Thus the \emph{actual} map $\sigma\mapsto\Theta_\sigma$ is
$\kappa$-Borel in its Boolean-matrix code.

\end{proof}

\begin{theorem}[Initial frame]\label{thm:single-cohen-initial-frame}
Let $\kappa$ be regular uncountable with $\kappa^{<\kappa}=\kappa$,
and let $0<|I|\leq\kappa$. For every local diagonal-equivariant
action of $\Gamma\rtimes\Sigma$ on $C=B^I$, the atoms of
$C^\Gamma$ form a frame. Every member of this frame is fixed
by $\Gamma\rtimes\Sigma$.
\end{theorem}
\begin{proof}
By \cref{lem:single-cohen-fixed-atomic}, $C^\Gamma$ is atomic
and $\Sigma$ fixes it pointwise. Fix an atom $U$. Its scalar support
is $1$, because $\Gamma$ is weakly homogeneous on $B$. Moreover,
its fiber cardinality is forced to be a constant ground cardinal
$0<|J|\leq\kappa$. Indeed, the Boolean values of the possible cardinalities
are $\Gamma$-invariant, and $B$ preserves cardinals at most $\kappa$.
Choose a Boolean enumeration of its fibers and identify
$C\restr U$ with $B^J$. Write $e_j$ for the selectors of this
chosen frame and $1$ for $U$. This choice need not be invariant.
The restricted $\Gamma$-action is ergodic, meaning that its fixed
algebra is $\{0,1\}$. The restricted $\Sigma$-action is defined
because $\Sigma$ fixes $U$.

By \cref{lem:single-cohen-small-kernel}, the set-action hypothesis
of \cref{lem:cohen-borel-lift} holds. Hence the actual permutation
action on this ergodic cover is $\kappa$-Borel.

\emph{An open stabilizer.}
Fix $i\in J$. The $\kappa$-Borel sets
\[
 E_{p,j}=
 \{\sigma:\Theta_\sigma(\delta(p)\wedge e_i)
                         =\delta(\sigma p)\wedge e_j\}
\]
cover $\Sigma$. To see that $\Sigma$ is $\kappa$-Baire, start
in any basic open set and extend a short partial permutation through
$\kappa$ steps. At these steps avoid the prescribed nowhere dense sets
and put every coordinate into its domain and range. At every
proper stage the union is still a short partial permutation;
the final union is a permutation. Hence some $E_{p,j}$ is
nonmeager. Subdivide it by the at most $\kappa$ possible values
$q=\sigma p$, and choose a nonmeager piece $E_{p,j,q}$ and
$\sigma_0$ in that piece. If $e=\delta(p)\wedge e_i$, then
\[
 \sigma_0^{-1}E_{p,j,q}
 \subseteq H_e:=\{\sigma:\Theta_\sigma(e)=e\}.
\]
The subgroup $H_e$ is $\kappa$-Borel and nonmeager, so it is open.
Indeed, its Baire property makes it comeager on some nonempty
open set $O$. For all $\sigma$ in a sufficiently small identity
neighborhood, $O\cap\sigma O$ contains a nonempty open set.
The two translates of $H_e$ are comeager there, so they intersect;
hence $\sigma\in H_eH_e^{-1}=H_e$.

It follows that $H_e$ contains every permutation fixing some
$E\subseteq\Omega$ pointwise, where $|E|<\kappa$.
Enlarge $E$ to contain $\operatorname{dom}(p)$ and choose a full
$E$-pattern $b\leq p$. The partial selector
$w=\delta(b)\wedge e_i$ is fixed by all permutations of
$\Omega\setminus E$.

\emph{A selector fixed by all bounded translations.}
Let $\tau_t$ be a bounded translation with $t\cap E=\varnothing$.
Below any basic $q\leq b$, choose a short set
$F\subseteq\Omega\setminus E$ containing
$t\cup(\operatorname{dom}(q)\setminus E)$, with two disjoint fresh sets
of a common infinite size $\mu<\kappa$ at least as large as this
required set. Extend $q$ to a full pattern $a$ on $E\cup F$, assigning
zeros to one fresh set and ones to the other. Both $a\restr F$ and
$(\tau_t a)\restr F$ then have
exactly $\mu$ zeros and $\mu$ ones. There is a permutation
$\rho$ supported on $F$ taking the first pattern to the second.
The scalar maps $\rho$ and $\tau_t$ agree on the entire
principal algebra below $a$: they have the same output pattern
and fix every undecided coordinate. Locality therefore makes
their lifts agree below $\delta(a)$. Since $\rho$ fixes $w$,
\[
 \Theta_{\tau_t}(w)\wedge\delta(\tau_t a)
       =w\wedge\delta(\tau_t a).
\]
Such $a$ are dense below $b$, and $\tau_t b=b$. Hence
$\Theta_{\tau_t}(w)=w$.

For each full $E$-pattern $c$, let $t_c\subseteq E$ take $b$
to $c$. The join
\[
 v=\bigvee_{c\in 2^E}\Theta_{\tau_{t_c}}(w)
\]
is a selector: its summands are partial selectors whose
pairwise disjoint scalar supports $c$ cover $1$.
Tail translations fix every summand by commutation, and
translations on $E$ permute the summands. Thus $v$ is
$\Gamma$-invariant. Ergodicity gives $v=1$, showing that the
original atom $U$ is itself a selector.

Every atom of $C^\Gamma$ is consequently a selector.
These atoms form a frame, fixed also by $\Sigma$ by
\cref{lem:single-cohen-fixed-atomic}. Their ground set has
cardinality at most $\kappa$, by the density of $C$.
Forcing with $B$ evaluates the atom partition to a bijection
between that ground set and $I$. Preservation of the relevant
cardinals therefore makes its ground cardinality $|I|$, so
the frame can be indexed by $I$.
\end{proof}

\subsection{Comparison of small automorphism groups}
\label{subsec:single-cohen-comparison}

We continue to identify each condition of $P$ with its basic Boolean
value in $B$. The meet of a nonempty decreasing sequence of basic
conditions of length less than $\kappa$ is positive and basic,
represented by the union of their partial functions. Every positive
interval contains an antichain of
size $\kappa$, obtained below a basic condition by specifying the
position of the first $1$ on a fresh coordinate set of size $\kappa$.

\begin{lemma}\label{lem:single-cohen-basic-images}
If $L\leq\Aut(B)$ and $|L|<\kappa$, then
\[
 Q_L=\{p\in P:(\forall h\in L)\ h(p)\in P\}
\]
is dense in $B^+$, is $L$-invariant, and is closed under nonempty
decreasing meets of length less than $\kappa$.
\end{lemma}

\begin{proof}
Enumerate $L$ in order type $\mu<\kappa$ and repeat this enumeration
$\omega$ times. Start with a basic condition below any prescribed
positive element. At a step assigned to $h$, with current basic
condition $p$, choose basic conditions
\[
 q\leq h(p),\qquad p'\leq p\wedge h^{-1}(q),
\]
and continue with $p'$. At each proper limit take the union of the
source conditions. The recursion has length $\mu\cdot\omega<\kappa$,
so these unions and the final condition $p_*$ are positive and basic.
For a fixed $h$, the recorded target conditions $q_n$ decrease, and
the steps assigned to $h$ are cofinal in the recursion.
The inequalities at those steps and completeness give
\[
 h(p_*)=\bigwedge_{n<\omega}q_n\in P.
\]
This proves density.  The group law gives $L$-invariance.  If
$(p_\xi:\xi<\delta)$ decreases in $Q_L$, where $0<\delta<\kappa$,
then $p=\bigwedge_{\xi<\delta}p_\xi$ is basic and, for every $h\in L$,
\[
 h(p)=\bigwedge_{\xi<\delta}h(p_\xi)
\]
is likewise a positive basic condition.
\end{proof}

For $h\in\Aut(B)$, the positive $p$ satisfying
\begin{equation}\label{eq:single-cohen-displacement}
 h\restr(B\restr p)=\id
 \quad\text{or}\quad p\wedge h(p)=0
\end{equation}
form a dense downward-open class.  To see this, if $h$ is not
the identity below $b$, choose $e\leq b$ with $h(e)\ne e$.
If $e\setminus h(e)>0$, this difference is disjoint from its
image.  Otherwise $e<h(e)$, and the same is true of the positive
element $e\setminus h^{-1}(e)$.

Call $p$ an \emph{admissible $L$-root} if $p\in Q_L$ and
\eqref{eq:single-cohen-displacement} holds for every $h\in L$.
Its distinct translates form a disjoint family of basic conditions,
of size less than $\kappa$.  Moreover,
\begin{equation}\label{eq:single-cohen-return}
 h(p)\wedge k(p)>0
 \quad\Longrightarrow\quad
 h(p)=k(p),\quad h\restr(B\restr p)=k\restr(B\restr p).
\end{equation}
We call this orbit a \emph{tower}, and its members its columns.
Thus any group element that maps a column to itself fixes its
whole principal algebra pointwise.

\begin{lemma}\label{lem:single-cohen-tower-refinement}
Let $L\leq\Aut(B)$ have size less than $\kappa$.
Admissible $L$-roots are dense.  In choosing one, we may require
each of its $L$-translates to lie below some member of each of
fewer than $\kappa$ specified Boolean partitions, and to decide
fewer than $\kappa$ specified Boolean elements.
\end{lemma}

\begin{proof}
For a partition, the positive elements below one of its members
form a dense downward-open class; the same is true of the class
deciding a Boolean element.  Pull back these classes by all members
of $L$ and add the requirements
\eqref{eq:single-cohen-displacement}.  There are fewer than
$\kappa$ requirements.  Meet them by successive basic refinements,
taking basic unions at proper limits, and then refine into $Q_L$
using \cref{lem:single-cohen-basic-images}.  All the earlier
requirements persist under refinement.
\end{proof}

\begin{theorem}\label{thm:single-cohen-comparison}
For every $H\leq\Aut(B)$ with $|H|\leq\kappa$, there is
$a\in\Aut(B)$ such that
\[
 aHa^{-1}\subseteq[\Gamma].
\]
\end{theorem}

\begin{proof}
Work with a source and a target copy of $B$.  Choose a continuous
increasing sequence $(H_\alpha:\alpha<\kappa)$ of subgroups of
size less than $\kappa$, with $H_0=\{\id\}$ and union $H$.
Enumerate the source and target basic conditions as
$(u_\alpha:\alpha<\kappa)$ and $(v_\alpha:\alpha<\kappa)$.

We construct refining basic partitions $\mathcal A_\alpha$ and
$\mathcal E_\alpha$ of the source and target, together with compatible
bijections $\psi_\alpha:\mathcal A_\alpha\to\mathcal E_\alpha$.
We also construct homomorphisms $\pi_\alpha:H_\alpha\to\Aut(B)$,
each extending its predecessors. These maps satisfy:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $H_\alpha$ and $\pi_\alpha(H_\alpha)$ permute the respective
partitions, and $\psi_\alpha$ is equivariant.
\item On either side, every return stabilizer fixes its whole
column pointwise.
\item For $h\in H_\alpha$ and $q\in\mathcal E_\alpha$, the
restriction of $\pi_\alpha(h)$ to $B\restr q$ is a translation
whose mask is contained in $\operatorname{dom}(q)$.
\end{enumerate}
In particular, all source cells belong to $Q_{H_\alpha}$.  Target
columns in one orbit have the same domain.  At a successor
$\alpha+1$, all source cells will decide $u_\alpha$ and all target
cells will decide $v_\alpha$.  At stage zero take the one-cell
partitions and the trivial action.  Each partition has at most
$\kappa$ cells, whereas each individual group orbit has fewer
than $\kappa$ columns.

\emph{Successor extension.}
Fix a stage, write $L=H_\alpha$, $M=H_{\alpha+1}$ and
$\pi=\pi_\alpha$, and refer to the stage-$\alpha$ partitions
and their $L$-orbits as old cells and old towers.
In each old source tower choose a root column $c$ and a reserve
$r_c\in Q_L$ below it.  Propagate the reserve to the other columns
using $L$.  This is well-defined by the pointwise return condition,
and all propagated reserves are basic.  In each paired target
tower choose a basic root reserve and propagate it using $\pi(L)$;
these reserves are basic by (iii).

An inner recursion of length $\kappa$ will select disjoint paired
new towers.  The used source region will be $M$-invariant and the
used target region $\pi(L)$-invariant.  Each old tower retains a
positive coherent reserve in its unused region.  On the source
its root reserve is always in $Q_L$, and on the target it is always
basic.  Reserves only decrease.  At an inner limit below $\kappa$,
take their meets, separately for each old tower.  These are positive
basic conditions by \cref{lem:single-cohen-basic-images}; the old
charts transport them coherently. We do not intersect reserves from
different towers.

At each step we meet a request on the source or target side.
A source-directed request is a positive basic condition in the unused source.  Restrict it to
an old source column.  For a target-directed request, first restrict
the unused target request to an old target column, and use the
paired source column for the source choice; its reserve guarantees
a positive unused part.  By \cref{lem:single-cohen-tower-refinement}
choose an admissible $M$-root $p$ there whose translates refine
the old source partition and decide $u_\alpha$.  All translates
remain unused, since the unused source is $M$-invariant.

We may choose this new source tower without exhausting any reserve.
Take $\kappa$ pairwise disjoint basic refinements of $p$ and refine
each further into $Q_M$, obtaining $(p_\xi:\xi<\kappa)$.  Their
saturations
\[
 F_\xi=\bigvee_{h\in M}h(p_\xi)
\]
are pairwise disjoint: distinct translates of $p$ are disjoint,
and coincident translates have identical charts by
\eqref{eq:single-cohen-return}.  All these towers visit the same
fewer-than-$\kappa$ old $L$-towers.  A given positive root reserve
can be contained in at most one $F_\xi$, so fewer than $\kappa$
indices are forbidden.  Choose another index and replace $p$ by
that $p_\xi$.  Shrink each affected source root reserve inside its
positive remainder, choosing it in $Q_L$.  The selected saturation
is $L$-invariant, so the propagated reserves remain coherent.

Let $J$ be the set of columns of this selected $M$-tower.  Split
$J$ into its $L$-orbits, and choose a representative $j_\ell$ in
each; in a target-directed step take $p$ itself as the representative
of its orbit.  If $c_\ell$ is the old source column containing $j_\ell$,
put $b_\ell=\psi_\alpha(c_\ell)$.  Choose a basic representative
$q_\ell\leq b_\ell$ on the target and propagate it by $\pi(L)$.
Choose these representatives successively so their orbits are
unused, pairwise disjoint, and leave every old target reserve
positive.  There are fewer than $\kappa$ choices, even when an old
tower occurs repeatedly.  In a target-directed step choose the
representative corresponding to the column $p$ first, below the
given target request.

For completeness, the reserve choice can be made inside any
prescribed positive available region.  Pull that region back to
its old root column by the old translation chart.  Split a basic
subregion into two disjoint positive basic candidates.  At most
one contains the entire current root reserve.  Choose the other
and replace the reserve by a basic condition in its positive
complement.  Exclude previously selected new orbits when forming
the available region.  At a limit in this short recursion the
decreasing basic reserves remain positive, so the next choice is
still possible.  Propagating through the old charts preserves
these disjointness and reserve requirements.

The paired old columns have the same stabilizer. For an old source
column $c$ and $0<d\leq c$,
\begin{equation}\label{eq:single-cohen-subcell-stabilizer}
 \{h\in L:h(d)=d\}=\{h\in L:h(c)=c\}.
\end{equation}
An element moving $c$ sends it to a disjoint column. An element
preserving $c$ fixes its whole principal algebra.  The paired target column
has precisely the same stabilizer, also pointwise.  Thus the orbit
of $q_\ell$ under $\pi(L)$ has exactly the same indexing as the
source orbit of $j_\ell$.

Choose $D\subseteq\Omega$ of size less than $\kappa$ containing
all domains of the representatives $q_\ell$, all old translation
masks used to propagate them, and $\operatorname{dom}(v_\alpha)$.
Masks between propagated columns are differences of these root
masks and are therefore also contained in $D$.
Refine each $q_\ell$ to a full $D$-pattern.  Its propagated images
are full $D$-patterns as well.  They remain disjoint and unused,
and shrinking them preserves all reserves.  Pair these patterns
with $J$ equivariantly under $L$, and denote the pattern paired
with $j$ by $w_j$.  For $g\in M$ define its action on this target
tower by
\begin{equation}\label{eq:single-cohen-copied-action}
 \pi'(g)\restr(B\restr w_j)
   =\tau_{\{\xi\in D:w_j(\xi)\ne w_{g(j)}(\xi)\}}
      \restr(B\restr w_j).
\end{equation}
Pattern differences add modulo two, so these maps satisfy all
group identities.  For $h\in L$ the old mask on $w_j$ is contained
in $D$ and equals the displayed pattern difference.  Hence
\eqref{eq:single-cohen-copied-action} agrees with the old $\pi(h)$
on the whole principal algebra below $w_j$.
It therefore extends the old action and preserves its return stabilizers.
All target columns decide $v_\alpha$, and (iii) holds.

Enumerate all basic source and target requests in the inner
$\kappa$ recursion.  Skip a request only if its unused part is
zero; otherwise refine its unused part to a basic condition in
an old cell and perform the appropriate directed step above.
Before every inner stage fewer than $\kappa$ columns have been
used, and the coherent reserves are positive.  At the end the source and target joins are both $1$.
Any positive omitted part would contain a basic request that
could not have been skipped.
The selected columns thus form refining paired partitions.
Pasting \eqref{eq:single-cohen-copied-action} over their target
towers gives complete automorphisms $\pi'(g)$, satisfying the
group law and extending $\pi$ on all of $B$.  They belong to
$[\Gamma]$.  The source cells are in $Q_M$, all return stabilizers
are pointwise identities, and the prescribed source and target
tests have been decided.  This completes the successor extension.

\emph{Proper limits.}
Let $0<\lambda<\kappa$ be a limit.  Form the common refinement
of the partitions below $\lambda$ by taking the meet along every
coherent descending lineage.  On either side every such meet is
a positive basic condition, represented by the union of its
short patterns.  Distinct lineages have disjoint meets, and there
are at most $\kappa^{|\lambda|}=\kappa$ lineages.  They cover $1$:
below any positive basic condition, refine successively through
the earlier partitions, using basic unions at proper limits.
The final positive condition lies below a lineage meet.  The
compatible earlier bijections pair the lineages, so they pair
their nonzero meets and extend all earlier partition bijections.

Take the union of the previously fixed homomorphisms on
$H_\lambda=\bigcup_{\alpha<\lambda}H_\alpha$.  For $h\in H_\lambda$
choose $\beta<\lambda$ with $h\in H_\beta$.  It and $\pi(h)$
permute all later partitions equivariantly, and hence permute
the limit partitions.  If either returns a limit cell to itself,
it returns every sufficiently late containing cell and was
pointwise the identity there.  On a target limit cell its chart
is the translation assigned on a containing stage-$\beta$ cell;
that mask is still contained in the enlarged domain of the limit
condition.  Completeness shows that each source image is the meet
of the corresponding basic images.  Thus all inductive properties
hold at $\lambda$.

\emph{The complete isomorphism.}
Each basic partition generates the complete atomic subalgebra
of all joins of its cells.  Refinement and the compatible
bijections give an isomorphism between the unions of these
subalgebras on the two sides.  Every source and target basic condition belongs to its respective
union. At the successor stage assigned to it, all cells decided it,
so it is the join of the cells below it.  Both unions are therefore order-dense in their
Boolean completions.  Their isomorphism extends uniquely to a
complete isomorphism $a:B\to B$.

For $h\in H$, equivariance holds on every sufficiently late
partition algebra, so density and completeness give
$aha^{-1}=\pi(h)$.  The construction fixed $\pi(h)$ when $h$ first entered a stage group
and preserved it at every later stage.  It belongs
to $[\Gamma]$, as required.
\end{proof}


\subsection{Full invariance and ordinal definitions}

\begin{theorem}\label{thm:single-cohen-full-frame}
Every local diagonal-equivariant action of $G$ on $B^I$,
$0<|I|\leq\kappa$, has a frame fixed pointwise by $G$.
The frame is exactly the atom set of $C^\Gamma$ and of $C^G$.
\end{theorem}
\begin{proof}
Let $F$ be the frame supplied by
\cref{thm:single-cohen-initial-frame}. Let $[\Gamma]$ denote the
automorphisms piecewise given by $\Gamma$, as in
\cref{thm:single-cohen-comparison}. Every $t\in[\Gamma]$ fixes $F$
pointwise. Indeed, on each piece $b$ where $t$ agrees with a
translation $\gamma$, locality makes $\Theta_t$ agree with
$\Theta_\gamma$ below $\delta(b)$. The images of these pieces
partition $1_C$, and $\Theta_\gamma$ fixes each member of $F$.

Every $\Gamma$-fixed selector $q$ is a member of $F$.
Atomicity expresses $q$ as a nonempty join of atoms in $F$.
Two distinct atoms cannot lie below $q$. Both have scalar support
$1$, and the support map is injective below the selector $q$.
Thus any $\Gamma$-fixed frame is exactly $F$, as a set.

Fix $g\in G$, and set $H=\langle\Gamma,g\rangle$.
Since $|H|=\kappa$, \cref{thm:single-cohen-comparison} gives
$a\in G$ with $aHa^{-1}\subseteq[\Gamma]$. Hence
\[
 F'=\{\Theta_{a^{-1}}(f):f\in F\}
\]
is a frame fixed pointwise by $H$. It is therefore $\Gamma$-fixed,
so $F'=F$. In particular $g$ fixes every member of $F$.
As $g$ was arbitrary, $F$ is fixed pointwise by $G$.
Since $C^G\subseteq C^\Gamma$ and every atom of the latter is
fixed, the two fixed algebras coincide.
\end{proof}

\begin{proof}[Proof of \cref{thm:single-cohen-invariant}]
The Boolean truth values of the possible cardinalities of the invariant
family name $\dot A$ are invariant. Thus $1_B$ decides its cardinality.
The forcing preserves cardinals at most $\kappa$, so choose
a ground cardinal $0<\nu\leq\kappa$ and names
$(\tau_i)_{i<\nu}$ forced to enumerate $\dot A$ bijectively.
Exactly as in the proof of \cref{thm:cohen-invariant}, the matrices
\[
 m^g_{ij}=\|g\tau_i=\tau_j\|_B,\qquad
 (\Theta_gz)_j=\bigvee_{i<\nu}\bigl(g(z_i)\wedge m^g_{ij}\bigr)
\]
define a local diagonal-equivariant action on $B^\nu$.
The argument is an induction on names and imposes no rank bound.
By \cref{thm:single-cohen-full-frame}, the fixed algebra has a
frame. Its atoms can be indexed by $\nu$, since their row and column
partitions force a bijection with $\nu$ and $B$ preserves ground
cardinals at most $\kappa$. Mixing the $\tau_i$ along each row gives
pairwise distinct invariant names which exhaust $\dot A$.
Their enumeration name is invariant as well.
\end{proof}

\begin{proof}[Proof of \cref{thm:single-cohen-od}]
Work over a ground $W$ satisfying
$\mathsf{GCH}+\mathsf{GA}+(V=\operatorname{HOD})$, as permitted by
\cref{rem:ground-convention}. GCH gives
$\kappa^{<\kappa}=\kappa$. The empty family has its empty
enumeration. For a nonempty OD family of size at most $\kappa$,
weak homogeneity turns its defining formula and ordinal parameters
into an invariant name forced to have that size bound.
Apply \cref{thm:single-cohen-invariant}. The enumeration name and
the forcing have ordinal codes by \cref{lem:stable-ground-codes};
\cref{lem:invariant-od} makes the enumeration OD in the extension.
Each of its values is then definable from the enumeration and
its ordinal index. Taking $W=L$ proves the stated theorem.
\end{proof}

\begin{corollary}[Sharpness]\label{cor:single-cohen-sharp}
In the extension of \cref{thm:single-cohen-od}, there is an OD
family of size $\kappa^+$ with no OD member.
\end{corollary}
\begin{proof}
Take $A=\mathcal P(\kappa)^{L[c]}\setminus L$.
Homogeneity makes every OD subset of $\kappa$ belong to $L$,
so $A$ has no OD member. To see that its cardinality is $\kappa^+$, note that
$c\mathbin\triangle a\in A$ for each $a\in\mathcal P(\kappa)^L$
and that these subsets are distinct. Conversely, $P$ has size $\kappa$
and is $\kappa^+$-cc, so the number of nice names for subsets of
$\kappa$ is at most $(2^\kappa)^\kappa=\kappa^+$ in $L$.
\end{proof}

\section{Countable families in a random extension}\label{sec:random}

Automatic continuity gives an enumeration invariant under
probability-preserving automorphisms. By comparing equivalent probability
measures, we then show that all nonsingular automorphisms fix it.

Work in a ZFC ground $W$ on a standard atomless probability space
$(X,\mu)$, and let $\B$ be its measure algebra. Put
$\G=\Aut(\B)$, $\Hh=\Aut(\B,\mu)$, and $K=\Sym(I)$, for
nonempty finite or countably infinite $I$. The first group consists of
all nonsingular transformations modulo null sets. The topology on $K$
is pointwise convergence; $L^0(X,K)$ has convergence in measure.

Suppose $\dot e:I\to\dot A$ is a name for a bijective enumeration
of an invariant family. For $g\in\G$, define its comparison
permutation by
\[
 1\Vdash g\dot e=\dot e\circ\dot\pi_g.
\]
Permutation names are identified with measurable $K$-valued functions.
Write $(g\cdot u)(x)=u(g^{-1}x)$. Then
\begin{align}
 \pi_{gh}&=\pi_g(g\cdot\pi_h),\label{eq:crossed}\\
 \pi_g&=1_K\text{ on }\fix(g).\label{eq:local}
\end{align}
Here $\fix(g)$ is the largest Boolean region on which $g$ acts
identically on every subregion. The first identity follows by comparing
$gh\dot e$ to $\dot e$ in two steps; the second follows from induction
on names. Replacing $\dot e$ by $\dot e\circ\dot\beta^{-1}$ changes
the comparison to
\[
 \pi_g^\beta=\beta\pi_g(g\cdot\beta^{-1}).
\]
It suffices to choose a relabeling that makes every comparison
permutation the identity. We make all subsequent choices and measure
calculations in $W$.

\subsection{The probability-preserving subgroup}\label{sec:pp-subgroup}

The following lemma gives a relabeling fixed by every
probability-preserving automorphism.

\begin{lemma}[Probability-preserving frames]\label{lem:random-mp-frame}
Let $Q$ be a standard atomless probability algebra and
$0<|J|<2^{\aleph_0}$. Every full local diagonal action of
$\Aut(Q,\mu)$ on $Q^J$ has an invariant frame.
\end{lemma}
\begin{proof}
For selectors set
\[
 d(u,v)=1-\mu\Bigl(\bigvee_{j\in J}(u_j\wedge v_j)\Bigr),
 \qquad F_u(h)=d(\Theta_hu,u).
\]
The action is isometric, and $F_u$ is a seminorm:
$F_u(1)=0$, $F_u(h^{-1})=F_u(h)$, and
$F_u(hk)\leq F_u(h)+F_u(k)$. Its orbit pseudometric has density
at most $\max(|J|,\aleph_0)$. Indeed, every selector has countably
many nonzero entries. Truncate to finitely many labels and approximate
their partition in a countable dense algebra of $Q$.
Locality gives
\[
 d(\Theta_hu,\Theta_ku)
 \leq\mu(\supp(k^{-1}h)).
\]
The standard probability-algebra group has ample metric generics,
and its uniform metric is equivalent to this support metric
\cite[Section~6.3]{BYBM}. Thus $F_u$ is Baire measurable for
that finer metric. The small-density seminorm theorem
\cite[Theorem~3.6]{BYMGrey} makes $F_u$ continuous for the weak
Polish topology. Every selector orbit is consequently weakly
continuous.

We now use continuity to construct a frame. This argument also applies
when $J$ is uncountable. The counting measure over the fibers is
\[
 \nu(z)=\sum_{j\in J}\mu(z_j),\qquad
 L^2(Q^J,\nu)=\bigoplus_{j\in J}L^2(Q,\mu).
\]
The action preserves $\nu$, since its Boolean permutation
matrices have partition rows and columns. Also
$\|\one_u-\one_v\|_2^2=2d(u,v)$. Continuity supplies a finite
Boolean partition $\mathcal P$ such that its stabilizer
$H_{\mathcal P}$ moves $\one_u$ by less than $1/8$ in norm.
The minimum-norm vector $v$ in the closed convex hull of this
orbit is nonnegative, $H_{\mathcal P}$-fixed, and within $1/8$
of $\one_u$. Its positive coordinates have countable support.
Select the unique strictly positive maximum among its fiber
coordinates wherever it exists. This defines an invariant partial selector $w$.
It is nonzero because, outside a set of measure at most
$16\|v-\one_u\|_2^2\leq1/4$, the coordinate chosen by $u$
exceeds $3/4$ and all others are below $1/4$.
The scalar support of $w$ is fixed by $H_{\mathcal P}$, hence
is a union of cells of $\mathcal P$. Restricting to one such
positive cell $p$ gives a partial selector of support $p$
fixed by every automorphism supported on $p$.

Its translates agree on overlaps. For $h,k$, put
$r=h(p)\wedge k(p)$, $b=h^{-1}(r)$ and $c=k^{-1}(r)$.
The partial map $k^{-1}h:Q\restr b\to Q\restr c$ extends
to an automorphism supported on $p$, by matching the equal-measure
complements inside $p$. Locality and the invariance of $w$
therefore identify $\Theta_hw$ and $\Theta_kw$ over $r$.
Their join over all $h$ is an invariant selector, since its
scalar support is $1$ by ergodicity. A maximal disjoint family
of invariant selectors covers $1$. Indeed, a nonzero invariant
complement $z$ would also have scalar support $1$; disjointifying
its coordinates supplies a selector $u\leq z$. Repeat the averaging and maximum selection within $z$, then join
the translates as above. This gives another invariant selector,
a contradiction. This is an invariant
frame, of cardinality $|J|$ by ccc and the corresponding bijection
in a generic fiber.

\end{proof}

\begin{proposition}\label{prop:normalize-H}
Let $\pi$ be the comparison permutations of an enumeration as above.
There is a permutation name $\beta$ such that, after replacing
$\dot e$ by $\dot e\circ\beta^{-1}$, the comparison permutations
\[
 \pi^\beta(g)=\beta\pi(g)(g\cdot\beta^{-1})
\]
satisfy $\pi^\beta(h)=1_K$ for every $h\in\Hh$.
\end{proposition}

\begin{proof}
Apply \cref{lem:random-mp-frame} to the local diagonal action on
$\B^I$ induced by the enumeration. Its invariant frame has cardinality $|I|$ and gives the required
relabeling. Locality persists because $g\cdot\beta=\beta$
on every identity region.
\end{proof}

For the remaining comparison argument, $I$ may be any nonempty
ground set. Let $\pi_g$ be the Boolean permutation matrix of a full
local diagonal $\G$-action on $\B^I$ relative to a fixed frame.
Products use Boolean joins and meets; the identities
\eqref{eq:crossed}--\eqref{eq:local} remain valid, with $1$ denoting
the identity matrix. When $I$ is countable these are the preceding
measurable permutation functions. Every row and column has only
countably many positive entries, by ccc. Assume the frame is
probability-preserving invariant, so
\begin{equation}\label{eq:alpha-H-zero}
 \pi\restr\Hh=1.
\end{equation}

\subsection{Equivalent probability measures}\label{sec:equivalent-measures}

Let $\Pp$ be the set of probability measures on $\B$ equivalent to $\mu$. For each $\nu\in\Pp$, choose $k_\nu\in\G$ such that $(k_\nu)_*\mu=\nu$, with $k_\mu=1$, and put
\[
 b_\nu=\pi(k_\nu).
\]
This does not depend on the choice of $k_\nu$. If
$k_\nu=\ell_\nu h$ for $h\in\Hh$, then \eqref{eq:alpha-H-zero}
gives $\pi(k_\nu)=\pi(\ell_\nu)$. Define
\[
 C(\nu,\lambda)=b_\nu^{-1}b_\lambda.
\]

\begin{lemma}\label{lem:comparison-identities}
For $\nu,\lambda,\rho\in\Pp$ and $g\in\G$,
\begin{align}
 C(\nu,\lambda)C(\lambda,\rho)&=C(\nu,\rho),\label{eq:C-compose}\\
 C(g_*\nu,g_*\lambda)&=g\cdot C(\nu,\lambda),\label{eq:C-covariance}\\
 \pi(g)&=C(\mu,g_*\mu).\label{eq:alpha-C}
\end{align}
\end{lemma}

\begin{proof}
The first identity is immediate. Since $gk_\nu$ carries $\mu$ to $g_*\nu$,
\[
 b_{g_*\nu}=\pi(g)(g\cdot b_\nu).
\]
The copies of $\pi(g)$ cancel in $b_{g_*\nu}^{-1}b_{g_*\lambda}$, giving \eqref{eq:C-covariance}. Taking $\nu=\mu$ gives \eqref{eq:alpha-C}.
\end{proof}

Let $\lambda=e^Z\nu$. We say that the ambient probability algebra is \emph{conditionally atomless over $\sigma(Z)$} if almost every conditional measure in the disintegration over $Z$ is atomless.

\begin{lemma}\label{lem:relative-fixed}
Let $K$ be a Polish space. Suppose the probability algebra is conditionally atomless over $\sigma(Z)$. If a measurable $K$-valued function is fixed by every $\nu$-preserving automorphism which fixes $Z$, then it is $\sigma(Z)$-measurable.
\end{lemma}

\begin{proof}
The relative product theorem identifies the extension over $\sigma(Z)$
with a product with an atomless probability space. The full group of the
fiber acts ergodically on almost every fiber, so its fixed algebra is
the base algebra. Apply this to the inverse images of a countable basis
of open subsets of $K$. Delbaen also characterizes conditional
atomlessness by independent atomless subalgebras \cite{Delbaen}.
\end{proof}

\begin{lemma}[Pexider rigidity]\label{lem:pexider}
Let $J$ be nonempty and countable, and put $K=\Sym(J)$.
Let $\xi$ be a probability measure on $\mathbb R$ which is not
countably supported, and let $\eta$ be equivalent to Lebesgue measure.
Suppose Borel maps $A,B,D:\mathbb R\to K$ satisfy
\[
 A(x)B(y)=D(x+y)
\]
for $\xi\times\eta$-almost every $(x,y)$. Then $A$ is constant $\xi$-almost everywhere.
\end{lemma}

\begin{proof}
Replacing $\eta$ by Lebesgue measure does not change its null sets.
By Fubini, there is a $\xi$-conull set $E$ such that the equation holds
for every $x\in E$ and almost every $y$. For $x,x'\in E$, compare
the equation at $(x,y)$ and $(x',y+x-x')$ to obtain
\begin{equation}\label{eq:pexider-difference}
 B(y+x-x')B(y)^{-1}=A(x')^{-1}A(x)
\end{equation}
for almost every $y$.

Let $D_0$ be the set of $h\in\mathbb R$ for which
$B(\cdot+h)B(\cdot)^{-1}$ is almost everywhere constant. This is a
subgroup of $\mathbb R$ containing $E-E$. The corresponding constants
define a homomorphism $\chi:D_0\to K$ with abelian range.

The group $D_0$ is dense. Otherwise its closure would be discrete, so
$D_0$ would be countable. For any fixed $x\in E$, the countable coset
$x+D_0$ would then contain $E$. Translation is continuous on
$L^0_{\mathrm{loc}}(\mathbb R,K)$, hence $\chi$ is continuous.

Every neighborhood of the identity in $K=\Sym(J)$ contains an open
subgroup. Its inverse image under $\chi$ is an open subgroup of $D_0$,
so it contains $D_0\cap(-\varepsilon,\varepsilon)$ for some
$\varepsilon>0$. Given $h\in D_0$, choose $n\geq2$ with
$|h|/n<\varepsilon/2$ and $d\in D_0$ with
$|d-h/n|<\varepsilon/(2n)$. Both $d$ and $h-(n-1)d$ lie in that
interval, so $h=(n-1)d+[h-(n-1)d]$ belongs to the subgroup. Thus every
such inverse image is all of $D_0$, and $\chi$ is trivial.
Equation \eqref{eq:pexider-difference} gives $A(x)=A(x')$ for all
$x,x'\in E$.
\end{proof}

\begin{proposition}\label{prop:conditional-atomless-comparison}
Let $\lambda=e^Z\nu$. Suppose the law of $Z$ is not countably supported and the probability algebra is conditionally atomless over $\sigma(Z)$. Then $C(\nu,\lambda)$ is almost everywhere constant.
\end{proposition}

\begin{proof}
An automorphism preserves both $\nu$ and $\lambda$ exactly when it
preserves $\nu$ and $Z$. By covariance and \cref{lem:relative-fixed},
each entry of $C(\nu,\lambda)$ is $\sigma(Z)$-measurable.
Conditional atomlessness supplies two atomless coordinates independent
of $Z$. On the first choose $Y$ with law equivalent to Lebesgue measure
and $\mathbb E_\nu e^Y=1$. Leave the second coordinate unused.
Put $d\rho=e^Yd\lambda$. Independence makes $\rho$ a probability.
The unused atomless coordinate gives conditional atomlessness over
$Z$, $Y$, and $Z+Y$ for the relevant measures. Entrywise, therefore,
\[
 C(\nu,\lambda)=A(Z),\qquad C(\lambda,\rho)=B(Y),\qquad
 C(\nu,\rho)=D(Z+Y).
\]
The composition identity gives $A(Z)B(Y)=D(Z+Y)$.

To apply \cref{lem:pexider}, close any label under the possible
positive entries of rows and columns of these three matrices. The
resulting ground component $J$ is countable and is preserved by all
three matrices. On $J$, choose simultaneous Borel representatives,
replacing them by the identity on their exceptional null sets.
They are $\Sym(J)$-valued functions of the displayed variables.
The laws of $Y$ and $Z+Y$ are equivalent to Lebesgue measure.
Changing between the three equivalent ambient probabilities preserves
null sets, so we may use the natural laws of these variables. Independence
of $Z,Y$ and \cref{lem:pexider} make $A(Z)|J$ constant. Varying the
initial label makes every entry of $C(\nu,\lambda)$ constant.
The partition rows and columns therefore determine one ground
permutation. This does not require an uncountable intersection of
conull sets.
\end{proof}

\subsection{Arbitrary likelihood ratios}\label{sec:arbitrary-likelihood}

\begin{proposition}\label{prop:ac-likelihood}
Let $\lambda=e^Z\nu$. If the law of $Z$ is absolutely continuous with respect to Lebesgue measure, then $C(\nu,\lambda)$ is almost everywhere constant.
\end{proposition}

\begin{proof}
Write $Z=n+r$, where $n\in\mathbb Z$ and $0\leq r<1$, and use the nonterminating binary expansion
\[
 r=\sum_{m\geq1}\varepsilon_m2^{-m}
\]
off the dyadic rationals. Put
\[
 P=n+\sum_{m\text{ odd}}\varepsilon_m2^{-m},\qquad Q=\sum_{m\text{ even}}\varepsilon_m2^{-m}.
\]
Then $Z=P+Q$, $Q\geq0$, and $P\leq Z$.

Under Lebesgue measure, the joint law of $(P,Q)$ is a product of two
atomless measures, with a counting factor in the first coordinate.
The law of $Z$ is absolutely continuous, so the actual joint law of
$(P,Q)$ is absolutely continuous with respect to this product.
The conditional law of $Q$ given $P$ is atomless, as is the conditional
law of $P$ given $Q$. Thus the ambient algebra is conditionally atomless
over both $\sigma(P)$ and $\sigma(Q)$, and both marginal laws are
nonatomic.

Let
\[
 d\eta=\frac{e^P}{\int e^P\,d\nu}\,d\nu.
\]
This is well-defined because $P\leq Z$. The log-likelihood from
$\nu$ to $\eta$ is $P$ plus a constant, and the log-likelihood from
$\eta$ to $\lambda$ is $Q$ plus a constant. Equivalent reweighting
preserves conditional atomlessness. By
\cref{prop:conditional-atomless-comparison}, both $C(\nu,\eta)$ and
$C(\eta,\lambda)$ are constant. Equation \eqref{eq:C-compose} then
makes $C(\nu,\lambda)$ constant.
\end{proof}

\begin{lemma}[Uniform perturbation]\label{lem:uniform-perturbation}
Let $Z$ be a finite real-valued random variable on a standard atomless probability space. There is a uniform random variable $T:(X,\nu)\to(0,1)$ such that $Z+T$ has an absolutely continuous law.
\end{lemma}

\begin{proof}
Let $\{z_m:m\in J\}$ be the at most countable set of atoms of the
law of $Z$, and put $E_m=\{Z=z_m\}$. On each positive-measure $E_m$,
choose a conditionally uniform random variable $T_m$. Then $Z+T_m$
has a shifted uniform law on $E_m$.

On the complement $E_*$, the conditional law of $Z$ is nonatomic.
Let $F$ be its continuous distribution function and $Q$ its quantile
function. Put $T_*=F(Z)$. Then $T_*$ is uniform and $Z=Q(T_*)$
almost surely. The map
\[
 h(t)=Q(t)+t
\]
is strictly increasing and satisfies $h(t)-h(s)\geq t-s$ for $s<t$. Its inverse on its range is $1$-Lipschitz. Hence the pushforward of Lebesgue measure by $h$ is absolutely continuous, so $Z+T_*=h(T_*)$ has an absolutely continuous law.

Patch the $T_m$ and $T_*$. Conditional on each member of the resulting countable partition, $T$ is uniform. Thus $T$ is uniform, and the law of $Z+T$ is a countable mixture of absolutely continuous laws.
\end{proof}

\begin{theorem}[Nonsingular invariance]\label{thm:random-nonsingular-frame}
For any ground set $I$, every probability-preserving invariant frame
for a full local diagonal $\Aut(\B)$-action on $\B^I$ is fixed
by the whole group. No cardinality bound on $I$ is required.
\end{theorem}
\begin{proof}
With the preceding notation, we show that $C(\nu,\lambda)$ is constant for every $\nu,\lambda\in\Pp$.

Write $\lambda=e^Z\nu$. By \cref{lem:uniform-perturbation}, choose a uniform $T\in(0,1)$ such that $U=Z+T$ has an absolutely continuous law. Put $V=-T$ and
\[
 d\eta=\frac{e^U}{\int e^U\,d\nu}\,d\nu.
\]
The integral is finite because $e^U\leq e e^Z$. The first log-likelihood is $U$ plus a constant, so \cref{prop:ac-likelihood} makes $C(\nu,\eta)$ constant. The second log-likelihood is $V$ plus a constant. Its law under $\eta$ is absolutely continuous because $\eta\ll\nu$ and the law of $V$ under $\nu$ is uniform. Hence $C(\eta,\lambda)$ is constant, and \eqref{eq:C-compose} shows that $C(\nu,\lambda)$ is constant.

By \eqref{eq:alpha-C}, each $\pi(g)$ is a constant permutation.
Thus $\pi$ is an ordinary homomorphism to $\Sym(I)$. Locality makes it trivial on any map with a positive identity region.
Every member of $\G$ is a product of two such maps.
For $g\ne1$, choose $b>0$ with
$b\wedge g(b)=0$, and $0<a<b$. Let $t$ exchange $a,g(a)$ by
$g,g^{-1}$ and fix their complement. Both $t$ and $tg$ have positive
identity regions, and $g=t(tg)$. Hence every comparison permutation is the identity.
\end{proof}

\begin{theorem}[Invariant enumeration for random forcing]\label{thm:random-invariant}
In any ground model $W$ of ZFC, let $\B$ be its standard atomless
probability algebra. Every invariant name forced to be a nonempty
countable set has an invariant bijective enumeration by a finite
ordinal or by $\omega$.
\end{theorem}
\begin{proof}
Normalize a bijective enumeration by \cref{prop:normalize-H}, then
apply \cref{thm:random-nonsingular-frame} to its fixed frame.
\end{proof}


\begin{corollary}\label{thm:one-random}
If $a$ is a set of ordinals and $r$ is random over $L[a]$, then
every countable $\OD(a)$ set in $L[a,r]$ has an $\OD(a)$ bijective
enumeration. In particular every countable OD set in $L[r]$ consists
of OD elements, without a rank restriction.
\end{corollary}
\begin{proof}
The argument also works over any ground $W$ satisfying
$\mathsf{ZFC}+\mathsf{GCH}+\mathsf{GA}+V=\mathrm{HOD}$, with
$a\in W$ a set of ordinals.  The ground and its names then have
stable ordinal codes by \cref{lem:stable-ground-codes}; for the
stated ground $L[a]$, use its canonical $a$-definable order instead.
In either case the definition supplies an invariant name. Apply
\cref{thm:random-invariant} and then \cref{lem:invariant-od} to
the resulting enumeration name.  No cardinal arithmetic is used
in this argument.
\end{proof}

\begin{remark}
This relative statement requires the parameter to belong to the ground
$L[a]$. The conclusion for every set-of-ordinals parameter in $L[c]$
used the Cohen intermediate-extension lemma. We need no such extension
of the parameter assertion here.
\end{remark}

\section{Countable families in collapse extensions}
\label{sec:finite-collapse}
\label{sec:countable-collapse}

We compare three coordinate factors to obtain an enumeration, then factor
automorphisms into recodings of finite words to prove its invariance.
The enumeration is indexed by a ground cardinal, which may become
countable in the extension.

Throughout this section, we work in a fixed ground model
$W\models\mathsf{ZFC}$. All algebras, names, groups, and choices belong
to this model.  Let $\mu$ be an
infinite cardinal, put
\[
 \Omega=\mu\times\mu,\qquad
 B=\RO(\Omega^\omega),\qquad G=\Aut(B),
\]
and identify the individual component coordinates with
$I=\omega\times\{0,1\}$.  This is a presentation of
$\Coll(\omega,\mu)$, fixed before any enumeration is chosen.  For
$A\subseteq I$, let $C_A$ be the complete subalgebra generated by the
coordinates in $A$. For any complete subalgebra $C\subseteq B$, write
\[
 G_C=\{g\in G:g\upharpoonright C=\id\}.
\]

\begin{theorem}\label{thm:countable-collapse-invariant}\label{thm:finite-collapse-invariant}
Let $\dot X$ be a
$B$-name invariant under $G$. If $1_B$ forces that $\dot X$ is
nonempty and countable, there are a ground cardinal $\nu\le\mu$ and
invariant names $(\dot y_\xi:\xi<\nu)$ which are forced to enumerate
$\dot X$ bijectively. If $1_B\Vdash|\dot X|=m<\omega$, then $\nu=m$.
\end{theorem}

The members may have arbitrary rank. The theorem holds over any ZFC
ground. We state the definability consequence after its proof.
The fixed algebra of $G$ is $\{0,1\}$, so homogeneity decides whether
$\dot X$ is infinite or has a particular finite size. For the proof,
fix a nonempty finite ordinal or $\omega$, denoted by $J$, such that
$1_B\Vdash|\dot X|=|\check J|$.

\subsection{Relative enumerations and three-factor comparison}

\begin{lemma}\label{lem:fc-relative}
Suppose $C\subseteq B$ is complete and forces that the quotient of $B$
is a Cohen algebra. Then $\dot X$ has a bijective $J$-indexed
enumeration by ground $B$-names invariant under $G_C$. Moreover,
\[
 B^{G_C}=C.
\]
\end{lemma}

\begin{proof}
In a $C$-generic extension $W[H]$, the quotient name for $\dot X$ is
invariant by \cref{lem:hc-quotient-invariance}.
Theorem~\ref{thm:cohen-invariant},
applied in that intermediate model, gives an invariant enumeration by
quotient names.  The maximum principle chooses $C$-names for such an
enumeration.  Flatten them to ground $B$-names.  A ground member of $G_C$
acts trivially on the intermediate model and descends to a quotient
automorphism, so it fixes the flattened enumeration.  This proves the
first assertion.

For the second, an element of $B^{G_C}$ becomes, in every $C$-generic
extension, a Boolean element fixed by all automorphisms of the Cohen
quotient.  Its quotient value is therefore $0$ or $1$.  The $C$-Boolean value deciding which case holds is the original element
of $B$. Thus that element belongs to $C$.  The reverse inclusion is immediate.
\end{proof}

We use regular forcing products below. These differ from direct products
of Boolean algebras. Represent a name forced to belong to a ground set
$S$ by the $S$-indexed Boolean partition of its possible values.

\begin{lemma}\label{lem:fc-separation}
Let $P_0,P_1,P_2$ be forcing notions and $S$ a ground set. Suppose
$u$ is a $P_0\times P_1$-name and $v$ a $P_1\times P_2$-name, both
forced to belong to $S$.  If the threefold product forces $u=v$, there
is a $P_1$-name $w$ such that the respective twofold products force
$u=w$ and $v=w$.  The assertion remains true after restricting any
unshared factor to a nonzero cone.
\end{lemma}

\begin{proof}
Force first with $P_1$.  In the intermediate extension the remaining
forcing is the product of the old posets $P_0$ and $P_2$.  A condition
forcing $u=s$ and one forcing $v=t$ are compatible in this product, and
therefore $s=t$.  Conditions deciding a value are dense in each factor.
Thus there is one value which both names are forced to take.  The forcing
theorem and maximum principle provide a $P_1$-name for it.  The argument also applies after restriction to a nonzero cone.
When we use quotient completions, we take the new completions of the
unchanged dense posets.
\end{proof}

\begin{lemma}[Effective descent]\label{lem:fc-effective-descent}\label{lem:fc-three-comparisons}
Let $J$ be a nonempty ground set. For $i<j<3$, let $t_{ij}$ be a
$P_i\times P_j$-name for a permutation of $\check J$ in the
extension. Suppose
\[
 1\Vdash t_{02}=t_{12}t_{01}.
\]
There are a ground set $E$, with $|E|\le |P_0|\cdot|J|$, and
$P_i$-names $e_i$ for bijections $\check E\to\check J$ such that
\[
 1\Vdash t_{ij}=e_je_i^{-1}\qquad(i<j<3).
\]
The bound may use any dense presentation of $P_0$.
\end{lemma}

\begin{proof}
Call $(p,n)\in P_0\times J$ \emph{stable} if there is a $P_1$-name
$\sigma$ for an element of $\check J$ such that
\[
 p\times1\Vdash t_{01}(n)=\sigma.
\]
The witness is unique modulo $P_1$-forced equality, since we can remove
the nonzero $P_0$-cone from the equality of two witnesses.
Represent names for elements of $\check J$ by Boolean partitions,
and choose one witness for each stable pair.

For every fixed $n\in J$, stable conditions are dense in $P_0$.
Indeed, below any given condition choose $p\in P_0$, $q\in P_2$,
and $m\in J$ such that $p\times q\Vdash t_{02}(n)=m$.
The triangle identity gives
$t_{01}(n)=t_{12}^{-1}(m)$ on $p\times P_1\times q$.
Lemma~\ref{lem:fc-separation}, with shared factor $P_1$, supplies
a witness $\sigma$ on the whole product $p\times P_1$.
Only one value in $J$ has been decided.

If $\sigma$ witnesses stability of $(p,n)$, the unrestricted triangle
identity, restricted only to $p$, gives $t_{12}(\sigma)=t_{02}(n)$.
These two names share only $P_2$. Separation supplies a $P_2$-name
$\rho$ such that
\[
 1_{P_1\times P_2}\Vdash t_{12}(\sigma)=\rho.
\]
For two such witnesses $\sigma,\tau$ with counterparts $\rho,\eta$,
injectivity of $t_{12}$ gives
\[
 1_{P_1\times P_2}\Vdash
       (\sigma=\tau)\ \longleftrightarrow\ (\rho=\eta).
\]
The Boolean values on the two sides belong to independent factors,
so their common value is $0$ or $1$. Thus distinct forced-equality
classes of stable witnesses are forced unequal. Let $E$ be their
ground set of classes, and choose representatives $\sigma_a,\rho_a$
for $a\in E$. There are at most $|P_0|\cdot|J|$ classes. Put
$e_1(a)=\sigma_a$ and $e_2(a)=\rho_a$. Both names are forced
injective, and $t_{12}e_1=e_2$.

To prove that $e_1$ is onto, fix $l\in J$ and $q\in P_1$. Choose a rectangle
$p_0\times q'$, with $q'\le q$, deciding
$t_{01}^{-1}(l)=k\in J$. Refine $p_0$ to a stable condition $p$
for this fixed $k$, and let $a\in E$ be its witness class. Then
$p\times q'\Vdash e_1(a)=t_{01}(k)=l$. Because $e_1(a)$ is a
$P_1$-name, already $q'\Vdash e_1(a)=l$. These conditions are dense
for every $l$, so $e_1$ is forced onto. The equation $t_{12}e_1=e_2$
makes $e_2$ onto as well; a statement about its range can be read
in $P_2$ alone.

Finally, for each $a\in E$, the triangle identity gives
\[
 t_{01}^{-1}(\sigma_a)=t_{02}^{-1}(\rho_a).
\]
Separation over the shared factor $P_0$ gives a $P_0$-name $e_0(a)$
for this value. Hence $t_{01}e_0=e_1$ and $t_{02}e_0=e_2$.
These equations make $e_0$ a bijection and prove every asserted
comparison. All chosen arrays of names are sets in the ground model.
\end{proof}

\begin{lemma}\label{lem:fc-coordinate-enumeration}
There are a ground cardinal $\nu\le\mu$ and a bijective enumeration
$\bar y=\langle\dot y_\xi:\xi<\nu\rangle$ of $\dot X$ invariant
under $G_{C_A}$ for every infinite coinfinite ground $A\subseteq I$.
\end{lemma}

\begin{proof}
Partition $I$ into three infinite sets and denote their coordinate
algebras by $C_0,C_1,C_2$. Each makes $\mu$ countable, so the remaining
coordinate forcing has a countable atomless dense poset.
Its completion is a Cohen algebra.  The same is true of the quotient over
$C_{ij}=C_i\vee C_j$, where the join denotes the generated complete
subalgebra.

Choose, by Lemma~\ref{lem:fc-relative}, enumerations
$\bar x^i=\langle\dot x^i_n:n\in J\rangle$ invariant under $G_{C_i}$.
Let $t_{ij}$ be the comparison permutation, so that
\[
 1\Vdash \dot x^i_n=\dot x^j_{t_{ij}(n)}\qquad(n\in J).
\]
Every automorphism fixing $C_{ij}$ fixes both enumerations. Thus every
coefficient $\|t_{ij}(n)=m\|$ lies in $B^{G_{C_{ij}}}=C_{ij}$.
These are names over the genuine twofold coordinate products, and they
satisfy $t_{02}=t_{12}t_{01}$.  Apply
Lemma~\ref{lem:fc-effective-descent}. Each coordinate forcing has a
dense presentation of size $\mu$, so its ground index set $E$ has
size at most $\mu$. Relabel $E$ by its ground cardinal $\nu\le\mu$,
and retain the notation $e_i$ for the resulting bijections
$\nu\to J$. Mixing names, put
\[
 \dot y_\xi=\dot x^i_{e_i(\xi)}\qquad(\xi<\nu).
\]
This does not depend on $i$ modulo forced equality, and it is invariant
under all three groups $G_{C_i}$.

A transposition of component coordinates fixes at least one of the
three coordinate sets pointwise.  Hence $\bar y$ is invariant under all finitary
coordinate permutations. For an arbitrary infinite coinfinite $A$,
choose a $G_{C_A}$-invariant $J$-indexed enumeration $\bar z$ by
Lemma~\ref{lem:fc-relative}.  Compare it with $\bar y$.  Both are
invariant under finitary permutations supported on $I\setminus A$.
The scalar fixed algebra of that group is $C_A$: over $C_A$, these
finitary permutations act weakly homogeneously on the remaining
finite-function forcing, since finite domains can be moved apart.
The quotient truth values are therefore $0$ or $1$.
Mixing them over $C_A$ shows that the original invariant Boolean
elements belong to $C_A$.

Apply this to each comparison coefficient
$\|\dot y_\xi=\dot z_n\|$, for $\xi<\nu$ and $n\in J$.
Every coefficient lies in $C_A$. Every member of $G_{C_A}$ fixes
these coefficients and $\bar z$, and therefore fixes $\bar y$.
\end{proof}

Fix this enumeration for the rest of the proof, and let
\begin{equation}\label{eq:fc-K}
 K=\{g\in G:1\Vdash g\dot y_\xi=\dot y_\xi\text{ for all }\xi<\nu\}.
\end{equation}
This subgroup contains all pointwise coordinate-factor stabilizers
from Lemma~\ref{lem:fc-coordinate-enumeration}.
We will prove $K=G$ by factoring automorphisms into maps defined by
finite words.

\subsection{Prefix codes and whole-space factorizations}

For a finite word $s\in\Omega^{<\omega}$, write $[s]$ for its
cylinder and $s^\frown t$ for concatenation.  A \emph{front} is a
maximal prefix antichain of nonempty finite words.  It is a \emph{bar}
if it meets every infinite branch.  An internal node of a front is a
proper prefix of one of its leaves.  By maximality, every immediate
child of an internal node is either a leaf or an internal node.

A history-dependent prefix code is a family of bijections
\[
 c_s:\Omega\longrightarrow S_s\qquad(s\in\Omega^{<\omega}),
\]
where the $S_s$ are fronts.  Its map replaces the next input letter by
its $c_s$-word and concatenates the successive words.  For each $n$,
the union of the cylinders obtained after $n$ complete input letters
is dense open.  The intersection of these sets is exactly the range of the map.
Unique parsing gives the inverse, and the codewords are nonempty,
so the output lengths tend to infinity.  Thus the map is a
homeomorphism onto a dense $G_\delta$ subset, and is onto the whole
space if every $S_s$ is a bar.  It induces a complete Boolean
automorphism.  We use forward images for these Boolean maps and compose
from right to left.  If the code is independent of $s$, call it
\emph{stationary}.

\begin{lemma}\label{lem:fc-tree}
Every level-preserving tree automorphism of $\Omega^{<\omega}$ belongs
to $K$.
\end{lemma}

\begin{proof}
Write its point map as $T(x)(n)=t_{x\upharpoonright n}(x(n))$, where
each $t_s$ permutes $\Omega$.  Let $U$ keep the even paired positions
unchanged and replace the odd paired positions by their $T$-images.
With the even positions fixed, this is an invertible triangular
recoding of the odd positions. Recover the old odd symbols successively,
using the symbols already reconstructed.  Both directions are
continuous.  The map $V=TU^{-1}$ fixes every odd paired position.
Thus $T=VU$, and each factor fixes an infinite coinfinite set of
component coordinates pointwise.  Both belong to $K$.
\end{proof}

\begin{lemma}[Reservoirs for a bar]\label{lem:fc-reservoir}
Let $S$ be a bar on $\Omega=\mu\times\mu$.  There is a coloring
$\lambda:S\to\mu$ such that, for each $b<\mu$, first-coordinate
projection maps $\lambda^{-1}\{b\}$ bijectively onto a bar
$P_b\subseteq\mu^{<\omega}$ of cardinality $\mu$.
\end{lemma}

\begin{proof}
Assign reservoirs $B_w\subseteq\mu$ to the nodes of the front tree,
starting with $B_\varnothing=\mu$.  Internal reservoirs have size
$\mu$ and terminal reservoirs are singletons.  At an internal node
$w$, for each first-coordinate symbol $p<\mu$ separately, partition
\[
 B_w=\bigsqcup_{q<\mu}B_{w^\frown(p,q)},
\]
giving a child reservoir size $1$ if that child is terminal and size
$\mu$ if it is internal.  The sum of these $\mu$ prescribed nonzero cardinalities is $\mu$,
so the partition exists.  The partitions
for different $p$ are independent partitions of the same reservoir.
Recursion on finite node length defines all reservoirs.  Color a leaf
by the unique member of its reservoir.

Fix $b<\mu$.  At an internal node containing $b$, each next first
symbol determines exactly one second symbol whose child reservoir
contains $b$.  Every infinite first-coordinate branch therefore determines a unique
paired path until it reaches a leaf. It must reach a leaf, since
otherwise it would avoid the bar $S$.
The resulting first-coordinate leaf words form a bar.  Two leaves of
color $b$ with comparable first projections must follow the same
second-coordinate choices on the shorter word.  Neither can properly
extend the other, so they are equal.  This proves both prefix-freeness
and injectivity of projection.  Finally the projected bar has size
$\mu$: it has a leaf above every possible first symbol, and there are
at most $\mu$ finite words.
\end{proof}

\begin{lemma}\label{lem:fc-bars}
Every history-dependent bar-front code map belongs to $K$.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:fc-reservoir} to each state front $S_s$.  Choose
bijections $e_{s,b}:\mu\to P_{s,b}$.  There is a unique bijection
\[
 \theta_s:\mu\times\mu\longrightarrow S_s
\]
whose value at $(a,b)$ has color $b$ and first projection $e_{s,b}(a)$.
For an originally given labeling $c_s$, the next-letter bijections
$\theta_s^{-1}c_s$ define a level-preserving input tree map $T$.
Reindex the old states by the recursively recovered inverse of $T$.
The original code map is then $PT$, where $P$ uses the normalized
labels $\theta_s$.  From now on the labels and their reservoirs are
indexed by these transferred input histories.  By
Lemma~\ref{lem:fc-tree}, $T\in K$.

For an input $(a,b)\in\mu^\omega\times\mu^\omega$, let
$s_i=(a,b)\upharpoonright i$.  Write the two projections of
$\theta_{s_i}(a(i),b(i))$ as $u_i,v_i$, and put
\[
 A=u_0u_1\cdots,\qquad B'=v_0v_1\cdots.
\]
Thus $P(a,b)=(A,B')$.  Define $U(a,b)=(A,b)$.  For fixed $b$, its
inverse parses the next word in the bar $P_{s_i,b(i)}$ of the remaining
$A$-stream, then uses $e_{s_i,b(i)}^{-1}$ to recover $a(i)$.  The earlier reconstruction determines the state, and the unchanged
second coordinate supplies the current $b(i)$.  This procedure works on every $(A,b)$.  The codewords are nonempty, so the forward map is continuous.
The inverse is jointly continuous because each finite reconstruction
uses only finitely many letters of $A,b$.

Define $V(A,b)=(A,B')$ by first recovering $a$ through $U^{-1}$ and
then producing the second projection words.  To invert $V$ on an
arbitrary pair $(A,B')$, reconstruct $a,b$ together.  At the current state $s_i$, pair the remaining tails of $A,B'$
coordinatewise and parse their next leaf $w\in S_{s_i}$.
Since the front is a bar, this takes finitely many steps.  Recover
\[
 b(i)=\lambda_{s_i}(w),\qquad
 a(i)=e_{s_i,b(i)}^{-1}(\operatorname{pr}_1 w).
\]
The normalized code of this pair is exactly $w$.  Continue after
consuming its length in both streams.  This constructs a unique inverse
on the whole space.  Both directions of $V$ are jointly continuous, since each finite
stage reads finitely many symbols and determines its next state.  No uniform bound on that reading is needed.

Now $P=VU$, the map $U$ fixes the second set of component coordinates
pointwise, and $V$ fixes the first.  Both belong to $K$, so $PT\in K$.
\end{proof}

\begin{lemma}\label{lem:fc-stationary}
Every individual prefix swap belongs to $K$.  If $h$ is a stationary
prefix-code automorphism, then it permutes the names $\bar y$ by one
ground permutation, and $hKh^{-1}=K$.
\end{lemma}

\begin{proof}
For incompatible nonempty words $s,t$, take a finite-height bar
containing them by filling the complement of $[s]\cup[t]$ with words
at a common depth at least $\max(|s|,|t|)$.  This bar has cardinality
$\mu$.  Let $F_1$ expand the first input letter to a chosen enumeration
of this bar and use identity letter codes thereafter.  Let $F_2$ use
the same enumeration with the leaves $s,t$ exchanged.  Both maps belong
to $K$ by Lemma~\ref{lem:fc-bars}.  Their product $F_2F_1^{-1}$ is
the desired prefix swap.

Let $f$ swap two input words of the same length.  Its conjugate
$hfh^{-1}$ swaps their concatenated stationary codewords and preserves
the tails.  The two output prefixes are incompatible, though their
lengths may differ.  Thus both $f$ and $hfh^{-1}$ belong to $K$.
For every $\xi<\nu$, the name $h^{-1}\dot y_\xi$ is consequently invariant
under all such $f$.  Since $\dot X$ is invariant under $G$, these
names form another enumeration of $\dot X$.  Compare it with $\bar y$.
Every comparison coefficient
\[
 b_{\xi\eta}=\|h^{-1}\dot y_\xi=\dot y_\eta\|
       \qquad(\xi,\eta<\nu)
\]
is fixed by all equal-length prefix swaps. This group is weakly homogeneous:
any two basic conditions have appropriate equal-length extensions
interchanged by a prefix swap.  Hence the comparison values are all
$0$ or $1$. Each row and column of the comparison matrix is a
partition of unity. Thus each has exactly one nonzero entry, even
if $\nu$ is uncountable in the ground. The matrix defines a ground permutation of $\nu$.
Thus $h$ permutes $\bar y$ modulo forced equality and normalizes
its pointwise stabilizer $K$.

If the stationary front is not a bar, the conjugacy holds on its dense
parsing domain, which is invariant under the indicated prefix swap.
It is therefore the asserted identity of complete Boolean automorphisms.
\end{proof}

\subsection{Truncation and alternating fronts}

\begin{lemma}\label{lem:fc-truncation}
For every history-dependent maximal prefix-code map $F$, there
are a bar-front code map $R$, a stationary prefix-code map $h$,
and a level-preserving tree map $v$ such that
\begin{equation}\label{eq:fc-truncation}
 Rh=Fhv.
\end{equation}
\end{lemma}

\begin{proof}
Let $c_s:\Omega\to S_s$ be the state codes of $F$.
We first partition $\Omega$ into continuation and stopping
sets $C,D$ so that every internal raw node has $\mu$ witnesses
of each color in distinct immediate-child cones.

Enumerate the requests $(s,t,e,\alpha)$, where $t$ is an
internal node of $S_s$, $e\in\{C,D\}$, and $\alpha<\mu$,
in order type $\mu$. At a stage below $\mu$, fewer than $\mu$
labels have been assigned. Discard every immediate child of
$t$ whose cone contains $c_s(a)$ for a previously assigned
label $a$. Fewer than $\mu$ children are discarded. Choose a
leaf above a remaining child and assign its label to $e$.
Maximality supplies the leaf, and its label is fresh by the
choice of child. Color all remaining labels arbitrarily.

Thus $|C|=|D|=\mu$. For each $(s,t)$ we have disjoint
sets of child symbols $A^C_{s,t},A^D_{s,t}\subseteq\Omega$,
both of size $\mu$, and for $x\in A^e_{s,t}$ a chosen
label $a^e_{s,t,x}\in e$ such that
\[
 t^\frown x\subseteq c_s(a^e_{s,t,x}).
\]
All chosen child symbols for this fixed $(s,t)$ are distinct,
including across the two colors. A previously chosen witness
would have excluded that child at every later request.

Choose a stationary bijection $d:\Omega\to C^{<\omega}D$
and let $h$ be its concatenation map. The composite $Fh$ has
maximal fronts obtained by encoding raw letters through $F$
until the first stopping letter has been encoded. They are
maximal because an unfinished raw code can be completed and,
if necessary, followed by a stopping letter.

At any internal node $q$ of a composite front, finite parsing
determines a current raw state $s$ and an internal raw node
$t$, possibly empty. The suffix of $c_s(a^D_{s,t,x})$ after
$t$ leads from $q$ to a composite terminal node. The suffix
of $c_s(a^C_{s,t,x})$ leads to a composite internal node,
since it finishes a continuation code. Denote these nonempty
suffixes by $w^e_{q,x}$. Their first symbols are the distinct
chosen child symbols.

Choose a positive finite cutoff for each immediate-child cone:
\[
 k_q(x)=
 \begin{cases}
 |w^e_{q,x}|,&x\in A^e_{s,t},\ e\in\{C,D\},\\
 1,&x\notin A^C_{s,t}\cup A^D_{s,t}.
 \end{cases}
\]
Starting at $q$, stop at the first composite terminal or at
distance $k_q(x)$, whichever comes first, where $x$ is the
first new symbol. The resulting relative front $T_q$ is a bar.
Each branch has a first symbol, which determines a finite cutoff. Each selected witness survives, since no composite
terminal properly precedes it. Consequently $T_q$ has exactly
$\mu$ terminal pieces and exactly $\mu$ internal pieces.
No bound uniform over its first-child cones is required.

At each state match $C$ bijectively to the internal pieces
and $D$ bijectively to the terminal pieces of $T_q$.
Reading a letter emits its matched piece. An internal piece
advances within the current composite front; a terminal piece
finishes it, recovers its composite input label, and starts
the next front with that recovered history. This defines a
history-dependent bar-front code $R$ and hence a whole-space
homeomorphism.

Each input $d$-word produces exactly one composite terminal
leaf. Conversely, parse any composite leaf into successive
$T_q$ pieces. The terminal stopping rule stops parsing at the leaf.
Since the pieces are nonempty, parsing terminates after finitely many
steps. The corresponding input letters lie in $C$ until
the final letter in $D$, so form one unique $d$-word. Thus the
matching of composite input labels is bijective at every state.
These bijections define a level-preserving tree map $v$.
For a source stream $x$, the successive composite labels
emitted by $Rh(x)$ are $v(x)$; therefore $Rh(x)=Fhv(x)$.
This is also the asserted identity of Boolean automorphisms.
\end{proof}

\begin{corollary}\label{cor:fc-all-codes}
Every history-dependent maximal
prefix-code automorphism belongs to $K$.
\end{corollary}

\begin{proof}
In \eqref{eq:fc-truncation}, Lemmas~\ref{lem:fc-tree} and
\ref{lem:fc-bars} give $v,R\in K$, and
Lemma~\ref{lem:fc-stationary} gives $hKh^{-1}=K$.  Therefore
$F=Rh v^{-1}h^{-1}\in K$.
\end{proof}

\begin{lemma}\label{lem:fc-alternating}
Every $g\in\Aut(B)$ has a factorization
\[
 g=P\,b^{-1}o\,Q^{-1},
\]
where $P,Q$ are history-dependent maximal prefix-code automorphisms,
$b$ replaces every input letter by two letters using one fixed
bijection $\Omega\to\Omega^2$, and $o$ keeps the first input letter
single and uses that bijection thereafter.
\end{lemma}

\begin{proof}
Let $\mathcal A_n$ be the complete atomic algebra with atoms the
length-$n$ cylinders, with $\mathcal A_0=\{0,1\}$.  Construct
increasing complete atomic algebras $\mathcal D_n$, starting at
$\mathcal D_0=\{0,1\}$, such that every atom has exactly $\mu$
children at the next level.  At even levels their atoms are ordinary
cylinders; at odd levels they are $g$-images of cylinders.

Both cylinder bases are order dense and have cardinality $\mu$.
Below a current atom choose a maximal antichain from the next required
basis.  Its join is that atom and its size is at most $\mu$.  Replace
one member by its $\mu$ immediate successors in that same basis.
The resulting antichain has exactly $\mu$ members.  Repeating this
below every atom gives the next level.  Each refinement is strict.
Between consecutive even levels, every cylinder is thus replaced by
proper subcylinders, so the atoms of $\mathcal D_{2n}$ have lengths
at least $n$.  Hence $\mathcal A_n\subseteq\mathcal D_{2n}$,
and the union of the $\mathcal D_n$ is order dense.  Similarly the
cylinders whose images are atoms of $\mathcal D_{2n-1}$ have lengths
at least $n$.

Choose bijections of corresponding sets of children.  They define
compatible atomic isomorphisms between $\mathcal A_n$ and
$\mathcal D_n$.  The isomorphism on their unions extends uniquely
to a complete Boolean automorphism $t$, since both unions are order
dense.  Thus $t[\mathcal A_n]=\mathcal D_n$.

The stated reblockings satisfy
\[
 b[\mathcal A_n]=\mathcal A_{2n},\qquad
 o[\mathcal A_n]=\mathcal A_{2n-1}\quad(n\ge1).
\]
Put $P=tb$ and $Q=g^{-1}to$.  Each sends input cylinders to actual
cylinders, and sends immediate successors to a maximal antichain of
proper subcylinders of the parent image.  Their nonempty relative
suffixes give the required state fronts and labels.  At the root the
images for $Q$ are proper as well.  The resulting code maps induce
the prescribed Boolean maps, since they agree on every cylinder.
Finally,
\[
 (tb)(b^{-1}o)(g^{-1}to)^{-1}=g.
\]
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:countable-collapse-invariant}]
Choose $\bar y$ by Lemma~\ref{lem:fc-coordinate-enumeration} and
form $K$ as in \eqref{eq:fc-K}.  For an arbitrary $g\in G$, apply
Lemma~\ref{lem:fc-alternating}.  Its maps $P,Q$ belong to $K$ by
Corollary~\ref{cor:fc-all-codes}. Both $b$ and $o$ are
history-dependent bar-front code maps, so Lemma~\ref{lem:fc-bars}
gives $b,o\in K$.
Every factor in the displayed factorization of $g$ therefore belongs
to $K$. Thus $K=G$. If $\dot X$ is forced to have finite size $m$,
the ground cardinal $\nu$ is also forced to have size $m$ and hence
equals $m$ already in $W$, since an infinite ground set cannot become
finite.
\end{proof}

\begin{corollary}\label{cor:countable-collapse-od}\label{cor:finite-collapse-od}
Let $a$ be a set of ordinals, $W=L[a]$, and let $V=W[H]$ be a
$\Coll(\omega,\mu)^W$-extension. Every countable
$\OD(a)$ set in $V$ admits an $\OD(a)$ bijection from some ground
cardinal $\nu\le\mu$. In particular, all its members are $\OD(a)$,
without a restriction on their ranks. For a finite set of size $m$,
one has $\nu=m$.
\end{corollary}

\begin{proof}
The same proof works with any ground $W$ satisfying
$\mathsf{ZFC}+\mathsf{GA}+V=\mathrm{HOD}$ and any
ground set of ordinals $a$.  For ground definability, we use only \cref{lem:stable-ground-codes}
or the canonical $a$-definable order when $W=L[a]$.
No GCH assumption is needed.
For the empty family, use the empty bijection with $\nu=0$.
Fix a formula and ordinal parameters uniquely defining the given
nonempty countable family. Weak homogeneity makes this assertion
forced by the top condition. A name for the unique family is invariant
under all ground Boolean automorphisms.
Theorem~\ref{thm:countable-collapse-invariant} supplies a bijective
enumeration by invariant names indexed by a ground cardinal
$\nu\le\mu$. The name for the whole enumeration is invariant too,
since its index ordinal is fixed. Lemma~\ref{lem:invariant-od}
makes this bijection $\OD(a)$ in $V$. Each of its values is then
$\OD(a)$ using its ordinal index.
\end{proof}

\begin{remark}
The theorem does not in general give an OD enumeration indexed by
$\omega$. If $\mu$ is uncountable in $L$, the ground set $\mu$ is
an OD countable set in the collapse extension, but an OD bijection
$\omega\to\mu$ would be a new OD set of ordinals, contrary to
homogeneity over $L$. The ground-cardinal index in the conclusion
is therefore necessary. The truncation fronts need not have bounded height.
The reservoir factorization requires only that parsing terminate
on each branch.
\end{remark}

\subsection{Rank-one consequences and invariant parameters}\label{sec:collapse}

\begin{corollary}\label{thm:collapse-rank-one}\label{cor:collapse-open-families}
Let $W=L[z]$, where $z$ is a real, let $\lambda$ be an infinite
cardinal of $W$, and let $V$ be a
$\Coll(\omega,\lambda)^W$-extension. Every countable $\OD(z)$
family of sets of ordinals consists of $\OD(z)$ members.
The same holds for a countable $\OD(z)$ family of open subsets
of an $\OD(z)$ set $D$ of generic filters, with basic opens
$N_p=\{H\in D:p\in H\}$.
\end{corollary}
\begin{proof}
Both are instances of \cref{cor:countable-collapse-od}. Equivalently,
an open set $U$ is recovered from the ordinal code
$c(U)=\{p:N_p\subseteq U\}$ by $U=\bigcup_{p\in c(U)}N_p$.
The stable-ground generalization is the same as in that corollary.
\end{proof}

The following variant permits a parameter fixed only by a subgroup.
For an infinite cardinal $\lambda$ in a ZFC ground $W$, use
the dense presentation $\mathbb P=\lambda^{<\omega}$.

\begin{lemma}[Generic graphs]\label{lem:generic-graph}
For $p\in\mathbb P$ and a family $\mathcal E$ of at most
$\lambda$ dense open subsets of $\mathbb P^2$, there is a
ground tree automorphism $h$ fixing $p$ such that
$\{q\supseteq p:(q,h(q))\in E\}$ is dense below $p$ for every
$E\in\mathcal E$.
\end{lemma}
\begin{proof}
In $\lambda$ steps meet every requirement $(E,s)$ with
$s\supseteq p$, and put every node into the domain and range.
Start by fixing the initial segments of $p$. For $(E,s)$,
extend the partial tree isomorphism to $s\mapsto t$, choose
unused successors $s^\frown\alpha,t^\frown\beta$, and find
$(q,r)\in E$ extending this pair. Extend one coordinate to
make their lengths equal, then match their fresh finite paths.
Domain and range requirements use the same back-and-forth
extension. At stage $\xi<\lambda$ the partial map is finite
if $\xi<\omega$, and otherwise has size at most
$\max(\aleph_0,|\xi|)<\lambda$. Fresh successors therefore
remain, even when $\lambda$ is singular. The union is the
required total automorphism.
\end{proof}

\begin{lemma}[Invariant parameters]\label{lem:collapse-invariant-parameter}
Suppose every ground set is $\OD(z)$ in every set-forcing
extension of $W$, and $\Gamma\leq\Aut(\mathbb P)$ has the
generic-graph property of \cref{lem:generic-graph} for every
countable $\mathcal E$ and every $p$. If $\dot C$ is fixed by
$\Gamma$, every countable $\OD(z,C)$ family of sets of ordinals
in a $\mathbb P$-extension is pointwise $\OD(z,C)$.
\end{lemma}
\begin{proof}
Otherwise its non-$\OD(z,C)$ subfamily is nonempty. Below some
$p$ choose names $(\tau_n)_{n<\omega}$ enumerating it.
For each $n,m$, $(p,p)$ forces $\tau_n^0\ne\tau_m^1$.
Otherwise a common value in two mutually generic extensions would
belong to their intersection $W$ and hence be $\OD(z)$,
a contradiction.
Since these are sets of ordinals, pairs deciding an ordinal
oppositely for $\tau_n,\tau_m$ form a dense open set below
$(p,p)$. Extend these dense sets outside that cone and apply
the stipulated property to obtain $h\in\Gamma$ fixing $p$.
For the actual generic $G$ containing $p$, it gives
$\tau_n^G\ne\tau_m^{h[G]}$ for all $n,m$.
But $W[G]=W[h[G]]$ and $C^G=C^{h[G]}$, so both enumerations
describe the same nonempty definable family, a contradiction.
\end{proof}


\section{Definable families modulo null and meager sets}
\label{sec:friedman-solovay}

Let $\mathcal N$ and $\mathcal M$ denote the ideals of Lebesgue-null
and meager subsets of $\mathbb R$, respectively. For either ideal
$\mathcal I$, write $A\equiv_{\mathcal I}B$ when
$A\mathbin\triangle B\in\mathcal I$, and let
$[A]_{\mathcal I}$ denote the resulting class in
$\mathcal P(\mathbb R)/\mathcal I$.
Friedman \cite{FriedmanNull} obtained the small-family conclusion below
in a further extension of Solovay's model, and asked whether its
single-class case already holds in the original collapse extension.
We prove the full conclusion there.

\begin{theorem}\label{thm:friedman-family}
Let $W\models\mathsf{ZFC}$, let $\kappa$ be inaccessible in $W$,
and let $G\subseteq\Coll(\omega,{<}\kappa)^W$ be generic.
In $W[G]$, an $\ODR$ family $K\subseteq\mathcal P(\mathbb R)$
containing a non-Lebesgue-measurable member represents exactly
$2^{\mathfrak c}$ classes modulo $\mathcal N$.
Consequently, if it represents fewer than $2^{\mathfrak c}$ classes,
all its members are Lebesgue measurable.
The same statements hold for the Baire property and $\mathcal M$.
\end{theorem}

The extension here is the full ZFC generic extension, rather than its
inner Solovay model. No additional hypothesis on $W$ is required.
The same proof applies to both ideals.

\subsection{Bounded collapse algebras and local separation}

Work in $W$ and use the finite-support presentation
\[
 P=\Coll(\omega,{<}\kappa),\qquad
 \B=\RO(P),\qquad
 \B_\alpha=\RO(\Coll(\omega,{<}\alpha)).
\]
The forcing is $\kappa$-cc, $|\B|=\kappa$, and
$|\B_\alpha|<\kappa$ for $\alpha<\kappa$.
Every Boolean value and every name for a real is supported on some
$\B_\alpha$. To see this, represent them by maximal antichains of
size less than $\kappa$ and bound their coordinates. There are $\kappa$ nice names for reals.
In the full extension, the reals of each bounded intermediate model
$W[G_\alpha]$ form a countable set. There are fewer than $\kappa$ nice
$\B_\alpha$-names for reals in $W$, and the full forcing makes
this set of names countable.

We will repeatedly use two properties of the initial algebras.
First, an automorphism of $\B_\alpha$ extends to every larger
$\B_\beta$ by acting identically on the independent tail.
Second, if $\langle\alpha_i:i<\eta\rangle$ is increasing with limit
$\delta$, then $\bigcup_{i<\eta}\B_{\alpha_i}$ is dense in
$\B_\delta$, since it contains the initial forcing posets.
Thus coherent automorphisms defined on whole initial algebras
extend to their completion at a limit stage.

\begin{lemma}[Bounded extension]\label{lem:friedman-extension}
If $E,F$ are complete subalgebras of $\B_\gamma$ and
$h:E\longrightarrow F$ is a complete isomorphism, then $h$ extends
to an automorphism of some $\B_\delta$, where $\delta<\kappa$
may be required to exceed any prescribed bound.
\end{lemma}

\begin{proof}
Choose a regular uncountable cardinal $\lambda<\kappa$ above that
bound and above $|\B_\gamma|$, and put $\delta=\lambda+1$.
In an $E$-generic extension, $\lambda$ remains regular and the quotient
$\B_\delta/E$ has density at most $\lambda$ and collapses $\lambda$
to countable. Its density is exactly $\lambda$, since a forcing of
smaller density would preserve the regular cardinal $\lambda$.
Its completion is therefore the collapse algebra
$\RO(\Coll(\omega,\lambda))$.
The same holds over $F$. Identify these quotients through $h$, then
use the complete-subalgebra factorization to lift $h$ to $\B_\delta$.
For the collapse characterization and relative factorization used here,
see \cite{Jech,KostanaCollapse}.
\end{proof}

Fix $\mathcal I\in\{\mathcal N,\mathcal M\}$.
For a set $A$ failing the corresponding regularity property, there
is a positive Borel region $D$ on which both $A$ and its complement
meet every positive Borel set even after removal of any member of
$\mathcal I$.
In the measure case, first restrict to a bounded interval on which
$A$ is nonmeasurable. Choose a Borel inner core $I\subseteq A$ and
outer hull $O\supseteq A$ attaining its inner and outer measures.
Then $D=O\setminus I$ has positive finite measure, and
$A\cap D$ and $D\setminus A$ both have full outer measure in $D$.
In the category case, let $U$ be the union of the basic open sets on which
$A$ is meager and $V$ the corresponding union for its complement.
The sets $U,V$ are disjoint. If their union were dense, $A$ would
have the Baire property. Otherwise choose a nonempty basic open
$D$ disjoint from its closure. Both sides are nonmeager in every
nonempty open subset of $D$, which gives the stated property for
positive Borel sets.

Now suppose a basic condition $p_0$ forces this property of names
$A,D$. Fix an initial stage $\B_{\alpha_0}$ supporting
$p_0$ and the Borel code of $D$. All our automorphisms fix this
algebra pointwise. Dots on names are suppressed.

\begin{lemma}[Persistent pair separation]
\label{lem:friedman-pair}
Let $\theta_s,\theta_t\in\Aut(\B_\alpha)$ fix
$\B_{\alpha_0}$ pointwise. Suppose $0<p\leq p_0$ belongs to
$\B_\alpha$ and $N$ is a $\B_\alpha$-name for a Borel member
of $\mathcal I$.
There are extensions $\sigma_s,\sigma_t\in\Aut(\B_\beta)$,
for some $\beta<\kappa$, and $0<f\leq p$ such that every pair
of full automorphisms $\pi_s,\pi_t\in\Aut(\B)$ extending them
satisfies
\[
 f\Vdash
 (\pi_s(A)\mathbin\triangle\pi_t(A))\setminus N
 \ne\varnothing.
\]
The two full extensions may be chosen independently.
\end{lemma}

\begin{proof}
We first construct a one-sided separation for an automorphism
$\theta$ of $C=\B_\alpha$, a condition $0<p\leq p_0$ in $C$,
and a $C$-name $N$ as above. Put $q=\theta^{-1}(p)$.
We will extend $\theta$ to $\tau\in\Aut(\B_\beta)$ and find
conditions $e,r\in\B_\beta$ and real names $x,y$ supported there
such that
\begin{equation}\label{eq:friedman-retained}
 \begin{gathered}
 \tau(e)=r,\qquad 0<r\leq p,\qquad
 e\Vdash x\in A,\qquad r\Vdash y\notin A,\\
 r\Vdash \tau(x)=y\ \text{ and }\ y\notin N.
 \end{gathered}
\end{equation}

In a $C$-generic extension, let $Q_D$ be the algebra of Borel
subsets of $D$ modulo $\mathcal I$.
This is the complete ccc random algebra in the measure case and
the complete ccc Cohen algebra in the category case.
In the full extension, the union of the Borel members of
$\mathcal I$ coded in this intermediate model belongs to
$\mathcal I$, since its family of codes has become countable.
Thus the reals generic for $Q_D$ over the intermediate model
form a conull, respectively comeager, subset of $D$.
The property of $A\cap D$ therefore allows us to choose a name $x$
such that $q$ forces that $x\in A\cap D$ is generic over that
model.

In a $C$-generic extension containing $q$, this name induces a
unital complete homomorphism into the remaining forcing algebra:
\[
 j_x:Q_D\longrightarrow\RO(\B/G_C),
 \qquad [H]_{\mathcal I}\longmapsto\|x\in H\|.
\]
Genericity makes the map well-defined, and Borel evaluation preserves
countable joins. Since $Q_D$ is ccc, each of its joins reduces to
a countable join, so the map is complete.
Its kernel is therefore a principal ideal. Let $u$ be the complement
of the largest element of the kernel. Then $u>0$, $j_x(u)=1$,
and $j_x$ is injective on $Q_D\restr u$.
Choose a $C$-name $U$ for a positive Borel representative of $u$.

On the range side, below $p$, choose $y\in D\setminus A$ generic
over the range intermediate model and belonging to $\theta(U)$.
This is possible by the same property of the complement.
Let $V$ be a $C$-name representing the positive support of $j_y$; thus
$[V]_{\mathcal I}\leq[\theta(U)]_{\mathcal I}$.
Restrict the domain side to
\[
 t=q\wedge\|x\in\theta^{-1}(V)\|.
\]
Its Boolean projection onto $C$ is $q$, because
$\theta^{-1}([V]_{\mathcal I})$ is positive below $u$ and $j_x$
is injective there. The restricted random or Cohen homomorphism
has exactly this smaller positive support.

All these real names and Boolean values occur at a bounded initial
stage. Since membership in each Borel set coded in the fixed intermediate
model is absolute, genericity over that model is absolute to further
forcing. Hence the homomorphisms above can already be computed
in the completion of a bounded quotient containing their real
names, with the same kernels and supports.
Beyond this stage choose fresh binary splits $c,d$, each of whose
two values has projection $1$ onto that stage, and put
\[
 e=t\wedge c,\qquad r=p\wedge d.
\]
Both $e$ and $q\setminus e$ project to $q$ in $C$; both $r$ and
$p\setminus r$ project to $p$. Because the splits are fresh, these restrictions send no nonzero
element of the respective support algebras to zero.

Form the complete subalgebra generated over $C$ by $e$ and the
membership values of $x$ restricted to $e$.
Over a $C$-generic filter containing $q$, it is the lottery sum
of $Q_D\restr[\theta^{-1}(V)]_{\mathcal I}$ and a trivial
branch. The first branch is embedded by
$[H]_{\mathcal I}\mapsto e\wedge j_x([H]_{\mathcal I})$;
the second is represented by $q\setminus e$.
Outside $q$ the quotient is trivial.
Similarly, the range algebra has branches
$Q_D\restr[V]_{\mathcal I}$ and the trivial branch below
$p\setminus r$. These descriptions give complete embeddings
into a common bounded initial collapse algebra.
They also give a complete isomorphism extending $\theta$,
sending $e$ to $r$, and matching the membership values of $x$
and $y$ on those branches. Every extension to an algebra supporting $x$ and $y$ therefore sends
$x$ to $y$ below $r$. After a fixed Borel coding of reals, the restricted
values match each binary digit.
Apply \cref{lem:friedman-extension} to extend this isomorphism
to $\tau\in\Aut(\B_\beta)$.

Our choices give $e\Vdash x\in A$ and $r\Vdash y\notin A$.
The genericity of $y$ over a model containing the code of $N$
also gives $y\notin N$. This proves \eqref{eq:friedman-retained}.
No bound on the support of $A$ was used.

To separate the given pair, apply this construction to
\[
 \theta=\theta_t^{-1}\theta_s,\qquad
 p'=\theta_t^{-1}(p),\qquad
 N'=\theta_t^{-1}(N).
\]
Obtain $\tau,e,r,x,y$ satisfying \eqref{eq:friedman-retained}
with $p',N'$ in place of $p,N$.
Extend $\theta_t$ to $\eta\in\Aut(\B_\beta)$ by the identity
on the tail, and set
\[
 \sigma_s=\eta\tau,\qquad \sigma_t=\eta,
 \qquad f=\eta(r)\leq p.
\]
These extend the required initial maps. For any independent full
extensions $\pi_s,\pi_t$, the equations for the bounded names give
\[
 \pi_s(e)=\pi_t(r)=f,
 \qquad f\Vdash\pi_s(x)=\pi_t(y).
\]
Applying the full automorphisms to the two membership inequalities
in \eqref{eq:friedman-retained}, this common real belongs to
$\pi_s(A)\setminus\pi_t(A)$ below $f$.
It lies outside $N$, since $\pi_t(N')=N$.
The equations matching the real names involve only bounded-stage
Boolean values and so persist under further extensions; the
membership inequalities are preserved by every full automorphism.
Thus the separation holds for every independent choice of the two
full extensions.
\end{proof}

\subsection{The maximal family of inequivalent images}

\begin{proof}[Proof of \cref{thm:friedman-family}]
Fix $\mathcal I\in\{\mathcal N,\mathcal M\}$.
Suppose a basic condition $p_0$ forces that $K$ is the family
defined by a fixed formula from real and ordinal parameters and
that $A\in K$ fails the corresponding regularity property.
Strengthen $p_0$ and choose $D$ as above. Take $\alpha_0<\kappa$
supporting $p_0$, the real parameters, and the code of $D$.
We retain the original $A\in K$; the region $D$ is used only
to choose the witnesses in \cref{lem:friedman-pair}.
Every full automorphism fixing $\B_{\alpha_0}$ pointwise preserves
the definition of $K$ below $p_0$. It therefore sends $A$ to a member
of $K$ below that condition.

Enumerate in $W$ all pairs $(p_i,N_i)$, $i<\kappa$, where $p_i$
is a basic condition below $p_0$ and $N_i$ is a name for a Borel
member of $\mathcal I$.
It suffices to use nice real names, replacing a name by the code
of the empty set on the Boolean region where it fails to code a
Borel member of $\mathcal I$.
This list includes a name for every possible Borel $\mathcal I$-cover.

We build a binary tree of partial automorphisms.
For each $\xi<\kappa$, assign to every node $s\in2^\xi$ an automorphism
$\theta_s$ of the same whole initial algebra $\B_{\alpha_\xi}$.
They fix $\B_{\alpha_0}$ pointwise and extend the maps at their
predecessors. Require the stages to increase cofinally in $\kappa$.
At a successor level initially give both children their parent's
map. At a limit level extend the coherent union along each node
to the completion of the common initial algebra.

At level $\xi$, first enlarge the common stage to support all
$(p_i,N_i)$ with $i<\xi$.
For every pair of distinct nodes $s,t\in2^\xi$ and every $i<\xi$,
apply \cref{lem:friedman-pair} to these two maps and $(p_i,N_i)$.
Replace them by the resulting extensions and extend all other
maps to the same larger initial stage by the identity on the tail.
Earlier separations persist under these independent extensions.
There are fewer than $\kappa$ tasks at this level, since
$2^{|\xi|}<\kappa$. At limits within the level take completions
of the coherent maps on whole initial algebras. Regularity of
$\kappa$ keeps the common stage below $\kappa$.
Finally enlarge it beyond $\xi$.
The recursion is now complete. At each limit step, the union of the
initial algebras is dense in the next initial algebra, as required.

Each ground-model branch $b\in(2^\kappa)^W$ now gives a full
automorphism $\pi_b$ of $\B$.
If $b\ne c$ and $i<\kappa$, choose a level $\xi>i$ after their
first split. The requirement for $b\restr\xi$, $c\restr\xi$,
and $i$ supplies $0<f\leq p_i$ forcing a point of
$\pi_b(A)\mathbin\triangle\pi_c(A)$ outside $N_i$.
Consequently,
\begin{equation}\label{eq:friedman-distinct-images}
 p_0\Vdash [\pi_b(A)]_{\mathcal I}
              \ne[\pi_c(A)]_{\mathcal I}
 \qquad(b\ne c).
\end{equation}
Indeed, a condition forcing equivalence would force the symmetric
difference into a Borel member of $\mathcal I$.
Take a name for that cover and a basic condition below the proposed
condition. The corresponding requirement gives a contradiction.

It remains to count these ground-model branches in the extension.
There $\mathfrak c=\kappa=\omega_1$ and
\[
 (2^\kappa)^{W[G]}=(2^\kappa)^W
 \quad\text{as cardinals}.
\]
For the latter equality, $|\B|^W=\kappa$, so subsets of $\kappa$
have at most $\kappa^\kappa=(2^\kappa)^W$ Boolean names.
The $\kappa$-cc preserves this ground cardinal, and the ground
subsets give the reverse inequality.
Thus the branch set used in \eqref{eq:friedman-distinct-images}
has size $2^{\mathfrak c}$ in the extension.
All images belong to $K$, so $K$ represents at least that many
classes, and it cannot represent more than
$|\mathcal P(\mathbb R)|=2^{\mathfrak c}$.
\end{proof}

\begin{corollary}\label{cor:friedman-class}
In the extension of \cref{thm:friedman-family}, if
$[A]_{\mathcal N}$ is $\ODR$, then $A$ is Lebesgue measurable.
If $[A]_{\mathcal M}$ is $\ODR$, then $A$ has the Baire property.
In either case the class has a Borel representative.
\end{corollary}

\begin{proof}
Apply the theorem to the family $K=[A]_{\mathcal I}$, which
represents just one class. Every measurable set, respectively every
set with the Baire property, agrees with a Borel set modulo the
corresponding ideal.
\end{proof}

\section{Small families in Cohen extensions}
\label{sec:higher-cohen}\label{sec:generalized-small-families}

We first prove a small-index theorem over any ZFC ground satisfying
the cardinal-arithmetic assumptions below, then apply it to definable
families. We distinguish the closure cardinal $\theta$ from the width
$\lambda$. Ordinary Cohen forcing is the case $\theta=\omega$.
The construction follows the
generic-automorphism and coset-tree method of
\cite{LS,GV,CohenSmallIndexNotes}.

Work in an arbitrary $W\models\mathsf{ZFC}$. Let $\theta$ be regular infinite,
let $\lambda>\theta$ be regular uncountable, and assume
\begin{equation}\label{eq:gcf-cardinal-assumptions}
 \theta^{<\theta}=\theta,\qquad
 \nu^{<\theta}<\lambda\quad(\nu<\lambda),\qquad
 \lambda^{<\lambda}=\lambda.
\end{equation}
Here $\nu$ ranges over cardinals. Put
\[
 P=\Add(\theta,\lambda)
   =\operatorname{Fn}(\lambda\times\theta,2,{<}\theta),\qquad
 B=\RO(P),\qquad G=\Aut(B).
\]
For $S\subseteq\lambda$, write $P_S=\Add(\theta,S)$ and let
$B_S$ be its specified complete coordinate subalgebra of $B$.
Write $b_{\alpha,\xi}$ for the Boolean value that coordinate bit
$(\alpha,\xi)$ is $1$. For a complete $C\subseteq B$, put
$G_C=\{g\in G:g\restr C=\id_C\}$.

\begin{theorem}[Cohen small index]\label{thm:cohen-small-index}
\label{thm:ordinary-small-index}\label{thm:generalized-small-index}
Every subgroup $H\leq G$ of index at most $\lambda$ contains
$G_{B_S}$ for some $S\in[\lambda]^{<\lambda}$.
Equivalently, it contains the pointwise stabilizer of a complete
subalgebra having an order-dense subset of size less than $\lambda$.
\end{theorem}

For the forward implication, $B_S$ has an order-dense Boolean subalgebra
of size at most $\bigl(\max(\theta,|S|)\bigr)^{<\theta}<\lambda$
when $S\ne\varnothing$, obtained by closing its conditions under
finite Boolean operations. Conversely, the supports of fewer than
$\lambda$ Boolean elements have union of size less than $\lambda$
by the counting lemma below. Thus a complete subalgebra with a
small dense subset is contained in a small $B_S$, and its
pointwise stabilizer contains $G_{B_S}$.
For $\theta=\omega$, the assumptions reduce to regular uncountable
$\lambda$ with $\lambda^{<\lambda}=\lambda$.
In particular the ordinary theorem holds over every ZFC+GCH ground.
For uncountable $\theta$, GCH implies the displayed assumptions when
$\lambda=\theta^+$; for other widths we retain all three assumptions.

\subsection{Coordinate factors and relative amalgamation}

For complete Boolean algebras $A,D$, let
$A\mathbin{\widehat\otimes}D$ denote the completion of the forcing
product $A^+\times D^+$, not their Boolean direct product. Put
$\mathbb C_\nu=\RO(\Add(\theta,\nu))$, with
$\mathbb C_0=\{0,1\}$. An isomorphism over $C$ fixes its
specified copy of $C$ pointwise.

\begin{definition}
Let
\[
 \mathcal F=\{g[B_S]:g\in G,\ S\in[\lambda]^{<\lambda}\}.
\]
For $C,D\in\mathcal F$, write $C\preceq D$ if some single
$g\in G$ and $S\subseteq T$ in $[\lambda]^{<\lambda}$ satisfy
$C=g[B_S]$ and $D=g[B_T]$.
\end{definition}

Both $\mathcal F$ and $\preceq$ are invariant under $G$.
The relation requires a simultaneous coordinate presentation, so it is
stronger than inclusion of complete subalgebras.

\begin{lemma}\label{lem:hc-factor-extension}
Every $C\in\mathcal F$ has a factorization
$B\cong C\mathbin{\widehat\otimes}\mathbb C_\lambda$ over $C$.
Every complete isomorphism between members of $\mathcal F$ extends
to an element of $G$.
\end{lemma}
\begin{proof}
For $C=B_S$, the unused coordinates have cardinality $\lambda$.
For an arbitrary member of $\mathcal F$, transport this product through
its coordinate presentation. An isomorphism between two such factors combines with
any isomorphism of their full Cohen complements to give an
automorphism of the two product presentations of $B$.
\end{proof}

\begin{lemma}\label{lem:hc-relative-product}
For $C,D\in\mathcal F$, one has $C\preceq D$ exactly when,
for some $\nu<\lambda$, there is an isomorphism
$C\mathbin{\widehat\otimes}\mathbb C_\nu\cong D$ over $C$.
In particular, $\preceq$ is transitive.
\end{lemma}
\begin{proof}
A simultaneous presentation uses the coordinates in $T\setminus S$
as the relative complement. Conversely, write $C=u[B_S]$ and
choose $U\subseteq\lambda\setminus S$ of cardinality $\nu$.
The relative product gives a complete isomorphism
$B_{S\cup U}\to D$ restricting to $u$ on $B_S$.
Lemma~\ref{lem:hc-factor-extension} extends it to a full
automorphism, which witnesses the required simultaneous presentation.
Two relative Cohen complements combine to a Cohen complement on
fewer than $\lambda$ coordinates, proving transitivity.
\end{proof}

\begin{lemma}\label{lem:hc-counting}
Every Boolean element has a column support of size at most $\theta$,
and $|B|=\lambda$. There are at most $\lambda$ members of $\mathcal F$,
complete isomorphisms between them, and tuples of such isomorphisms
of length less than $\lambda$.
\end{lemma}
\begin{proof}
The forcing has size $\lambda^{<\theta}=\lambda$ and is
$\theta^+$-cc. Indeed, the generalized delta-system lemma, using
regularity and $\theta^{<\theta}=\theta$, gives a delta system of
$\theta^+$ domains from any such family of conditions. There are
at most $\theta$ assignments on the common root, so two conditions
agree there and are compatible. For $\theta=\omega$ this is the
ordinary finite-set delta-system proof of ccc.

Every Boolean value is a join of a maximal antichain of conditions
below it, of size at most $\theta$. The union of their supports has
size at most $\theta$. Thus $|B|\leq\lambda^\theta=\lambda$,
and the coordinate bits give the reverse inequality.
A factor $B_S$ has $|S|\cdot\theta<\lambda$ complete generators.
An image factor and a complete isomorphism are determined by the
images of these generators. The assumption
$\lambda^{<\lambda}=\lambda$ gives the asserted bounds, including the
bound for tuples.
\end{proof}

\begin{lemma}[Invariant coordinate enlargement]
\label{lem:hc-invariant-enlargement}
Suppose $\mathcal E\subseteq\mathcal F$ and $\Gamma\subseteq G$
have cardinality less than $\lambda$, and $T\in[\lambda]^{<\lambda}$.
There is $S\in[\lambda]^{<\lambda}$ containing $T$ such that
\[
 g[B_S]=B_S\quad(g\in\Gamma),\qquad
 C\preceq B_S\quad(C\in\mathcal E).
\]
The factor may also contain any prescribed set of fewer than
$\lambda$ Boolean elements.
\end{lemma}
\begin{proof}
Choose presentations $C=u_C[B_{T_C}]$ for $C\in\mathcal E$.
Adjoin these $u_C$ and all inverses to $\Gamma$, obtaining a family
$\Gamma^*$ of size less than $\lambda$. Let $S_0$ contain $T$,
all $T_C$, and supports of size at most $\theta$ for the prescribed
Boolean elements. From $S_m$, form $S_{m+1}$ by adjoining
supports of size at most $\theta$ for
\[
 g(b_{\alpha,\xi})\qquad
 (g\in\Gamma^*,\ \alpha\in S_m,\ \xi<\theta).
\]
There are fewer than $\lambda$ maps and bits at every stage.
By regularity, $S=\bigcup_{m<\omega}S_m$ has size less than
$\lambda$. The coordinate bits completely generate $B_S$, so
each $g\in\Gamma^*$ maps it into itself; inverse closure gives
equality. In particular $u_C[B_S]=B_S$ and $T_C\subseteq S$.
The same $u_C$ therefore witnesses
$C=u_C[B_{T_C}]\preceq u_C[B_S]=B_S$.
\end{proof}

The closure under the presentations $u_C$ ensures the relative factor
relation. Collecting only the supports of elements of $C$ would not
establish it.

\begin{lemma}[Relative amalgamation]\label{lem:hc-amalgamation}
Let $C\preceq D_0,D_1$ belong to $\mathcal F$. For $i$ in an
index set $J$, suppose $s_i\in\Aut(D_0)$ and
$t_i\in\Aut(D_1)$ preserve $C$ and agree there. There are
$a\in G_C$, $E\in\mathcal F$, and $r_i\in\Aut(E)$ such that
$D_0\preceq E$, $a[D_1]\preceq E$, and $r_i$ extends both
$s_i$ and $at_i a^{-1}$ on their respective domains. The placement
$a$ and factor $E$ are the same for all $i$.
\end{lemma}
\begin{proof}
Choose $C=v[B_S]$, $D_0=v[B_T]$, where $S\subseteq T$.
Write $D_1\cong C\mathbin{\widehat\otimes}\mathbb C_\nu$
over $C$, and choose $U\subseteq\lambda\setminus T$ of size $\nu$.
Set
\[
 D'_1=v[B_{S\cup U}],\qquad E=v[B_{T\cup U}].
\]
An isomorphism $D_1\to D'_1$ fixing $C$ extends to $a\in G_C$.
The three independent factors of $E$ are $C$, the coordinates in
$T\setminus S$, and those in $U$.

For each $i$, let $t'_i=at_i a^{-1}\restr D'_1$ and let
$\beta_i$ be its common restriction with $s_i$ to $C$.
Extend $\beta_i$ independently to $E$. Removing this extension
from $s_i,t'_i$ leaves maps fixing $C$ pointwise. Extend each
independently over the other's complementary factor. These
extensions commute. One fixes $C$ and the entire third factor,
and the other fixes $C$ and the entire second factor. Each therefore
fixes the images of the other's extra generators. Their product,
followed by the independent extension of $\beta_i$, is the
required $r_i$. The product factorizations make all these maps
complete automorphisms. The placement $a$ and factor $E$ were
chosen independently of $i$.
\end{proof}

\subsection{Coherent limits through coordinate factors}

For this subsection only, let $\Omega$ be any set and $\theta$
any regular infinite cardinal. Put $P=\Add(\theta,\Omega)$
and $B=\RO(P)$, with the same notation $P_S,B_S$ for
$S\subseteq\Omega$. No cardinal-arithmetic assumption is needed.

Write $[p]$ for the Boolean value of a condition. For a complete
subalgebra $C$, let $\pi_C(b)$ be the least element of $C$ above $b$.
Thus, for $c\in C$,
\begin{equation}\label{eq:gcf-projection-compatibility}
 b\wedge c>0\quad\Longleftrightarrow\quad\pi_C(b)\wedge c>0.
\end{equation}
If $C\subseteq D$ are complete, then $\pi_C\pi_D=\pi_C$.
A complete isomorphism commutes with these projections onto the
subalgebras it carries to one another. On conditions,
\begin{equation}\label{eq:gcf-coordinate-projection}
 \pi_{B_S}([p])=[p\restr(S\times\theta)].
\end{equation}
The projection identities follow from the definition of $\pi_C$,
and the formula on conditions follows from the product on $S$ and its
complement. For increasing
sets $S_i$, we also have
\begin{equation}\label{eq:gcf-condition-meet}
 \bigwedge_i[p\restr(S_i\times\theta)]
 =[p\restr((\bigcup_iS_i)\times\theta)].
\end{equation}
This meet is positive because all its prescribed bits belong to the
domain of $p$, which has size less than $\theta$.

Suppose increasing complete subalgebras satisfy
$C_i\subseteq A_i\subseteq C_{i+1}$, where $A_i=B_{S_i}$ are
literal coordinate factors. At a nonzero limit $j$, the complete
closure of $\bigcup_{i<j}C_i$ is $B_{\bigcup_{i<j}S_i}$.
This follows from complete generation by the individual coordinate
bits. The interleaving inequalities between the projections, followed
by \eqref{eq:gcf-condition-meet}, give
\begin{equation}\label{eq:gcf-interleaved-projections}
 \bigwedge_{i<j}\pi_{C_i}([p])
 =\bigwedge_{i<j}\pi_{A_i}([p])
 =\pi_{B_{\bigcup_{i<j}S_i}}([p]).
\end{equation}

\begin{lemma}[Extension through coordinate factors]
\label{lem:gcf-fusion}\label{lem:hc-coordinate-limits}
Let $\delta$ be a nonzero limit ordinal. Suppose
$\varphi_\xi:C_\xi\to D_\xi$, $\xi<\delta$, are coherent
complete isomorphisms between increasing complete subalgebras of
$B$. Suppose there are literal coordinate factors $A_\xi$
and $B'_\xi$ such that
\begin{equation}\label{eq:gcf-sandwich}
 C_\xi\subseteq A_\xi\subseteq C_{\xi+1},
 \qquad D_\xi\subseteq B'_\xi\subseteq D_{\xi+1}
 \qquad(\xi<\delta).
\end{equation}
Let $B_S,B_T$ be the complete closures of the respective unions.
There is a unique complete isomorphism $\Phi:B_S\to B_T$
extending every $\varphi_\xi$.
\end{lemma}
\begin{proof}
If $\operatorname{cf}(\delta)\geq\theta$, each condition on the
union of the source supports belongs to some coordinate stage,
because it uses fewer than $\theta$ columns. The same holds on the target
side. The unions are therefore order-dense in their completions,
and the coherent isomorphism extends uniquely.

This proves the case $\theta=\omega$, since every nonzero
limit ordinal has infinite cofinality. Suppose instead that
$\rho=\operatorname{cf}(\delta)<\theta$.
Pass to a strictly increasing cofinal sequence of length $\rho$.
Between each pair of successive retained stages, keep an inserted
coordinate factor on each side. Relabel so that
\eqref{eq:gcf-sandwich} holds for $i<\rho$. In particular, every
nonzero limit prefix of either chain still has a literal coordinate
complete closure.

For $q\in P_S$, define
\begin{equation}\label{eq:gcf-image-condition}
 a_i=\varphi_i(\pi_{C_i}([q])),
 \qquad F(q)=\bigwedge_{i<\rho}a_i.
\end{equation}
The $a_i$ decrease. We prove that their meet is positive by
constructing a decreasing sequence of target conditions.

Take preimages of the target coordinate factors at the next stage. Set
\[
 E_i=\varphi_{i+1}^{-1}[B'_i],\qquad
 r_i=\varphi_{i+1}\restr E_i:E_i\longrightarrow B'_i,
 \qquad d_i=r_i(\pi_{E_i}([q])).
\]
Then $C_i\subseteq E_i\subseteq C_{i+1}$, the $E_i$ increase,
and the $r_i$ are coherent. Moreover
\[
 a_i\geq d_i\geq a_{i+1},\qquad
 F(q)=\bigwedge_{i<\rho}d_i.
\]
Write $B'_i=B_{T_i}$. Construct conditions $t_i\in P_{T_i}$
extending all their predecessors and satisfying $[t_i]\leq d_i$.
At stage $i$, put
\[
 t_{<i}=\bigcup_{j<i}t_j,\qquad T_{<i}=\bigcup_{j<i}T_j,
\]
with the empty condition at $i=0$. Regularity of $\theta$ and
$i<\rho<\theta$ ensure that $t_{<i}$ is a condition in
$P_{T_{<i}}$.

Let $U_i$ be the complete closure of $\bigcup_{j<i}E_j$, taking
$U_0=\{0,1\}$. The map $r_i$ sends $U_i$ onto $B_{T_{<i}}$.
We claim
\begin{equation}\label{eq:gcf-prefix-meet}
 \pi_{U_i}([q])=\bigwedge_{j<i}\pi_{E_j}([q]).
\end{equation}
At zero this is the empty meet. At a successor, the earlier $E_j$
have a largest member. At a nonzero limit, the chains $(E_j)$ and
$(C_j)$ have cofinal coordinate factors between them. For example,
$E_j\subseteq C_{j+1}\subseteq A_{j+1}\subseteq C_{j+2}
\subseteq E_{j+2}$. Thus $U_i$ is the corresponding coordinate
factor, and \eqref{eq:gcf-interleaved-projections} proves the claim.
Commuting projections with the complete map $r_i$ now gives
\begin{align}
 \pi_{B_{T_{<i}}}(d_i)
 &=r_i(\pi_{U_i}([q]))\notag\\
 &=\bigwedge_{j<i}r_i(\pi_{E_j}([q]))
 =\bigwedge_{j<i}d_j.\label{eq:gcf-fusion-identity}
\end{align}
By induction, $[t_{<i}]$ lies below the last meet. Since this
condition belongs to $B_{T_{<i}}$, equation
\eqref{eq:gcf-projection-compatibility} shows that
$[t_{<i}]\wedge d_i>0$. Choose $t_i\in P_{T_i}$ below that
meet. Since its Boolean value lies below $[t_{<i}]$, it extends $t_{<i}$
as a partial function. This completes the recursion. The union
$t=\bigcup_{i<\rho}t_i$ is still a condition, and
\[
 0<[t]\leq\bigwedge_{i<\rho}d_i=F(q).
\]
Thus $F(q)$ is positive.

Interchanging the two sides and applying the same argument to
the inverse chain proves, for every $t\in P_T$,
\begin{equation}\label{eq:gcf-inverse-condition}
 F^-(t)=\bigwedge_{i<\rho}
 \varphi_i^{-1}(\pi_{D_i}([t]))>0.
\end{equation}
The map $F$ preserves order and incompatibility. For the latter,
a conflicting bit of two conditions belongs to some source stage;
their projections there are incompatible, as are their images.
Conversely, compatible $q,q'$ have the common extension $q\cup q'$,
whose positive image lies below both $F(q)$ and $F(q')$.

The image of $F$ is order-dense in $B_T$. Given $t\in P_T$,
choose $q\in P_S$ with $[q]\leq F^-(t)$. For every $i$,
\[
 \pi_{C_i}([q])\leq
 \varphi_i^{-1}(\pi_{D_i}([t])),
\]
and consequently
\[
 F(q)\leq\bigwedge_i\pi_{D_i}([t])=[t].
\]
The last equality is \eqref{eq:gcf-interleaved-projections} on the
full target coordinate union. Order preservation, incompatibility
preservation in both directions, and density give an isomorphism
of Boolean completions with $\Phi([q])=F(q)$.

If $b\in C_i$ and $[q]\leq b$, then $F(q)\leq\varphi_i(b)$.
Taking the join of all conditions below $b$ gives
$\Phi(b)\leq\varphi_i(b)$. Apply the same argument to the
complement of $b$ to obtain equality. Finally, the $C_i$ completely
generate $B_S$, so a complete extension is unique.
\end{proof}

\subsection{Generic families and conjugacy}

We return to the assumptions on $\theta,\lambda$ in
\eqref{eq:gcf-cardinal-assumptions}.

\begin{definition}\label{def:hc-generic}
A family $(g_i)_{i\in I}$ in $G$, with $|I|\leq\lambda$, is
\emph{generic} if the following holds for every
$J\in[I]^{<\lambda}$. Whenever $C\preceq D$ belong to $\mathcal F$,
all $g_i$ for $i\in J$ preserve $C$, and $t_i\in\Aut(D)$ extend
$g_i\restr C$, there is $a\in G_C$ such that
\begin{equation}\label{eq:hc-generic-placement}
 g_i\restr a[D]=at_i a^{-1}\restr a[D]\qquad(i\in J).
\end{equation}
A generic family of length less than $\lambda$ is called a generic tuple.
\end{definition}

Genericity is preserved by restriction to a subfamily, by bijective reindexing,
and by simultaneous conjugation. For conjugation, transport the two
factors and their prescribed automorphisms, and use the invariance of
$\mathcal F$ and $\preceq$.

\begin{lemma}[Conjugacy of generic tuples]\label{lem:hc-conjugacy}
Suppose $(g_i)_{i\in J}$ and $(h_i)_{i\in J}$ are generic tuples,
$|J|<\lambda$, and $C,D\in\mathcal F$ are invariant under the
respective tuples. If $p:C\to D$ is a complete isomorphism with
\[
 p(g_i\restr C)=(h_i\restr D)p\qquad(i\in J),
\]
then $p$ extends to $u\in G$ with $ug_i u^{-1}=h_i$ for all $i$.
\end{lemma}
\begin{proof}
Construct coherent intertwining isomorphisms
$p_\alpha:C_\alpha\to D_\alpha$, for $\alpha<\lambda$, starting
with $p$. Domains and ranges belong to $\mathcal F$, are invariant
under the appropriate tuples, and increase in $\preceq$. At each
successor, extend first the domain and then the range so that both
contain coordinate $\alpha$.

For the forth step, Lemma~\ref{lem:hc-invariant-enlargement} gives
a fixed coordinate factor $A_\alpha$ with $C_\alpha\preceq A_\alpha$,
invariant under all $g_i$, and containing coordinate $\alpha$.
Extend $p_\alpha$ to $v\in G$. Then
$D_\alpha\preceq v[A_\alpha]$, and
$v(g_i\restr A_\alpha)v^{-1}$ extends $h_i\restr D_\alpha$.
Genericity of $(h_i)$ gives $a\in G_{D_\alpha}$ such that
$p^*=av\restr A_\alpha$ intertwines the tuples. It extends
$p_\alpha$ and has range $R_\alpha$ with
$D_\alpha\preceq R_\alpha$.

Choose a fixed coordinate factor $D_{\alpha+1}$ strongly extending
$R_\alpha$, invariant under all $h_i$, and containing coordinate
$\alpha$. Apply the forth argument to
$(p^*)^{-1}:R_\alpha\to A_\alpha$, now using genericity of
$(g_i)$. Its extension maps $D_{\alpha+1}$ onto a factor
$C_{\alpha+1}$ with $A_\alpha\preceq C_{\alpha+1}$.
Invert it to obtain $p_{\alpha+1}$. Thus
\begin{equation}\label{eq:hc-two-sided-sandwich}
 C_\alpha\preceq A_\alpha\preceq C_{\alpha+1},
 \qquad
 D_\alpha\preceq R_\alpha\preceq D_{\alpha+1},
\end{equation}
where $A_\alpha$ and $D_{\alpha+1}$ are fixed coordinate factors.

At a nonzero limit $\delta<\lambda$, the inserted $A_\alpha$
are cofinal coordinate factors on the source side; the successor
ranges $D_{\alpha+1}$ have the same property on the target side.
Apply \cref{lem:hc-coordinate-limits} to obtain coordinate closures
$C_\delta,D_\delta$ and the unique complete extension $p_\delta$.
Regularity of $\lambda$ ensures that both supports have size less than $\lambda$.
Invariance and intertwining extend by complete generation. Every
earlier factor remains a strong subfactor of the limit: pass through
$A_\alpha$ on the source side and $D_{\alpha+1}$ on the target side.
At stage $\lambda$, every coordinate has been included. The same lemma
therefore gives a full automorphism of $B$.
\end{proof}

\subsection{Simultaneous construction and the coset tree}

\begin{lemma}[Simultaneous multipliers]\label{lem:hc-simultaneous}
Let $|I|\leq\lambda$, let $(q_{ij})_{i\in I,j<\lambda}$ be a matrix
in $G$, and fix $C_i\in\mathcal F$ for each row. There are
$u_i\in G_{C_i}$ such that, for every $f:I\to\lambda$, the
selected family $(u_iq_{i,f(i)})_{i\in I}$ is generic.
\end{lemma}
\begin{proof}
For each row construct automorphisms $u_i^\xi$ of fixed coordinate
factors $E_i^\xi$, for $\xi<\lambda$. On a fixed row the domains
increase and the maps extend one another. Start with the identity
on a coordinate factor containing $C_i$.

Schedule all requirements
\begin{equation}\label{eq:hc-matrix-requirement}
 (J,f_0,C,D,(t_i)_{i\in J}),
\end{equation}
where $\varnothing\ne J\in[I]^{<\lambda}$, $f_0:J\to\lambda$,
$C\preceq D$ belong to $\mathcal F$, and $t_i\in\Aut(D)$ preserve
$C$. Also schedule coordinate coverage for each $(i,\alpha)\in
I\times\lambda$. Lemma~\ref{lem:hc-counting} and
$\lambda^{<\lambda}=\lambda$ bound the total number of requirements
by $\lambda$. Each is processed once in a recursion of
length $\lambda$.

At a requirement \eqref{eq:hc-matrix-requirement}, put
$q_i=q_{i,f_0(i)}$ for $i\in J$ and extend the current row maps
to $\widehat u_i\in G$. Choose a fixed coordinate factor $E$
containing the current $E_i^\xi$ for $i\in J$, with $C\preceq E$, invariant
under every $q_i$ and $\widehat u_i$. This follows from
Lemma~\ref{lem:hc-invariant-enlargement}, since fewer than $\lambda$
rows participate. Set
$s_i=(\widehat u_iq_i)\restr E\in\Aut(E)$.

If $s_i\restr C\ne t_i\restr C$ for some $i$, replace the partial
map on each participating row by $\widehat u_i\restr E$ and leave
the other rows unchanged. The disagreement persists under later extensions. Since
$q_i[C]\subseteq E$, every later extension $u_i$ satisfies
\[
 (u_iq_i)\restr C=(\widehat u_iq_i)\restr C.
\]
Thus the antecedent of this requirement fails for the eventual
selected family. This argument also covers failure of $s_i$ to
preserve $C$.

Otherwise all $s_i$ preserve $C$ and agree there with $t_i$.
Apply Lemma~\ref{lem:hc-amalgamation} to $C\preceq E,D$.
It supplies $a\in G_C$, a common factor $F$ with
$E\preceq F$ and $a[D]\preceq F$, and $r_i\in\Aut(F)$
extending both $s_i$ and $at_i a^{-1}$. Extend $r_i$ to
$\widehat r_i\in G$, and put $w_i=\widehat r_iq_i^{-1}$.
For $b\in E$, the invariance $q_i[E]=E$ gives
\begin{equation}\label{eq:hc-matrix-coherence}
 w_i(b)=r_i(q_i^{-1}b)
       =(\widehat u_iq_i)(q_i^{-1}b)=\widehat u_i(b).
\end{equation}
Choose a fixed coordinate factor $E'$ containing $F$ and invariant
under all $\widehat r_i,q_i$. Then $w_i\restr E'$ is an
automorphism of $E'$. Use it as the new partial map on row $i$;
\eqref{eq:hc-matrix-coherence} proves coherence.

The witness also persists. Since $q_i[a[D]]\subseteq E'$,
every subsequent row extension satisfies, for $b\in a[D]$,
\begin{equation}\label{eq:hc-matrix-witness}
 (u_iq_i)(b)=w_i(q_i(b))=r_i(b)=(at_i a^{-1})(b).
\end{equation}
Thus every later extension has the required restriction on $a[D]$.

A coordinate-coverage requirement is met by extending the current
row map to $G$, choosing a coordinate enlargement invariant under
that extension and containing the requested coordinate, and restricting
the extension to it. At a nonzero limit $\delta<\lambda$, handle each
row separately. Its increasing supports have a union of size less
than $\lambda$, and Lemma~\ref{lem:hc-coordinate-limits} completes
the union map to an automorphism of that coordinate factor. No union
of the supports of the different rows is taken.

At the end, each row includes every coordinate and hence defines a full
$u_i\in G$. It fixes $C_i$ because all maps extend the initial identity.
Fix any selector $f$ and any nonempty short $J$ with data to which
Definition~\ref{def:hc-generic} applies. The requirement with
$f_0=f\restr J$ was processed. Had it been rejected, disagreement on $C$ would persist. It was
therefore accepted, and its witness in
\eqref{eq:hc-matrix-witness} is precisely
\eqref{eq:hc-generic-placement}. The empty-index case is immediate.
Neither the construction nor the verification requires $f$ to be
injective.
\end{proof}

To prove the small-index theorem, suppose that $[G:H]\leq\lambda$
but $H$ contains no $G_{B_S}$ with $|S|<\lambda$. It then contains
no $G_C$ with $C\in\mathcal F$, since invariant coordinate
enlargement puts $C$ inside a small coordinate factor.
For $p\in\Aut(C)$, its full extensions meet at least two left
cosets of $H$: if $q$ extends $p$ and $v\in G_C\setminus H$,
then $q,qv$ both extend $p$, while $qH\ne qvH$.

\begin{lemma}\label{lem:hc-coset-family}
Under this contrary assumption there is a generic family of length
$\lambda$ containing $\lambda$ entries in $H$ and, for each
$C\in\mathcal F$ and $p\in\Aut(C)$, $\lambda$ entries outside
$H$ extending $p$. All entries are distinct.
\end{lemma}
\begin{proof}
Partition the row set into two sets $I_0,I_1$ of cardinality
$\lambda$. Assign every pair $(C,p)$ to $\lambda$ rows of $I_0$.
There are at most $\lambda$ such pairs. On an assigned row choose
matrix entries among the full extensions of $p$, including two from
different left cosets, and prescribe $C_i=C$. On every row of
$I_1$ include representatives of all left cosets and prescribe
$C_i=\{0,1\}$.

Apply Lemma~\ref{lem:hc-simultaneous}. On a row in $I_0$, left
multiplication by $u_i$ preserves distinctness of left cosets, so
some product $u_iq_{ij}$ is outside $H$. It still extends $p$,
since $q_{ij}[C]=C$ and $u_i$ fixes $C$. On a row in $I_1$,
choose an entry in $u_i^{-1}H$; its product with $u_i$ lies in
$H$. The selected family is generic.

No generic family has two equal entries. Otherwise use those two
indices, base $\{0,1\}$, and a nontrivial small coordinate factor
$D$, prescribing the identity for one index and a bit flip for the
other. Definition~\ref{def:hc-generic} would make the same
automorphism have two different restrictions on the same placed
factor. Thus all the indicated multiplicities count distinct entries.
\end{proof}

\begin{proof}[Proof of \cref{thm:cohen-small-index}]
Under the contrary assumption, fix the family of
Lemma~\ref{lem:hc-coset-family}. We construct a tree indexed by $2^{<\lambda}$. Since
$2^{<\lambda}\leq\lambda$, it has only $\lambda$ nodes before its
final level. At each node $s$, choose a fixed coordinate factor
$C_s$ and $p_s\in\Aut(C_s)$. Along its path select source
and target lists of family entries $h_t,k_t$, each without repetition.
Require that these entries preserve $C_s$ and
\begin{equation}\label{eq:hc-tree-intertwining}
 p_s(h_t\restr C_s)p_s^{-1}=k_t\restr C_s.
\end{equation}
Factors and maps extend along branches, and $C_s$ contains all
coordinates below the length of $s$. The two outgoing edges satisfy
\begin{equation}\label{eq:hc-tree-split}
 h_{s,0}\in H,\qquad h_{s,1}\notin H,\qquad k_{s,0}=k_{s,1}.
\end{equation}
Choices on unrelated branches may reuse entries. Start with
$C_\varnothing=\{0,1\}$ and its identity.

At a node $s$ of length $\alpha$, the earlier lists are generic
tuples of length less than $\lambda$. By Lemma~\ref{lem:hc-conjugacy},
$p_s$ extends to $v\in G$ intertwining those lists globally.
Choose a fresh family entry $h_{s,0}\in H$. Enlarge to a fixed
coordinate factor $C'$ containing $C_s$ and coordinate $\alpha$,
invariant under $v,h_{s,0}$ and all earlier source and target entries.
Then $p'=v\restr C'$ is an automorphism of $C'$.
Choose a fresh $h_{s,1}\notin H$ agreeing with $h_{s,0}$ on
$C'$, and a further fresh family entry $k$ extending
\[
 p'(h_{s,0}\restr C')(p')^{-1}.
\]
Lemma~\ref{lem:hc-coset-family} supplies $\lambda$ possibilities
for each required extension, while fewer than $\lambda$ entries
have been excluded along the path. Give both children $C',p'$
and put $k_{s,0}=k_{s,1}=k$. The new entries preserve $C'$
because their restrictions are automorphisms of it. The previous
intertwining equations follow from $v$, and the new ones from the
chosen restrictions.

At a node of nonzero limit length $\delta<\lambda$, take the
coordinate factor on the union of the preceding supports.
Regularity keeps it small. Lemma~\ref{lem:hc-coordinate-limits}
completes the coherent union map to its automorphism. For each
earlier edge, invariance and \eqref{eq:hc-tree-intertwining}
hold on a cofinal tail, hence on the completion.

For every branch $\eta\in2^\lambda$, coordinate coverage makes
the complete closure equal to $B$. The limit lemma gives its
complete extension $p_\eta\in G$. It satisfies $p_\eta h_t p_\eta^{-1}=k_t$
for each edge on that branch. If $\eta,\zeta$ first split after
$s$, taking edges $0,1$, respectively, then
\[
 p_\eta h_{s,0}p_\eta^{-1}
 =k_{s,0}=k_{s,1}
 =p_\zeta h_{s,1}p_\zeta^{-1}.
\]
If $p_\eta H=p_\zeta H$, then $z=p_\eta^{-1}p_\zeta\in H$
and this equation gives $h_{s,1}=z^{-1}h_{s,0}z\in H$, contrary
to \eqref{eq:hc-tree-split}. Thus there are $2^\lambda$ distinct
left cosets, contradicting $[G:H]\leq\lambda$.
\end{proof}

The proof does not require $H$ to be normal. It applies to index at most
$\lambda$. To treat every index below $2^\lambda$, one would need a
different construction of the family, since the present construction
places representatives of all cosets in $\lambda$ columns.

\subsection{Invariant names of small families}

\begin{proposition}\label{prop:cohen-name-reduction}\label{prop:ordinary-name-reduction}
Under \eqref{eq:gcf-cardinal-assumptions}, suppose $\dot A$ is a
$B$-name fixed modulo forced equality by every element of $G$, and
$1_B\Vdash0<|\dot A|\leq\check\lambda$. Every name $\tau$ forced
to belong to $\dot A$ is fixed by $G_{B_S}$ for some
$S\in[\lambda]^{<\lambda}$. For a generic $F\subseteq B$, its
quotient interpretation over $W[F\cap B_S]$ is invariant under
every automorphism of the quotient completion computed there.
The ranks of the members are unrestricted.
\end{proposition}
\begin{proof}
Coordinate permutations make $P$ weakly homogeneous, so the full
fixed algebra is $\{0,1\}$ and the cardinality of the invariant
family is decided. Choose names $(\tau_i)_{i<\eta}$ forced to
enumerate it bijectively, where $0<\eta\leq\lambda$.
Every member name is equivalent to a mixture of these names:
its possible index values form a maximal antichain, and the
$\theta^+$-chain condition leaves at most $\theta$ nonzero pieces.

Fix a mixing construction and take its equivalence classes under
forced equality. This set $Z$ has size at most
\[
 |B|^\theta\lambda^\theta=\lambda.
\]
There is an ordinary action $g[\sigma]=[g\sigma]$ on $Z$, representing
the result again by an equivalent mixture. Invariance of $\dot A$
makes the action well-defined. A member-name stabilizer has index
at most $|Z|$, so \cref{thm:cohen-small-index} puts $G_{B_S}$
inside it for some $|S|<\lambda$.
Apply \cref{lem:hc-quotient-invariance} to this fixed-factor invariance.
That lemma includes new quotient automorphisms named and mixed over
$B_S$, and computes the completion in the quotient ground.
\end{proof}

\subsection{Definability in Cohen extensions of \texorpdfstring{$L$}{L}}
\label{subsec:cohen-family-applications}

The name-reduction theorem gives the following two definability
results.  As in the introduction, $\ODlt{\kappa}$ allows ordinal
parameters and a sequence of fewer than $\kappa$ reals.
For any regular $\lambda$, $\OD_{H_\lambda}$ allows ordinal parameters
and one parameter whose transitive closure has cardinality less than
$\lambda$.  Cardinalities and definability are computed in the
extension.

\begin{theorem}[Ordinary higher-Cohen definability]
\label{thm:regular-cohen}
Let $\kappa$ be regular uncountable in $L$, and let
$V=L[G]$ for $G$ generic on $\Add(\omega,\kappa)^L$.  Then
\[
 A\in\ODlt{\kappa}^V,\qquad |A|^V\leq\kappa
 \quad\Longrightarrow\quad A\subseteq\ODlt{\kappa}^V.
\]
\end{theorem}

\begin{theorem}[Generalized Cohen definability]
\label{thm:generalized-small-family}
Let $\kappa<\lambda$ be regular uncountable cardinals of $L$
satisfying \eqref{eq:gcf-cardinal-assumptions} there with
$\theta=\kappa$, and let $V=L[G]$ for $\Add(\kappa,\lambda)^L$.
Then
\[
 A\in\OD_{H_\lambda}^V,\qquad |A|^V\leq\lambda
 \quad\Longrightarrow\quad A\subseteq\OD_{H_\lambda}^V.
\]
\end{theorem}

Neither theorem imposes a rank bound on the members of $A$.
We prove them together and treat the two parameter classes separately.

\begin{proof}[Proof of \cref{thm:regular-cohen,thm:generalized-small-family}]
We prove both conclusions with $L$ replaced by an arbitrary original
ground $W_0\models\mathsf{ZFC}+\mathsf{GCH}+\mathsf{GA}
+V=\mathrm{HOD}$, retaining the displayed cardinal assumptions.
By \cref{lem:stable-ground-codes}, $W_0$ and its ground-name order
are definable in all its set-forcing extensions. The intermediate
grounds below need not satisfy GA or $V=\mathrm{HOD}$.
Write $(\theta,\lambda)=(\omega,\kappa)$ in the ordinary case,
and retain $(\theta,\lambda)=(\kappa,\lambda)$ in the generalized
case.  Thus in both cases
$V=W_0[G]$ for $\Add(\theta,\lambda)^{W_0}$, where $\theta<\lambda$
are regular, $\lambda$ is uncountable, and
\eqref{eq:gcf-cardinal-assumptions} holds in $W_0$.
In the ordinary case these assumptions follow from GCH and
regularity of the width.  Assume $A\ne\varnothing$.

\emph{Capturing the defining parameter.}
In the ordinary case, let
$\vec r=\langle r_i:i<\delta\rangle$, $\delta<\lambda$, define
$A$ together with ordinals.  Choose a name for the whole sequence
and a condition in $G$ deciding its length.  Nice names for its
entries each use countably many columns.  The union of their
supports, together with the support of the deciding condition,
is contained in some $S\in W_0$ with $|S|^{W_0}<\lambda$.
Consequently $\vec r\in W=W_0[G\restr S]$.

In the generalized case, let $z\in H_\lambda^V$ and ordinals
define $A$.  Code the transitive closure of $\{z\}$ by a
well-founded extensional relation on an ordinal $\delta<\lambda$,
with a distinguished point representing $z$.
Choose a name for this code and a condition in $G$ deciding
$\delta$ and the distinguished point.  Each membership bit
has a deciding antichain of size at most $\theta$.
There are fewer than $\lambda$ bits, and every condition uses
fewer than $\theta$ columns.  Regularity of $\lambda$ therefore
places the code and the deciding condition on a ground set
$S$ of fewer than $\lambda$ columns.
The code belongs to $W=W_0[G\restr S]$.
Its well-foundedness is downward absolute, and its transitive
collapse in $W$ agrees with the ambient collapse; hence $z\in W$.

In either case enumerate $S$ in $W_0$ in length $|S|^{W_0}$ and
code the resulting sequence of generic columns by a set of
ordinals $a$ so that $W=W_0[a]$.
For $\theta=\omega$, this code is recoverable from a sequence of
fewer than $\lambda$ reals and ordinals.  For uncountable $\theta$,
choose the code with hereditary size at most
$\max(\theta,|S|)<\lambda$, so $a\in H_\lambda^V$.
The original defining parameter has an ordinal index in the
$a$-definable ground-name order of $W_0[a]$ supplied by
\cref{lem:stable-ground-codes}. Substituting this definition shows
that $A$ is $\OD(a)$ in $V$.

\emph{The remaining forcing over $W$.}
The forcing on $S$ is $<\theta$-closed and $\theta^+$-cc.
It preserves cardinals and adds no ordinal sequences of length
less than $\theta$.  Thus $W$ still satisfies
$\theta^{<\theta}=\theta$ and
$\nu^{<\theta}<\lambda$ for every cardinal $\nu<\lambda$.
When $\theta=\omega$, this uses only the absoluteness of finite
sequences.  To verify the remaining cardinal assumption, let
$0<\rho<\lambda$.  A name for a function $\rho\to\lambda$
is specified by $\rho$ antichains, each of size at most $\theta$,
with ordinal labels below $\lambda$.  The forcing on $S$ has
size at most $\lambda$, so in $W_0$ there are at most
\[
 (\lambda^\theta)^\rho=\lambda
\]
such specifications.  These ground collections retain size at
most $\lambda$ in $W$, giving
$W\models\lambda^{<\lambda}=\lambda$.
The case $\rho=0$ is immediate.

The untouched columns have cardinality $\lambda$.
Since no short ordinal sequences were added, their conditions
are unchanged; after reindexing, the residual forcing is
$\Add(\theta,\lambda)^W$.  Its Boolean completion and
automorphisms are now computed in $W$.

\emph{Descent of an individual member.}
Weak homogeneity gives an invariant residual name $\dot A$,
forced nonempty and of size at most $\lambda$, from the
definition using $a$ and ordinals.
Fix $x\in A$.  Mix a name for $x$ below a condition in the
residual generic with a fixed member name off that condition.
The resulting $\tau$ is forced to belong to $\dot A$ and has
actual value $x$.
By \cref{prop:cohen-name-reduction}, there is a residual
coordinate set $T\in W$, $|T|^W<\lambda$, such that $\tau$
is fixed by its pointwise factor stabilizer.
Its quotient interpretation over $U=W[G\restr T]$ is fully
invariant by \cref{lem:hc-quotient-invariance}.
Here $T$ is a coordinate set in the new ground $W$; it need
not belong to $W_0$.

Enumerate $T$ in $W$ and code the additional columns by a
set of ordinals $b$, obtaining $U=W_0[a,b]$.
The enumeration of $T$ is specified by $a$ and its ordinal
index in the same relative ground-name order. Thus for
$\theta=\omega$, the pair $(a,b)$
is recoverable from fewer than $\lambda$ reals and ordinals.
For uncountable $\theta$, both codes have hereditary size
less than $\lambda$, as does their pair.

Code the pair as one set of ordinals and apply
\cref{lem:invariant-od} to the quotient forcing over $W_0[a,b]$,
whose names have ordinal-and-$(a,b)$ codes by
\cref{lem:stable-ground-codes}.
It follows that $x$ is $\OD(a,b)$ in $V$.
In the ordinary case this is $\ODlt{\lambda}$, and in the
generalized case it is $\OD_{H_\lambda}$.
Since $x\in A$ was arbitrary, both theorems follow.
\end{proof}

\begin{corollary}\label{cor:omega1-cohen}
In the extension of $L$ by $\omega_1$ Cohen reals, every $\ODR$
family of cardinality at most $\aleph_1$ consists of $\ODR$ elements.
\end{corollary}
\begin{proof}
A sequence of fewer than $\omega_1$ reals is coded by one real.
Apply \cref{thm:regular-cohen}.
\end{proof}

\begin{corollary}\label{cor:gcf-successor}
Let $\kappa$ be regular uncountable in $L$, let
$\lambda=(\kappa^+)^L$, and let $V=L[G]$ for
$\Add(\kappa,\lambda)^L$.  Every $\OD_{H_\lambda}^V$ family
of cardinality at most $\lambda$ consists of
$\OD_{H_\lambda}^V$ elements.
\end{corollary}
\begin{proof}
GCH in $L$ implies \eqref{eq:gcf-cardinal-assumptions} with
$\theta=\kappa$.  Apply \cref{thm:generalized-small-family}.
\end{proof}

\subsection{A chain without an order-dense union}

A chain may have Cohen successor quotients and Cohen complements in
the ambient algebra without having an order-dense union. In any ZFC ground let $\lambda$ be
regular uncountable and work in
$B=\RO(2\times(2^\omega)^\omega\times(2^\omega)^\lambda)$,
with coordinates $(t,(x_i)_{i<\omega},z)$ and $r=[t=0]$.
Keep the tail of each $x_i$ and replace its first bit by
\[
 y_i(0)=\begin{cases}0,&t=0,\\x_i(0),&t=1.\end{cases}
\]
For $n\geq1$, let $C_n$ be the complete algebra generated by
$y_0,\ldots,y_{n-1}$.
The output map is a continuous open surjection onto $n$ reals;
these maps commute with projection, so $C_{n+1}$ is a Cohen
product extension of $C_n$.

Each $C_n$ also has a full $\lambda$-Cohen complement in $B$.
Where all output first bits vanish, its fiber consists of one
copy of the unused product for $t=1$ and $2^n$ copies for $t=0$;
elsewhere only the $t=1$ copy occurs. A finite disjoint sum of
copies of $(2^\omega)^\lambda$ is homeomorphic to that space,
using a finite clopen partition of a spare real. These fiber
identifications give the required complement.

Let $A_n$ be the image of the countable clopen algebra in $C_n$.
The $A_n$ are increasing and order dense there, but
\[
 p_n=\bigwedge_{i<n}[y_i(0)=0]\in A_n,
 \qquad \bigwedge_{n\geq1}p_n=r>0.
\]
On $t=1$ the infinite intersection has empty interior, since
finite conditions leave some rows unrestricted. Every positive
element of $C_n$, however, meets $t=1$, where the output map is
ordinary projection. Hence no positive element of $\bigcup_nC_n$
lies below $r$, and $\bigcup_nA_n$ is not order dense in its
complete closure.

The coordinate condition \eqref{eq:hc-two-sided-sandwich} excludes
this example. Any raw factor containing $C_n$ contains $t$:
otherwise its coordinate flip would fix the factor while changing
$y_i(0)$. It therefore contains $r\notin C_{n+1}$, so no such
factor lies between the consecutive stages.


\section{Higher random extensions}\label{sec:higher-random}

Let $\kappa$ be an uncountable cardinal in $L$ and put
\[
 \lambda=(\kappa^{\aleph_0})^L.
\]
\begin{theorem}\label{thm:singular-random-od}
Suppose $\operatorname{cf}^L(\kappa)=\omega$. In the random
extension of $L$ of Maharam type $\kappa$, every OD family of
cardinality at most $\kappa$ has an OD bijective enumeration
by a cardinal at most $\kappa$. In particular, it consists entirely of OD members.
The members may have arbitrary rank.
\end{theorem}

\begin{theorem}\label{thm:higher-random}
In the random extension of $L$ of Maharam type $\kappa$, there is
an OD family $\mathcal A\subseteq\mathcal P(\mathcal P(\lambda))$
of cardinality $\lambda$ with no member definable from ordinals and
fewer than $\kappa$ reals.
\end{theorem}

At singular $\kappa$ of countable cofinality, these theorems give
OD bijective enumerations for families of size at most $\kappa$ and a
counterexample of size $\kappa^+$. At uncountable cofinality, the
counterexample has size exactly $\kappa$.
We first construct the counterexample and prove the countable-cofinality
theorem. We then prove uniform enumeration below $\kappa$ at regular
width in \cref{cor:regular-random-enumeration}. The corresponding
singular strong-limit result is \cref{thm:ssf-random-enumeration}.
The invariant-enumeration argument works over every ZFC ground at
uncountable Maharam type of countable cofinality. Stable ground codes
then give the OD conclusion.

\subsection{Orbits of maximal random presentations}

The construction and orbit calculations work over any ZFC ground $W$,
with $\lambda=(\kappa^{\aleph_0})^W$. For the OD conclusion, take
$W\models\mathsf{ZFC}+\mathsf{GCH}+\mathsf{GA}+V=\mathrm{HOD}$;
\cref{lem:stable-ground-codes} makes $W$ and its internal HOD order
definable in the extension. This includes the stated case $W=L$.
We use GCH only to evaluate $\lambda$ in the final corollary.

Let $\Rk$ be the measure algebra of the product probability space
$(2^\omega)^\kappa$ computed in $W$. Counting coordinate supports and Borel
codes gives $|\Rk|\leq\kappa^{\aleph_0}$. For the reverse bound,
write $r_\alpha$ for the $\alpha$th coordinate and, for each
countably infinite $S=\{\alpha_n:n<\omega\}\subseteq\kappa$, put
\[
 b_S=\bigvee_{n<\omega}
 [r_{\alpha_n}\restr(n+2)=0^{n+2}].
\]
For each $n$, the other cylinders are independent of $r_{\alpha_n}$
and their union has measure at most $1/2$. Flipping the first bit at
$\alpha_n$ therefore changes $b_S$ on a positive set. Thus $b_S$
cannot be supported on a set omitting any member of $S$, and
distinct $S$ give distinct elements. A single coordinate factor
already has $2^{\aleph_0}$ elements. Consequently
$|\Rk|\geq\max(2^{\aleph_0},|[\kappa]^{\aleph_0}|)
=\kappa^{\aleph_0}=\lambda$.

Fix a bijection $e:\lambda\to\Rk$ in $W$, using the first one in
its internal HOD order when proving the OD conclusion.
For a filter $F\subseteq\Rk$, put
\[
 x_F=\{\xi<\lambda:e(\xi)\in F\}\subseteq\lambda.
\]
Let $G$ be $\Rk$-generic over $W$, put $M=W[G]$, and define
\[
 D=\{F\subseteq\Rk:F\text{ is }\Rk\text{-generic over }W\text{ and }W[F]=M\}.
\]
Let
\[
 K=\Aut^W(\Rk),\qquad H=\Aut^W(\Rk,\mu).
\]
For $F\in D$, set
\[
 [F]_H=\{x_{h[F]}:h\in H\}\subseteq\mathcal P(\lambda),
\]
and let
\begin{equation}\label{eq:random-orbit-family}
 \A_\kappa=\{[F]_H:F\in D\}.
\end{equation}
For the specified definable grounds, this family is OD in $M$, since
the ground, the algebra, $e$, and the two ground groups have ordinal
definitions there. The assertion $W[F]=M$ says that every ambient
set is the value of a ground name under $F$.

\begin{lemma}\label{lem:higher-random-maximal-orbit}
$D=\{\pi[G]:\pi\in K\}$.
\end{lemma}

\begin{proof}
This is the arbitrary-ground same-extension theorem stated in
\cref{sec:names}, applied directly to $\Rk$. The fixed coding gives
$W[F]=W[x_F]$.
\end{proof}

\begin{lemma}[Fixing a generic filter]\label{lem:fix-generic-region}
For a complete Boolean algebra $B$ and $g\in\Aut(B)$, the Boolean
value $b=\|g\dot G=\dot G\|$ is the largest region on whose
principal ideal $g$ is the identity.
\end{lemma}
\begin{proof}
For every $a\in B$, the assertions $a\in\dot G$ and
$g(a)\in\dot G$ are equivalent below $b$. Thus $b\wedge a=b\wedge g(a)$.
Taking $a=b$ and $a=g^{-1}(b)$ gives $b\leq g(b)$ and
$b\leq g^{-1}(b)$, hence $g(b)=b$. For $a\leq b$ the same
identity now gives $g(a)=a$. Conversely, an identity region forces
the two filters to agree, by the rank induction for the generic name.
\end{proof}

\begin{proposition}\label{prop:higher-random-cardinality}
$M\models|\A_\kappa|=\lambda$.
\end{proposition}

\begin{proof}
For $\pi\in K$, define $\nu_\pi(b)=\mu(\pi b)$. The assignment
$H\pi\mapsto\nu_\pi$ is injective. Equivalent probability measures have
strictly positive $L^1$ densities, and every measurable real-valued
function depends on countably many coordinates. There are therefore
at most $\lambda$ cosets, so $|\A_\kappa|\leq\lambda$.

For the reverse inequality, fix in $W$ a bijection between
$\omega\times\kappa$ and $\kappa$. Let $U_{n,\alpha}$ be a uniform
$[0,1]$ random variable read from the corresponding product coordinate.
For $s\in(\kappa^\omega)^W$, put
\[
 X_s=\sum_{n<\omega}2^{-n-1}U_{n,s(n)},
 \qquad
 f_s=Z^{-1}e^{X_s},
\]
where $Z=\int e^{X_s}\,d\mu$ is independent of $s$. If $s\neq t$,
choose $n$ with $s(n)\neq t(n)$. Conditional on all coordinates except
$U_{n,s(n)}$, the equation $X_s=X_t$ determines at most one value of
that continuously distributed variable. Thus
\begin{equation}\label{eq:density-separation}
 \mu(f_s=f_t)=0.
\end{equation}

The probability algebra $(\Rk,f_s\mu)$ is homogeneous of Maharam type $\kappa$. By Maharam's theorem, choose $\pi_s\in K$ such that
\[
 \mu(\pi_s b)=\int_b f_s\,d\mu
\]
for every $b\in\Rk$. Suppose $\pi_s[G]$ and $\pi_t[G]$ lie in the
same $H$-orbit. Then, for some $h\in H$, the automorphism
$\delta=\pi_t^{-1}h\pi_s$ fixes $G$. By
\cref{lem:fix-generic-region}, $\delta$ is the identity on a nonzero
principal ideal $\Rk\restr c$. For every $b\leq c$,
\[
 \int_b f_s\,d\mu=\mu(\pi_s b)=\mu(h\pi_s b)=\mu(\pi_t b)=\int_b f_t\,d\mu.
\]
Hence $f_s=f_t$ almost everywhere on $c$, contradicting
\eqref{eq:density-separation}. Thus the ground family of $\lambda$
presentations gives distinct orbit classes; random forcing preserves
this cardinal.
\end{proof}

\begin{proposition}\label{prop:no-od-less-kappa-random}
No member of $\A_\kappa$ is $\ODlt{\kappa}$ in $M$.
\end{proposition}

\begin{proof}
Fix a maximal presentation $F$ and let $\dot O$ be the canonical name
for its $H$-orbit class. Suppose this class is defined from a sequence
$\vec z$ of $\delta<\kappa$ reals and ordinal parameters. Nice names
for the entries of $\vec z$ use fewer than $\kappa$ coordinates in
total. Choose a condition $p\in F$ forcing uniqueness of the definition,
and include the countably many coordinates of $p$ in the support.
Since $\kappa$ is uncountable, choose a coordinate $\beta$ outside
this support.

On coordinate $\beta$, choose a nonsingular involution whose
Radon--Nikodym derivative is $2$ on a set of measure $1/3$ and $1/2$
on its complement. Extend it by the identity on all other coordinates,
obtaining $k\in K$. Then $k$ fixes $p$, every name in $\vec z$, and
all ordinal parameters. Symmetry gives
\[
 p\Vdash k(\dot O)=\dot O.
\]
But if this equality held below a nonzero condition, some
measure-preserving $h\in H$ would agree with $k$ or $k^{-1}$ on a
nonzero principal ideal. This contradicts their Radon--Nikodym derivatives. The derivative of
$h$ is $1$, while those of $k$ and $k^{-1}$ are nowhere $1$.
\end{proof}

\begin{proof}[Proof of \cref{thm:higher-random}]
The family in \eqref{eq:random-orbit-family} is OD, has size $\lambda$ by \cref{prop:higher-random-cardinality}, and has no $\ODlt{\kappa}$ member by \cref{prop:no-od-less-kappa-random}.
\end{proof}

\begin{corollary}\label{cor:higher-random-gch}
Assume $\kappa$ is uncountable in $L$.
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
 \item If $\operatorname{cf}^L(\kappa)>\omega$, then the extension by $\kappa$ random reals has an OD family of size $\kappa$ with no $\ODlt{\kappa}$ member.
 \item If $\operatorname{cf}^L(\kappa)=\omega$, the construction gives such a family of size $\kappa^+$.
\end{enumerate}
\end{corollary}

\begin{proof}
More generally, in every ZFC+GCH ground $W$,
\[
 (\kappa^{\aleph_0})^W=
 \begin{cases}
  \kappa,&\operatorname{cf}^W(\kappa)>\omega,\\
  \kappa^+,&\operatorname{cf}^W(\kappa)=\omega.
 \end{cases}
\]
Apply the construction over $L$, or over the definable grounds used
in the proof of \cref{thm:higher-random}.
\end{proof}

\subsection{An initial invariant enumeration}\label{subsec:sr-frame}

We work in a ZFC ground $W$. Let $\kappa$ be an uncountable
cardinal of cofinality $\omega$. Write $\B=\B_\Omega$ for
the fair product probability algebra on $2^\Omega$, where
$|\Omega|=\kappa$, and put
\[
 H=\Aut(\B,\mu),\qquad K=\Aut(\B),\qquad C=\B^I,
 \qquad 0<|I|\leq\kappa.
\]
The full $K$-action $\Theta$ on $C$ is assumed to satisfy diagonal
equivariance and locality, as in
\eqref{eq:cohen-diagonal}--\eqref{eq:cohen-locality}.
A \emph{selector} is an element $u\in C$ whose coordinates partition
$1_\B$; a \emph{frame} is a partition of $1_C$ into selectors.
An invariant frame amounts to an invariant bijective enumeration.
Let $\Gamma$ be the finite coordinate translations and let
$\Sigma=\Sym(\Omega)$ act by coordinate permutations.
Write $K_{\mathrm{cs}}$ for the subgroup of $K$ consisting of
automorphisms fixing $\B_{\Omega\setminus S}$ pointwise for
some countable $S\subseteq\Omega$; their action on $S$ may
depend on the complementary coordinates.

We first obtain a frame fixed by $\Gamma$ and $K_{\mathrm{cs}}$.
We then use diagonal actions of standard probability-algebra groups
to prove that $\Sigma$ fixes this frame. The following subsections extend its invariance to the full group.

\begin{lemma}\label{lem:sr-countable-kernel}
Let $D$ be a complete Boolean subalgebra of $\B^I$. Every
homomorphism $\Sigma\to\Aut(D)$ which kills the finitary
permutations also kills the countably supported permutations.
\end{lemma}
\begin{proof}
The space $\mathcal M(D)$ of normal finite signed measures on $D$,
with the total-variation norm, has density at most $\kappa$.
A measure is normal if it preserves arbitrary increasing joins.
To verify the bound, restrict the coordinate measures
$\nu_i(d)=\mu(d_i)$ to $D$. If $e_i$ is the complement of the
join of the zero sets of $\nu_i$, then $\nu_i$ is strictly positive on $D\restr e_i$,
and these $e_i$ cover $1$. That interval embeds in $\B$ by its
$i$th coordinate, so its measure-metric density is at most $\kappa$.
Disjointify the $e_i$ into a partition $(f_i)$ with $f_i\leq e_i$.
Every normal finite measure is concentrated on countably many
$f_i$; on each, the Radon--Nikodym theorem represents it by an
$L^1(\nu_i)$ function. Truncating to finitely many intervals and approximating by simple
functions gives the asserted density bound.

Pushforward gives a faithful isometric representation of $\Aut(D)$
on $\mathcal M(D)$. Faithfulness follows by restricting a coordinate
measure to a nonzero part of a moved Boolean element. The strong
topology on this isometry group has weight at most $\kappa$, since
a dense set of that many vectors determines it. A Hausdorff group
of weight at most $\kappa$ has at most $\kappa$ pairwise commuting
nonabelian subgroups: cover its open noncommutation relation by
$\kappa$ basic rectangles and assign to each subgroup a rectangle
containing a noncommuting pair. Two commuting subgroups cannot
receive the same rectangle.

Now apply \cref{lem:branch-permutation-kernel}(i).
\end{proof}

\begin{lemma}\label{lem:sr-fixed-atomic}
The algebra $D=C^\Gamma$ is atomic with at most $\kappa$ atoms.
\end{lemma}
\begin{proof}
Locality makes every finitary coordinate permutation fix $D$:
on each pattern of its finite support it agrees, on the whole
principal algebra, with a finite translation. Since $\Sigma$
normalizes $\Gamma$, \cref{lem:sr-countable-kernel} gives
\[
 D\subseteq C^{\Sigma_c},
 \qquad
 \Sigma_c=\{\pi\in\Sigma:|\supp(\pi)|\leq\aleph_0\}.
\]

We first show that a $\Sigma_c$-invariant name for a family of at most
$\kappa$ reals is forced to consist of ground members. Suppose otherwise and choose a member name $\dot r$
which is nonground with positive Boolean value. Weak homogeneity of $\Sigma_c$ forces the family to be nonempty
everywhere, since countable event supports can be moved apart. Let $\eta$ be the
ground distribution of $\dot r$, and remove its countable set
of atoms, leaving a set $E$ of positive $\eta$-measure. Copy
the countable input support of $\dot r$ onto the nonempty
initial segments of each $s\in\kappa^\omega$. Each copy is
induced by a countably supported coordinate permutation; a
countable reserve extends the prescribed support bijection.
Denote these member names by $\dot r_s$.

For $s\ne t$, condition on their finitely many common input
bits. The remaining inputs are independent, and their laws on
$E$ are nonatomic, being absolutely continuous with respect
to $\eta\restr E$. Hence
\[
 \|\dot r_s=\dot r_t\text{ and }\dot r_s\in E\|=0.
\]
The set of branches $s$ for which $\dot r_s\in E$ is therefore
forced to have size at most $\kappa$. By ccc covering, it lies
in a ground set of at most $\kappa$ branches: use countable
antichains deciding the values of a $\kappa$-term enumeration.
Yet below any positive condition there are $\kappa^\omega$
branches whose supports avoid that condition's countable
support. Choose a first branch symbol unused by that support.
For each of these branches, membership in $E$ has the same
positive conditional probability $\eta(E)$. This contradicts the existence of the ground cover.

Now suppose $C^{\Sigma_c}$ has a nonatomic part. Restrict to
a nonzero interval $a$ carrying a strictly positive coordinate
measure $\nu_i$, as in the preceding proof, and split it into
a binary tree $(a_t:t\in2^{<\omega})$ with
$\nu_i(a_t)=2^{-|t|}\nu_i(a)$. Every branch has zero meet.
In a generic fiber, the labels belonging to $a$ determine a
set of at most $|I|$ branches through this invariant tree.
That real-family name is $\Sigma_c$-invariant, so all its
members must be ground. On the positive event $a_i$, however,
the branch selected by label $i$ differs from every ground
branch, because its Boolean equality value is a coordinate of
the corresponding zero meet. This is a contradiction.

Thus $C^{\Sigma_c}$ is atomic. A complete subalgebra of a
complete atomic Boolean algebra is atomic: the least upper
bound in the subalgebra of an ambient atom is again an atom.
Consequently $D$ is atomic. Every antichain in $\B^I$ has size
at most $\max(|I|,\aleph_0)\leq\kappa$, by ccc in each
coordinate, which bounds the number of atoms.
\end{proof}

\begin{lemma}[Standard factors below the continuum]\label{lem:sr-standard-frame}
Let $Q$ be a standard atomless probability algebra and let
$0<|J|<2^{\aleph_0}$. Every full local diagonal action of
$\Aut(Q,\mu)$ on $Q^J$ has an invariant frame. If the action
extends to $\Aut(Q)$, it has a frame fixed by that full group.
\end{lemma}
\begin{proof}
Use \cref{lem:random-mp-frame} for the probability-preserving frame
and \cref{thm:random-nonsingular-frame} for its full invariance.
\end{proof}

\begin{lemma}[Full relative actions]\label{lem:sr-relative-descent}
In any ZFC ground, let $\B=\B_\Omega$ be a product probability
algebra, without a restriction on $|\Omega|$ or cardinal
arithmetic. Let $\Omega=S\sqcup T$, let $A=\B_S$, and force first with
$A$. In that extension let $Q$ be the recomputed probability
algebra on $2^T$. A full local diagonal action on $\B^I$
descends to a full local diagonal action on the original
$Q^I$, including every new relative automorphism. This holds
both for the measure-preserving and the nonsingular groups,
and for every ground set $I$.
\end{lemma}
\begin{proof}
Apply \cref{lem:full-quotient-action} to the product/Fubini
factorization $\B=\B_S*Q$. Its measure-preserving clause applies
because conditional probabilities integrate to the original product
probability.
\end{proof}

\begin{lemma}\label{lem:sr-countable-frame}
The atoms of $C^\Gamma$ form a frame fixed pointwise by
$\Gamma$ and $K_{\mathrm{cs}}$.
\end{lemma}
\begin{proof}
By \cref{lem:sr-fixed-atomic}, $C^\Gamma$ is atomic, with at
most $\kappa$ atoms. Every such atom $U$ has scalar support $1$
and a constant fiber cardinal by scalar $\Gamma$-ergodicity.
We show that this cardinal is one.

For any countably infinite $S\subseteq\Omega$, first force
with $\B_{\Omega\setminus S}$. This ccc forcing preserves
$\kappa$ and $\operatorname{cf}(\kappa)=\omega$ and supplies
$\kappa$ distinct reals. Since the continuum has uncountable
cofinality, the intermediate continuum is strictly larger
than $\kappa$. Hence \cref{lem:sr-relative-descent,lem:sr-standard-frame}
give a full invariant frame for the original relative cover
$Q^I$, where $Q$ is the recomputed standard algebra on $S$.

Suppose first that $U$ has infinite degree. Its counting measure
$\nu(U)$ is infinite, while its $\Gamma$-action is ergodic.
Its invariant $L^2$ subspace is therefore zero. Choose a selector
$u\leq U$. The closed convex hull of its orbit indicator
contains zero; choose finite rational convex averages $z_n$
with $\|z_n\|_2<2^{-n}$. The countably many selectors in these averages together have only
countably many positive coordinates, each with countable scalar support. Choose $S$ carrying all of them,
$u$, and the finite translation supports used. After forcing
the complementary factor, the norm inequalities are unchanged.
Some selector $v$ in the relative invariant frame has positive
overlap with $u$, whereas invariance gives
\[
 \langle\one_v,z_n\rangle
   =\langle\one_v,\one_u\rangle>0
 \quad(n<\omega),
\]
contradicting $\|z_n\|_2\to0$.

Suppose next that $U$ has finite degree $m$. Since
$\sum_i\mu(U_i)=m$, only countably many labels occur positively
in $U$. Enumerating them, select their first $m$ active labels
in each fiber. This identifies $C\restr U$ over $\B$ with
$\B^m$, with a countably supported choice of the $m$ selectors.
We construct a countable $S$ on which this finite
$\Gamma$-action is ergodic, with witnesses that remain valid
after forcing the complementary factor.

Start with a countably infinite $S_0$ carrying that identification,
adding a reserve if necessary.
Given $S_r$, include in $S_{r+1}$ the supports of all gauged
matrix entries for translations in $\Gamma_{S_r}$. For every
rational simple finite-cylinder function $f$ on $2^{S_r}\times m$
and positive rational $\varepsilon$, ergodicity supplies a
finite rational convex average of translates of $f$ within
$\varepsilon$ of the constant
$\bar f=m^{-1}\int f\,d\nu$. Include the supports of the
translations and their matrix entries in $S_{r+1}$. There are
only countably many requirements. Put $S=\bigcup_rS_r$.

Every finite translation on $S$ appears at some stage, and its
matrix is supported on $S$. Every rational finite-cylinder
function has the recorded arbitrarily accurate averages.
These functions remain dense in the recomputed $L^2(Q^m)$
after complementary forcing, and all recorded norm inequalities
are unchanged. If $v$ is any new invariant vector, taking inner
products with these averages gives
$\langle v,f\rangle=\langle v,\bar f\rangle$ on that dense
set. Thus the relative $\Gamma_S$-action on $U$ remains
ergodic. Intersect $U$ with the selectors of the full relative
invariant frame. These intersections are invariant partial
selectors and cover $U$. Ergodicity makes a nonzero one equal
to $U$, so $U$ is a selector and $m=1$. In both cases we descend the original full cover. We do not assume
that the relative full group preserves $U$.

The atoms of $C^\Gamma$ therefore form a $\Gamma$-fixed frame,
indexed by $I$.
Finally fix any countably infinite $S$. In the relative
nonsingular invariant frame on $Q^I$, a global $\Gamma$-fixed
selector has scalar coefficients fixed by $\Gamma_S$.
The latter acts ergodically on the recomputed standard $Q$,
so those coefficients are zero or one. The selector is therefore a member of the relative invariant frame,
so every relative nonsingular automorphism fixes it. This proves the
assertion in the ground, including for automorphisms whose action
depends on parameters from the entire complementary factor.
\end{proof}

\begin{lemma}[Initial frame]\label{lem:sr-initial-frame}
Under the hypotheses of this subsection, a full local diagonal
$K$-action on $C$ has a frame fixed pointwise by $\Gamma$,
$\Sigma$, and $K_{\mathrm{cs}}$.
\end{lemma}
\begin{proof}
Let $F=(f_i:i\in I)$ be the frame from
\cref{lem:sr-countable-frame}. We first show that it is fixed
by diagonal standard actions on disjoint countable coordinate
blocks with a $\kappa$-sized complement.

Fix an infinite ground set $E\subseteq\Omega$ with
$|\Omega\setminus E|=\kappa$, and force with
$A=\B_{\Omega\setminus E}$. In $W'=W[G_A]$, let $Q_E$ be
the recomputed probability algebra on $2^E$. By
\cref{lem:sr-relative-descent}, the full action descends to
$Q_E^I$ with the same frame $F$. Let $L_E\leq\Aut(Q_E,\mu)$
consist of all automorphisms fixing the coordinate
algebra outside some countable subset of $E$ pointwise.
Every member of $L_E$ fixes $F$. Indeed, ccc covering puts its
named countable support inside a ground countable $S\subseteq E$. Its ground lift fixes both $A$ and
$\B_{E\setminus S}$, hence belongs to $K_{\mathrm{cs}}$.
Names given only below a condition are mixed with the identity
elsewhere. This includes every new relative operator.

The $L_E$-fixed algebra of the cover is precisely the copy
of $\mathcal P(I)^{W'}$ formed by unions of frame selectors.
To see this, write an element uniquely as
\[
 z=\bigvee_{i\in I}(b_i\wedge f_i),\qquad b_i\in Q_E,
\]
where scalar events act diagonally. Since $L_E$ fixes each
$f_i$, an invariant $z$ has invariant coefficients $b_i$.
Finite coordinate translations belong to $L_E$ and act
ergodically on $Q_E$, so all these coefficients are zero or one.

Partition $E$ into countably infinite blocks $(D_t:t\in T)$,
with ground bijections $\omega\to D_t$. In $W'$, put
$P=\Aut(Q_\omega,\mu)$ for the recomputed standard algebra.
For $h\in P$, define $\Delta(h)$ by applying $h$ independently
on every block. Define this map first on finite tensor cylinders,
then extend by measure completion. Its inverse is $\Delta(h^{-1})$,
and the assignment is a homomorphism $P\to\Aut(Q_E,\mu)$.

Each $\Delta(h)$ normalizes $L_E$. If $\ell\in L_E$ fixes
the complement of a countable $S$, let
\[
 S^*=\bigcup\{D_t:D_t\cap S\ne\varnothing\}.
\]
This is countable. Both $\Delta(h)$ and its inverse preserve
each block algebra, so $\Delta(h)\ell\Delta(h)^{-1}$ fixes
the coordinate algebra outside $S^*$ pointwise. Applying the
same argument to $h^{-1}$ gives normalization. Thus the cover
action of $\Delta(P)$ preserves the $L_E$-fixed algebra
$\mathcal P(I)^{W'}$ and permutes its atoms $f_i$. We obtain
an ordinary homomorphism $P\to\Sym(I)$.

The forcing $A$ preserves $\kappa$ and its countable cofinality
and supplies $\kappa$ distinct reals. Hence
$W'\models |I|\leq\kappa<2^{\aleph_0}$.
Malicki's theorem \cite[Theorem~2.3]{Malicki} states that the
standard probability-algebra group has no proper subgroup
of index less than the continuum. Applied in $W'$ to each
point stabilizer, it makes this homomorphism trivial.
Consequently every ground diagonal map fixes the original frame,
since the factorization identification reflects equality of ground
cover elements. The theorem is applied to the full
recomputed group $P$ and requires no regularity of its action.

Now let $\tau$ be a coordinate involution whose moved support
$E$ has $\kappa$-sized complement. If $E$ is countable, then
$\tau\in K_{\mathrm{cs}}$. Otherwise group its transposed
pairs into countably infinite collections. Their unions form
blocks on which $\tau$ is the same standard automorphism,
interchanging successive pairs of bits. The preceding argument
therefore makes $\tau$ fix $F$.

Every coordinate involution is a product of at most two of
this kind. If its moved support lacks a $\kappa$-sized
complement, it has $\kappa$ transposed pairs; divide these
into two sets of size $\kappa$. Each restricted involution
has $\kappa$ fixed coordinates. Finally, every permutation is
a product of two involutions, by factoring the cyclic shift
on each finite or bilateral orbit into two reflections.
Thus all of $\Sigma$ fixes $F$.
\end{proof}

\subsection{Proper factors and relative frames}

The proper-factor lemmas apply both to Cohen and to random algebras.
Let $\kappa$ be uncountable, $|\Omega|=\kappa$, and let $B=B_\Omega$
be either the finite-condition Cohen algebra or the fair product
probability algebra. Let $H=\Aut(B)$ in the Cohen case and
$H=\Aut(B,\mu)$ in the random case. Assume a full local diagonal
$H$-action. Let $N$ be generated by independently extended
automorphisms in $H$ on banks $E$ satisfying
$|E|=|\Omega\setminus E|=\kappa$.

In a displayed frame $(f_i)_{i\in I}$, use target comparison
matrices $P_h$: their $(i,j)$-entry is the scalar event on which
$\Theta_h f_j$ agrees with $f_i$. Rows and columns are Boolean
partitions of $1$, and diagonal equivariance gives
\begin{equation}\label{eq:sr-comparison}
 P_{hk}=P_h\,h(P_k).
\end{equation}
Here matrix multiplication uses joins and meets, and $h$ acts
entrywise. These matrices are invertible. Locality computes the
comparison of a scalar pasting on its target pieces. None of the
following arguments requires a cardinality bound on $I$ once the
displayed frame is given.

\begin{lemma}\label{lem:sr-proper-fixation}\label{lem:sr-relative-frame}
\label{lem:scf-proper-fixation}
If every raw coordinate permutation fixes the frame, then every
element of $N$ fixes it. Whenever the frame is $N$-fixed, forcing
a raw factor $B_X$, $|X|<\kappa$, leaves its descended frame
fixed by the entire recomputed complementary proper-factor group,
including newly named banks and automorphisms.
\end{lemma}
\begin{proof}
Let $g$ act independently on a bank $E$ with
$|E|=|\Omega\setminus E|=\kappa$. Partition $\Omega$ into banks
$(E_j)_{j\in\mathbb Z}$ of size $\kappa$, with $E_0=E$, and fix
coordinate identifications between them. Let $R$ apply a copy of
$g$ on each bank $j\geq0$ and the identity on each bank $j<0$.
This defines an automorphism in $H$. In the Cohen case, the
compatible maps on finite products extend from their order-dense
union; in the random case they preserve measure and extend from
the metric-dense finite-product algebra. The copied inverses give
the inverse.

The map $R$ commutes with every permutation of the nonnegative
banks and, independently, every permutation of the negative banks.
These coordinate permutations fix the frame. Equation
\eqref{eq:sr-comparison} therefore makes every coefficient of $P_R$
invariant under both permutation groups. Their common scalar fixed
algebra is $\{0,1\}$. For random algebras, approximate a function by
finite coordinate cylinders, move the finitely many banks involved to
disjoint banks in their respective half-lines, and average;
independence makes the centered averages tend to zero in $L^2$.
For Cohen algebras, finite conditions below an invariant event
and its complement could be made compatible by moving their bank
supports apart, giving the same zero--one conclusion. It follows that $P_R$ is a constant ground permutation.

Let $S$ shift $E_j$ to $E_{j+1}$. It fixes the frame, so $P_S=1$.
On the subgroup generated by $R,S$, the constant comparison
matrices multiply without scalar transport. But
\[
 RSR^{-1}S^{-1}=g:
\]
the exponent of the copied $g$ on bank $j$ is
$1_{j\geq0}-1_{j-1\geq0}$, which is $1$ just at $j=0$.
The comparison of this commutator is the commutator of $P_R$ with
$1$, hence is $1$.

For the relative assertion, assume the frame is $N$-fixed and
write $A=B_X$. Put $Y=\Omega\setminus X$. By \cref{lem:full-quotient-action}, the full relative action includes
every new quotient automorphism in either setting.
For a prescribed ground bank $E\subseteq Y$ with
$|E|=|Y\setminus E|=\kappa$, a named independent relative
$E$-automorphism lifts to an independent automorphism in $H$ of
$B_{X\cup E}$. Its complementary bank has size $\kappa$, so the
lift belongs to $N$.

We must also allow newly named banks. Let $\dot\pi$ be an $A$-name
for a permutation of $Y$, and set
$a_{yz}=\|\dot\pi(y)=z\|_A$. Each row and each column of $(a_{yz})$
has countably many positive entries, by ccc. The undirected graph
joining $y,z$ when $a_{yz}>0$ or $a_{zy}>0$ therefore has countable
components. Each component is forced to be preserved by $\dot\pi$.
Split these components into two collections of cardinality $\kappa$,
giving ground unions $Y_0,Y_1$ of size $\kappa$.

The lift of $\dot\pi$ is the product of its restrictions to $Y_0$
and $Y_1$. Each restriction fixes $X$ and the other union, and is
an independent automorphism in $H$ on $X\cup Y_j$, hence belongs to
$N$. In the random case, partition $A$ by the images of each finite
list of coordinates: these images are distinct on every positive
piece, so their conditional distribution is still the fair product.
This verifies measure preservation. In both cases the named
permutation and its inverse extend to inverse automorphisms of the
completed product iteration.

Any named bank of size and complementary size $\kappa$ is the
image of a prescribed ground bank under such a named permutation.
Conjugating its relative map to that ground bank gives a lift in
$N$; the conjugating lift is itself in $N$. Thus every new
proper-factor generator fixes the descended frame. Only conjugation
by members of $N$ has been used.
\end{proof}

\begin{lemma}\label{lem:sr-short-adjustment}\label{lem:scf-adjustment}
Assume the frame is $N$-fixed. Suppose $P_h$ preserves a ground
set $J\subseteq I$ and restricts
there to a constant ground permutation $\gamma$. For every raw
factor $A=B_S$ with $|S|<\kappa$, some $n\in N$ makes $nh$ fix
$A$ pointwise, with the same comparison $\gamma$ on $J$.
\end{lemma}
\begin{proof}
Close $S$ under countable raw supports of the images of its
literals by $h$ and $h^{-1}$, repeating $\omega$ times. The result
$T$ has size at most $\max(|S|,\aleph_0)<\kappa$, and
$h[B_T]=B_T$. Independently extend $(h|B_T)^{-1}$ to $n$.
This map belongs to $H$ and is supported inside a proper
$\kappa$-bank, so $n\in N$. The comparison equation changes
$P_h$ only by scalar transport, leaving its constant restriction
$\gamma$ unchanged.
\end{proof}

\subsection{A common construction of commuting bit flips}

\begin{lemma}[Countable clocks]\label{lem:product-countable-clocks}
In either of the preceding two settings, suppose
$\operatorname{cf}(\kappa)=\omega$ and the frame is $N$-fixed.
If $P_h$ preserves a ground block $J$ and is constant there,
there are commuting involutions and clock bits satisfying
\eqref{eq:sr-bit-comparison}, with a complementary fixed algebra
$D$ and the product factorizations \eqref{eq:sr-clock-product}.
\end{lemma}
\begin{proof}
Write $\gamma$ for that restriction. We construct commuting
involutions $\tau_i\in H$ and independent bit events $b_i$,
$i<\omega$, such that each finite-stage map preserves $J$ and
\begin{equation}\label{eq:sr-bit-comparison}
 \tau_i(b_i)=\neg b_i,\qquad
 \tau_i(b_j)=b_j\quad(i\ne j),\qquad
 P_{\tau_i}|J=
 \begin{cases}
  \gamma&\text{on }b_i,\\
  \gamma^{-1}&\text{on }\neg b_i.
 \end{cases}
\end{equation}
The two comparison regions are target regions. We will arrange
that later involutions fix successively larger short factors
whose union is dense in $B$.

Suppose the first $n$ generators have been constructed. Denote
their group by $F=(\mathbb Z/2\mathbb Z)^n$, its elements by
$\sigma_s$, and the old bit phases by $d_s$, so
$\sigma_s(d_0)=d_s$. Equation \eqref{eq:sr-comparison} gives
\[
 P_{\sigma_s}|J=
 \gamma^{\,\sum_{\{i:s_i=1\}}(2b_i-1)},
\]
interpreted on the finite bit partition. These comparisons are
powers of $\gamma$ with coefficients in the old clock algebra.

Choose a short raw factor $A$ containing the old clocks, invariant
under $F$, and containing any prescribed short coordinate set.
To obtain it, close the desired coordinate set under the countable
supports of images of its literals by the finitely many
$\sigma_s^{\pm1}$, repeating $\omega$ times. This remains short.
Choose a fresh raw bit $x$, and let $v\in N$ be its independent
raw flip, fixing $A$. By \cref{lem:sr-short-adjustment}, choose
$h_0$ with comparison $\gamma$ on $J$ which fixes $A$ and $x$.

On phase $d_s$, use respectively
\[
 h_s=\sigma_s h_0\sigma_s^{-1},
 \qquad w_s=\sigma_s v\sigma_s^{-1}.
\]
Both maps fix $A$ pointwise, so they preserve $d_s$. Their comparisons
on $J$ are $\gamma$ and $1$. For example, put
$Q_s=P_{\sigma_s}|J$ and compare $h_s\sigma_s=\sigma_s h_0$:
\[
 (P_{h_s}|J)\,h_s(Q_s)=Q_s\gamma.
\]
The coefficients of $Q_s$ lie in $A$, and its values commute
with $\gamma$, proving the assertion. The calculation for $w_s$
has $1$ in place of $\gamma$.

Paste these maps on their preserved phases to obtain $u,w$.
They fix $A$, commute with $F$, and satisfy
$P_u|J=\gamma$, $P_w|J=1$, and $w^2=1$. Commutation follows
because conjugating the piece on $d_s$ by $\sigma_t$ gives
exactly the prescribed piece on $d_{s+t}$. The event
\[
 b=\bigvee_{s\in F}\sigma_s(d_0\wedge x)
\]
is $F$-invariant and independent of $A$. For random algebras,
its contribution on phase $d_s$ has measure $\mu(a\wedge d_s)/2$
for $a\in A$. For Cohen algebras, both halves meet every positive
$a\in A$, since the fresh bit does so before each conjugation. Moreover $u$
fixes $b$ and $w$ flips it.
Define the next generator on its source halves by
\[
 \tau=wu\text{ on }\neg b,\qquad
 \tau=u^{-1}w\text{ on }b.
\]
These restrictions are inverse maps in $H$. Thus $\tau$ is an
involution, fixes $A$, commutes with $F$, and satisfies
\eqref{eq:sr-bit-comparison} for the new bit. This proves the
induction step.

Now use $\operatorname{cf}(\kappa)=\omega$. At stage $n$ choose
the factor $A_n$ increasing, invariant under the old finite
group, and containing a prescribed increasing exhaustion of
$\Omega$ by sets of size less than $\kappa$. All $\tau_j$ with
$j\geq n$ fix $A_n$ pointwise, and the earlier generators
preserve $A_n$. For
$t\in T=(\mathbb Z/2\mathbb Z)^\omega$, define its action on
$A_n$ by the finite product of $\tau_i^{t_i}$, $i<n$.
These definitions and their inverses agree on overlaps and extend
to $B$: the union of the $A_n$ is order dense in the Cohen case
and metric dense in the random case. They define a $T$-action on $B$. In the random case its scalar orbit maps are continuous
by approximation in some $A_n$, where only finitely many group
coordinates matter. This construction requires no regularity of the comparison maps.

Let $D=B^T$. The bits give the product factorizations
\begin{equation}\label{eq:sr-clock-product}
 B\cong\begin{cases}
 D\widehat\otimes B_\omega&\text{(random)},\\
 \RO(D^+\times\operatorname{Fn}(\omega,2))&\text{(Cohen)}.
 \end{cases}
\end{equation}
To verify these factorizations, let $a\in A_n$ and saturate each
$a\wedge d_s$ under the first $n$ generators. Its saturation
belongs to $D$: the finite group permutes its terms, and the
tail fixes them all. Its intersection with $d_s$ is precisely
$a\wedge d_s$. Thus each $A_n$ lies in the
algebra generated by $D$ and the clock bits. In the random case, transitivity gives
$\mu(d\wedge d_s)=2^{-n}\mu(d)$ for $d\in D$; completion
proves the product identity. In the Cohen case it makes every
positive $d\in D$ meet every finite clock pattern. The displayed
product map therefore preserves incompatibility, and the saturation
calculation makes its image order dense, proving its completion
is $B$.

\end{proof}

We now specialize again to the random algebra, with
$H=\Aut(B,\mu)$ and $\operatorname{cf}(\kappa)=\omega$.

\begin{lemma}\label{lem:sr-pure-block}
In a full local $H$-action with an $N$-fixed frame, suppose one
comparison $P_h$ preserves a ground block $J\subseteq I$ and is
constant there. Its restriction to $J$ is the identity.
\end{lemma}
\begin{proof}
Write $\gamma$ for the restriction, and take the clocks of
\cref{lem:product-countable-clocks}. Let $C$ be the complete
algebra generated by their bits, so $B=D\widehat\otimes C$.
Each nonzero principal algebra of $D$ has Maharam type
$\kappa$. Its type is at most $\kappa$ as a metric subspace of
$B$; if it were smaller, adjoining the countable clock in
\eqref{eq:sr-clock-product} could not give a principal algebra
of the homogeneous $B_\kappa$. Maharam's classification
\cite[331I]{Fremlin} therefore gives an MP automorphism $q$
carrying a raw countable clock and its raw complement onto
$C$ and $D$, with its $i$th raw bit sent to $b_i$.
If $a_i\in N$ is the corresponding raw bit flip,
\begin{equation}\label{eq:sr-clock-conjugacy}
 \tau_iq=qa_i.
\end{equation}

Let $\rho_{ij}$ exchange clock coordinates $i,j$ and fix $D$.
It is also the scalar pasting of $\tau_i\tau_j$ where those
bits differ and the identity where they agree. Both maps fix
$D$ and agree on all clock generators, so
\eqref{eq:sr-clock-product} identifies them on $B$.
The signed exponents in \eqref{eq:sr-bit-comparison} cancel
on the differing-bit region. Hence $P_{\rho_{ij}}$ preserves
$J$ and is identity there. This remains true for every
finite clock permutation $\rho_\pi$.

Put $Q=P_q$. Since $\rho_\pi q=qa_\pi$ for a raw coordinate
permutation $a_\pi\in N$, we have
\begin{equation}\label{eq:sr-clock-rows}
 P_{\rho_\pi}\,\rho_\pi(Q)=Q.
\end{equation}
For $j\in J$, the $j$th row of $P_{\rho_\pi}$ is the $j$th
identity row. Therefore every $Q_{jk}$, $j\in J$, $k\in I$,
is fixed by all finite clock permutations.
Their common scalar fixed algebra is $D$. To see this
without a separability assumption on $D$, approximate an
invariant $L^2$ function using $D$ and finitely many clock
coordinates. Move the clock coordinates to many disjoint
blocks and average. After subtracting conditional expectation
onto $D$, these copies are conditionally orthogonal; their
averages tend to zero in $L^2$. The approximation error is
unchanged by the permutations, so the invariant function
belongs to $L^2(D)$.

Thus the rows of $Q$ indexed by $J$ are fixed by every
$\tau_i$. Applying \eqref{eq:sr-comparison} to
\eqref{eq:sr-clock-conjugacy}, and restricting to the target
half $b_i$, gives
\[
 Q_{\gamma^{-1}(j),k}=Q_{jk}
 \quad(j\in J,\ k\in I),\qquad\text{below }b_i.
\]
If $\gamma$ moved $j$, these would be two distinct rows.
Distinct rows are disjoint in each column, so every entry
of the $j$th row would vanish below $b_i$, contradicting that
the row has join $1$ and $\mu(b_i)=1/2$.
Consequently $\gamma=1$.

This argument requires only the constructed finite-stage maps and
finite clock permutations to preserve $J$. The conjugator $q$ need
not preserve $J$. We therefore use the rows of the whole matrix $Q$.
\end{proof}

\subsection{Full rigidity and ordinal definitions}

\begin{theorem}[Full frame rigidity]\label{thm:sr-frame-rigidity}
Let $\kappa$ be any uncountable cardinal of countable cofinality
in a ZFC ground. Every full local, diagonally equivariant
$\Aut(B_\kappa,\mu)$-action with an $N$-fixed frame fixes
that frame pointwise. The frame may have any ground set of
labels.
\end{theorem}
\begin{proof}
Fix $h\in H$ and a label $i$. Close $\{i\}$ under all possible
images and inverse images under $P_h$. Each row and each column
has only countably many positive entries, so this produces a
countable ground block $J$ containing $i$ which $P_h$ preserves.
The restricted matrix has countable raw support. Add a countable
reserve and close this support under the supports of images
of its literals by $h$ and $h^{-1}$, for $\omega$ steps.
The resulting countable raw factor $A$ satisfies $h[A]=A$
and carries every entry of $P_h|J$.

Let $k\in N$ independently extend $h|A$. The right residual
$r=hk^{-1}$ fixes $A$ pointwise and satisfies $P_r=P_h$.
Force first with $A$. By
\cref{lem:sr-relative-descent,lem:sr-relative-frame}, the
whole cover descends with its full recomputed MP action
and its proper-factor-fixed frame. The relative comparison
of $r$ on $J$ is now a constant ground permutation, since
its entries belonged to $A$. Apply \cref{lem:sr-pure-block}
in this relative ZFC ground. Its cardinal $\kappa$ is still
uncountable of countable cofinality, as required.

The relative restricted comparison is the identity. It follows that
$P_h|J$ was already the identity, so $P_h$ fixes $i$. Since $h$ and
$i$ were arbitrary, the whole group fixes the frame.
\end{proof}

\begin{proof}[Proof of \cref{thm:singular-random-od}]
We prove the result over any ground
$W\models\mathsf{ZFC}+\mathsf{GA}+(V=\operatorname{HOD})$.
The empty family has the empty enumeration.
Let $\kappa$ be uncountable with $\operatorname{cf}^W(\kappa)=\omega$,
and let $A$ be a nonempty OD family of size at most $\kappa$
in the $B_\kappa^W$-extension. Scalar homogeneity makes its
definition and cardinality hold with Boolean value $1$.
Choose an invariant name $\dot A$ and a name for a bijection
from a ground cardinal $I\leq\kappa$ onto it.
As in \cref{sec:names}, the algebra of named subfamilies is
$B_\kappa^I$, with a full local diagonal action of
$K=\Aut(B_\kappa)$. Locality follows by induction on names;
there is no rank bound on their values.

The hypotheses of \cref{lem:sr-initial-frame} hold in $W$.
It supplies a frame fixed by all coordinate permutations
and by every relative countable-coordinate nonsingular map.
By \cref{lem:sr-proper-fixation}, this frame is $N$-fixed,
and \cref{thm:sr-frame-rigidity} makes it fixed by
$H=\Aut(B_\kappa,\mu)$.

Finally every $g\in K$ has a factorization $g=ch$, where
$h\in H$ and $c$ changes only a countable coordinate factor.
Indeed the positive finite density
$f=d(g_*\mu)/d\mu$ has countable raw support. On a countably
infinite raw factor carrying it, the standard atomless
isomorphism theorem gives $c$ with $c_*\mu=f\mu$; extend
$c$ independently outside that factor. Then $c^{-1}g\in H$.
Both factors fix the frame, so the frame is $K$-fixed.

Relabeling the original enumeration by this frame gives an invariant
name for a bijection $I\to\dot A$, since every member-name and the
ground ordinal $I$ are fixed.
By \cref{lem:stable-ground-codes}, the ground and the function-name
are ordinal-definable in the extension. Thus
\cref{lem:invariant-od} gives an OD bijective enumeration of
$A$. In particular $A\subseteq\OD$.
\end{proof}




\subsection{Small families at regular random width}
\label{subsec:regular-random-families}

\begin{lemma}[The regular measure-preserving small-index theorem]
\label{lem:ra-small-index}
Let $\lambda$ be regular uncountable with $\lambda^{<\lambda}=\lambda$.
If $C$ is a Boolean subalgebra of $\mathbb B_\lambda$ of cardinality
less than $\lambda$ and
$H\leq\Aut(\mathbb B_\lambda,\mu)_{(C)}$ has index at most
$\lambda$, then $H$ contains the pointwise stabilizer of some coordinate
factor on fewer than $\lambda$ coordinates containing $C$.
In particular, CH gives countable supports at $\lambda=\omega_1$.
\end{lemma}

\begin{proof}
We verify the hypotheses of the small-index theorem for homogeneous
abstract elementary classes \cite[Theorem~4.1]{GV}.  Consider Boolean
algebras equipped with a strictly positive finitely additive probability,
in the countable language consisting of the Boolean operations and the
predicates $\mu(x)<q$, $q\in\mathbb Q$.  Strong substructure is Boolean
subalgebra with the restricted probability.  The rational cuts determine
the measure, so embeddings are exactly measure-preserving embeddings.
Increasing unions remain in the class: positivity and finite additivity
are checked at one stage.  The subalgebra generated by $X$ has size at
most $|X|+\aleph_0$.  Thus this is an abstract elementary class of
L\"owenheim--Skolem number $\aleph_0$.  Adding constants for $C$ gives
L\"owenheim--Skolem number at most $\max(|C|,\aleph_0)<\lambda$.

The metric completion of a probability Boolean algebra for
$d(x,y)=\mu(x\mathbin\triangle y)$ is a complete probability algebra,
and measure-preserving embeddings extend uniquely to completions.
An isomorphism between two subalgebras of $\mathbb B_\lambda$ of
cardinality less than $\lambda$ consequently extends to their
completions, of metric density less than $\lambda$.
Every positive principal algebra of $\mathbb B_\lambda$ has relative
Maharam type $\lambda$ over either such completion: fewer than
$\lambda$ relative generators would, together with the base generators,
make its absolute type less than $\lambda$.
The relative extension theorem
\cite[333C(b)]{Fremlin} therefore extends the isomorphism to an
automorphism of $(\mathbb B_\lambda,\mu)$.  This also proves the
required homogeneity after naming $C$.  Every probability Boolean
algebra of cardinality less than $\lambda$ embeds in
$\mathbb B_\lambda$, by Maharam's theorem applied to its completion;
homogeneity makes such embeddings extend a prescribed embedding of
a subalgebra of cardinality less than $\lambda$.

We now verify the simultaneous automorphism amalgamation required by
\cite[Definition~3.5]{GV}.  Let $A_0\subseteq A_1,A_2$ be probability
Boolean algebras of cardinality less than $\lambda$, and let fewer than
$\lambda$ paired automorphisms
of $A_1,A_2$ agree on $A_0$, each preserving $A_0$ setwise.
Complete the three algebras and take the relatively independent product
over the completion of $A_0$.  Its probability on rectangular generators is
\[
 \mu(x_1\wedge x_2)
 =\int \mathbb E(\mathbf1_{x_1}\mid A_0)
          \mathbb E(\mathbf1_{x_2}\mid A_0)\,d\mu.
\]
Each pair preserves this probability, since conditional expectation
commutes with its common base automorphism, and hence induces an
automorphism of the product.  The Boolean algebra generated by the two
images has cardinality less than $\lambda$ and is invariant under all
these maps and their inverses.
In forming this algebra, take the quotient by the null ideal. Strict
positivity of the marginals makes both original embeddings injective.  These are
the usual relative probability products
\cite[458N--458P]{FremlinProducts}.
Embed the amalgam into $\mathbb B_\lambda$ over $A_0$.  Homogeneity extends its two
embeddings to global automorphisms, giving precisely the simultaneous
amalgamation demanded by the cited theorem.  The construction retains
any named constants.

Finally $|\mathbb B_\lambda|=\lambda^{\aleph_0}=\lambda$.
The small-index theorem now supplies the stabilizer of a subalgebra
of cardinality less than $\lambda$ containing $C$.  The union of the
countable coordinate supports of its elements has size less than
$\lambda$.  Fixing that coordinate factor therefore fixes the subalgebra.
\end{proof}

\begin{corollary}[Regular random families]\label{cor:regular-random-enumeration}
Suppose $W\models\mathsf{ZFC}+\mathsf{GA}+(V=\mathrm{HOD})$ and
$\kappa$ is regular uncountable with $\kappa^{<\kappa}=\kappa$ in $W$.
In the extension by $\kappa$ random reals, every
$A\in\ODlt{\kappa}$ with $|A|<\kappa$ has a bijective enumeration
belonging to $\ODlt{\kappa}$.  There is no rank restriction.
\end{corollary}

\begin{proof}
Capture the short parameter defining a nonempty $A$ and a condition
deciding its definition and cardinality $\mu<\kappa$ in a short
coordinate extension $U$ of $W$.  The empty case is immediate.
The residual family name is fully invariant; choose a forced bijection
$(\tau_i:i<\mu)$ onto it.  The arithmetic
$\kappa^{<\kappa}=\kappa$ still holds in $U$: the original random
algebra has size $\kappa$, and nice names for sequences in $\kappa$
of any length $\xi<\kappa$ number at most
$\kappa^{\xi\cdot\aleph_0}=\kappa$.  Ccc preserves regularity.
The set of all member-name classes has size at most
$(\mu\cdot|\mathbb B_\kappa|)^{\aleph_0}=\kappa$,
by countable Boolean mixing.  Apply \cref{lem:ra-small-index} to
the stabilizer of each $\tau_i$ in the measure-preserving group.
It contains the fixer of a short coordinate factor $\B_{S_i}$.
Regularity makes $S=\bigcup_{i<\mu}S_i$ short.

Force $\B_S$ first. In the new ground $U_1$, the original
enumeration is fixed by the full recomputed measure-preserving tail
group, by \cref{lem:full-quotient-action}: every named operator lifts
to a ground map fixing $\B_S$, hence all $\tau_i$.

We show that the same original enumeration is fixed by nonsingular
maps as well.  First let $g$ fix the complement of a countably infinite
coordinate bank $T$ pointwise, and temporarily force that complement.
Its width is $\kappa$, so its extension has at least $\kappa$ reals.
The quotient on $T$ is the recomputed standard random algebra $Q$,
with the full relative action on $Q^\mu$ by
\cref{lem:sr-relative-descent}.  Since
$\mu<\kappa\leq2^{\aleph_0}$ there,
\cref{lem:sr-standard-frame} supplies a fully invariant frame.
Every original selector is still fixed by every new relative
measure-preserving map, by \cref{lem:full-quotient-action}.
Its coefficients in the invariant frame are consequently invariant
scalars for the standard measure-preserving group, hence are zero or
one.  It is therefore one of that frame's selectors, and the induced $g$
fixes it.
Returning to $U_1$, $g$ therefore fixes every original entry.
This temporary forcing verifies a Boolean equality. Its generic is
not needed as a parameter in the final definition.

For arbitrary nonsingular $g$, its Radon--Nikodym density has countable
support.  On a countable factor containing that support choose a
nonsingular $c$ having the same pushforward probability; then
$c^{-1}g$ is measure-preserving, as in the density correction in
\cref{lem:ssf-random-countable-frame}.  Both maps fix the enumeration.
Thus its single name is fully invariant, and
\cref{lem:stable-ground-codes,lem:invariant-od} give the required
definition.  The parameters are the generic from the initial short capture and the
generic on $S$. They have one combined short code.
\end{proof}



\section{Singular Cohen extensions}\label{sec:singular-cohen}

At singular width, families of size strictly below the width admit
uniform enumerations without a cardinal-arithmetic hypothesis. At
countable cofinality, the conclusion extends to families of size equal
to the width, and OD families have OD enumerations without an additional
short parameter. We first prove the countable-cofinality theorem, then
give the additional argument needed at other singular cardinals.
Throughout this section $B=\Coh_\Omega=\RO(\operatorname{Fn}(\Omega,2))$,
where $|\Omega|=\kappa$. A \emph{raw factor} is the complete algebra
$B_S$ generated by the coordinates in $S\subseteq\Omega$.
We use the full local diagonal actions on $B^I$ introduced in
\cref{sec:names}. In a frame $(f_i)_{i\in I}$ their target comparison
matrices satisfy
\begin{equation}\label{eq:scf-comparison}
 P_{gh}=P_g\,g(P_h).
\end{equation}
As in \eqref{eq:sr-comparison}, rows and columns are Boolean partitions
of $1$. A constant comparison is an ordinary permutation of the labels.

\subsection{Initial frames at countable cofinality}
\label{subsec:scf-initial-frame}

In this subsection all Cohen conditions are finite. Work in an arbitrary
ZFC ground $W$, let $\kappa$ be uncountable with
$\operatorname{cf}(\kappa)=\omega$, and put
\[
 B=\RO(2^\Omega),\qquad |\Omega|=\kappa,\qquad
 C=B^I,\qquad 0<|I|\leq\kappa.
\]
Suppose $\Theta:\Aut(B)\to\Aut(C)$ satisfies diagonal equivariance
and locality, as in
\eqref{eq:cohen-diagonal}--\eqref{eq:cohen-locality}.
Write $\delta(b)=(b)_{i\in I}$, $s(z)=\bigvee_i z_i$, and let
$\epsilon_i$ be the canonical coordinate selectors. Let $\Gamma$
be the finite bit translations and $\Sigma=\Sym(\Omega)$ the
coordinate permutations. As before, a selector has coordinates
partitioning $1_B$, and a frame is a partition of $1_C$ into selectors.

\begin{theorem}[Initial frame]\label{thm:scf-initial-frame}
The atoms of $C^\Gamma$ form a frame of cardinality $|I|$.
Every member of this frame is fixed by all coordinate permutations
and by every automorphism fixing $B_{\Omega\setminus S}$ pointwise
for some countable $S\subseteq\Omega$. Such an automorphism may
depend on all the complementary coordinates.
\end{theorem}

We first prove atomicity. We then use a standard-factor result to
show that the atoms are selectors. No cardinal arithmetic
hypothesis is used.

\begin{lemma}\label{lem:scf-knaster-real}
Every nonzero atomless complete Knaster algebra has a positive
condition forcing a nonground real.
\end{lemma}
\begin{proof}
Suppose no new real is added. Every countable sequence of maximal antichains then has a common
maximal refinement. Indeed, all antichains are countable, and the
generic sequence of their selected indices belongs to the ground. Equivalently, the nonzero meets of ground choices from
these antichains cover $1$.

Build refining maximal antichains through $\omega_1$, splitting
every node into two positive pieces at successor stages and taking
a common refinement at countable limits. Each level is countable.
Choose one node at each level. Knaster gives an uncountable
pairwise-compatible set of chosen nodes, hence an unbounded chain
in this refinement tree. Along that chain, the unused children at
uncountably many successive splits form an uncountable antichain,
a contradiction. If $b$ is the positive Boolean value that a
witnessing real is nonground, its binary digit tree below $b$ has
zero meet along every ground branch.
\end{proof}

\begin{lemma}\label{lem:scf-fixed-atomic}
The algebra $C^\Gamma$ is atomic with at most $\kappa$ atoms,
and it is fixed pointwise by every countably supported coordinate
permutation and every countably supported bit translation.
\end{lemma}
\begin{proof}
Every complete subalgebra $R\subseteq C$ has order density at most
$\kappa$. Indeed, the positive finite conditions in single
coordinates form an order-dense family in $C$ of that size, and
their upper projections
$\pi_R(p)=\bigwedge\{r\in R:p\leq r\}$ are order-dense in $R$.
Finitary permutations fix $C^\Gamma$ by locality: on each pattern
of their finite support they agree on the whole principal algebra
with a finite translation. Since $\Sigma$ normalizes $\Gamma$,
\cref{lem:scf-permutation-kernel} gives
\[
 C^\Gamma\subseteq C^{\Sigma_c},
 \qquad \Sigma_c=\{\sigma:|\supp(\sigma)|\leq\aleph_0\}.
\]

A $\Sigma_c$-invariant name for a family of at most $\kappa$ reals is
forced to consist of ground members. Suppose otherwise, and choose a
member name $\dot r$ which is nonground on a positive condition $b$.
Weak homogeneity allows us to take the family to be nonempty everywhere,
and we can extend the member name outside its original condition.
The values $\|\dot r=\check x\|$ for ground reals $x$ have only
countably many nonzero terms. Thus $\dot r$ and the positive
Boolean value
\[
 b=\neg\bigvee_{x\in(2^\omega)^W}\|\dot r=\check x\|
\]
have a common countable coordinate support $S$.
Copy $S$ onto the nonempty initial segments of each branch
$t\in\kappa^\omega$; each prescribed copy extends to a countably
supported permutation. Denote the copies by $\dot r_t,b_t$.
For distinct $t,u$,
\begin{equation}\label{eq:scf-separated-real-copies}
 b_t\wedge b_u\wedge\|\dot r_t=\dot r_u\|=0.
\end{equation}
Indeed, decide the finitely many common input bits. In the two remaining independent factors, a product condition forcing
equality must decide every digit of each real. Otherwise, two opposite
decisions on one side could both be combined with the fixed condition
on the other side. The product condition therefore forces a ground
real, contradicting $b_t$.
The branches for which $b_t$ holds inject into the given family,
so ccc covering puts them in a ground set of size at most $\kappa$.
Every $b_t$ is nonzero, whereas $\kappa^\omega>\kappa$, a
contradiction.

Put $E=C^{\Sigma_c}$. For each coordinate homomorphism $E\to B$,
let $e_i$ be the complement of the join of its kernel. The $e_i$
cover $1_E$, and $E\restr e_i$ embeds completely into
$B\restr(e_i)_i$. These intervals are Knaster: the Cohen algebra
is Knaster by the finite-condition $\Delta$-system argument,
and the property passes to complete subalgebras. If $E$ had an
atomless part, one such interval would have a positive binary tree
with zero meets along all ground branches, by
\cref{lem:scf-knaster-real}. Each label in the tree's root selects
a branch. The resulting family of at most $|I|$ reals is
$\Sigma_c$-invariant, so consists of ground reals. But on a
nonzero coordinate of the root its selected branch equals no
ground branch, a contradiction. Thus $E$ is atomic.
Upper projections of the atoms of a complete atomic algebra are
atoms of any complete subalgebra and cover its unit. Hence
$C^\Gamma$ is atomic. Its number of atoms is at most $\kappa$,
by ccc in each coordinate of $C$.

Finally, for a countably supported translation $t_u$, choose on
a countable supporting set a vector $v$ such that both $v$ and
$v+u$ have infinitely many zeros and ones. A permutation $\sigma$
of that set takes $v$ to $v+u$, and
$t_u=\sigma t_v\sigma^{-1}t_v^{-1}$.
The maps $\sigma$ and $t_v$ normalize $\Gamma$, and $\sigma$ acts
trivially on $C^\Gamma$. Their commutator therefore acts trivially
there as well.
\end{proof}

\begin{lemma}[Standard Cohen factors below the continuum]
\label{lem:scf-standard-frame}
Let $Q=\RO(2^\omega)$ and
$\max(|J|,\aleph_0)<2^{\aleph_0}$. Every full local
diagonal-equivariant action of $\Aut(Q)$ on $Q^J$ has a frame
fixed pointwise by $\Aut(Q)$.
\end{lemma}
\begin{proof}
Write $X=2^\omega$, let $\Gamma_0$ be the finite translations,
$T$ all translations, and $\Sigma_0=\Sym(\omega)$.
The proof of \cref{lem:scf-fixed-atomic}, using
\cref{lem:scf-permutation-kernel}(ii), shows that both $\Sigma_0$
and $T$ fix $(Q^J)^{\Gamma_0}$. To prove atomicity, we need a separate argument because a name may
use every coordinate of $X$.

Suppose a $T$-invariant family of reals has size at most
$\lambda<2^{\aleph_0}$ and has a nonground member $\dot r$
on a positive $b$. Read $\dot r$ by a Borel function $f:X\to X$
and represent $b$ by a nonempty regular open $E$.
Each $E\cap f^{-1}(y)$ is meager, so
$Z=\{(x,y)\in E^2:f(x)=f(y)\}$ is meager by Kuratowski--Ulam.
The set
\[
 F=\{(a,c,x):x+a,x+c\in E, f(x+a)=f(x+c)\}
\]
is meager as well, by the change of variables
$(a,c,x)\mapsto(x+a,x+c,x)$. Cover $F$ by closed nowhere dense
$F_n$. For a nonempty basic clopen $U\subseteq X$, the set
\[
 R_{n,U}=\{(a,c):\{(a,c)\}\times U\subseteq F_n\}
\]
is closed nowhere dense. Off their meager union $R$, each section
$F_{a,c}$ is meager. There is a perfect $P\subseteq X$ whose distinct pairs avoid $R$.
To construct it, recursively split finitely many clopen sets and shrink
the children so that every ordered product of distinct children avoids
the first finitely many closed nowhere dense relations. Choose the
children with diameters tending to zero.
For distinct $a,c\in P$, therefore,
\[
 t_a(b)\wedge t_c(b)\wedge\|t_a(\dot r)=t_c(\dot r)\|=0.
\]
The conditions $t_a(b)$ are all nonzero. As in
\eqref{eq:scf-separated-real-copies}, ccc covering contradicts
$|P|=2^{\aleph_0}>\lambda$. Thus all members of the family are
ground. Apply \cref{lem:scf-knaster-real} to the coordinate-kernel
intervals of $(Q^J)^T$, exactly as above, to obtain atomicity of
$(Q^J)^T$ and then of $(Q^J)^{\Gamma_0}$.

Fix an atom $U$ of the latter algebra. Its scalar support is $1$.
For any $j$ with $U_j>0$, the join of the countable orbit of
$U\wedge\epsilon_j$ is $U$. Each translate is a partial selector
and has only countably many nonzero coordinates, so $U$ has
countable coordinate support. Its nonzero fiber cardinality is
constant by scalar ergodicity. Choose a Boolean fiber enumeration,
identifying $Q^J\restr U$ over its diagonal with $Q^{J_0}$ for a
nonempty finite or countable $J_0$. The $\Gamma_0$-action on this
cover is ergodic, and $\Sigma_0,T$ preserve it.

Apply \cref{lem:cohen-borel-lift} with $\kappa=\omega$ to this
ergodic cover. Its set-action hypothesis follows from
\cref{lem:scf-permutation-kernel}(ii), applied to $\mathcal P(D)$
for any countable set $D$, since this complete algebra has countable
order density. Short-support permutations here are precisely the
finitary ones. The lemma therefore makes the actual map
$\sigma\mapsto\Theta_\sigma$ ordinary Borel in Boolean-matrix
codes, without any assumed regularity of the original action.

Choose $v_0,v_1\in X$ such that $v_0,v_1,v_0+v_1$ each have
infinitely many zeros and ones. The two countable sets
$\{u:v_i+u\text{ is finite or cofinite}\}$ are disjoint.
For each $u$, choose the first $i$ for which $v_i+u$ is neither finite
nor cofinite, and match the zeros and ones of $v_i$ increasingly with
those of $v_i+u$. The resulting
Borel $\sigma(u)$ satisfies
\[
 t_u=\sigma(u)t_{v_i}\sigma(u)^{-1}t_{v_i}^{-1}.
\]
The two fixed lifts $\Theta_{t_{v_i}}$ therefore make the actual
translation action Borel as well.

Choose jointly Borel representatives of its countable matrices,
replacing the matrix by the identity where it is not a permutation.
This gives $c:T\times X\to\Sym(J_0)$ with
$(\Theta_{t_u}z)(x)=c(u,x)z(x+u)$. For each fixed $u,v$ the
identity $c(u+v,x)=c(u,x)c(v,x+u)$ holds for comeager $x$.
Kuratowski--Ulam and the change of variables
$(u,v,x)\mapsto(x,x+u,x+u+v)$ show that
\[
 F(x,z)=F(x,y)F(y,z),\qquad F(x,y)=c(x+y,x),
\]
holds for comeager triples. Choose $a$ such that the identity holds
at $z=a$ for comeager pairs, and put $K(x)=F(x,a)$.
Then $c(u,x)=K(x)K(x+u)^{-1}$ for comeager $(u,x)$.
The selectors $q_j(x)=K(x)(j)$ are therefore fixed by a comeager
set of translations. Their common stabilizer is a subgroup of $T$ containing a comeager
set. Since a comeager set meets each of its translates, this subgroup
is all of $T$. Ergodicity of $\Gamma_0$ now forces this frame to
have a single member. Thus $U$ is a selector.

The atoms form a $\Gamma_0$-fixed frame $\mathcal F$. Given
$g\in\Aut(Q)$, apply \cref{lem:sww-comparison} to
$\langle\Gamma_0,g\rangle$ and $\Gamma_0$. It gives $q$ such
that every $qhq^{-1}$ is piecewise given by $\Gamma_0$.
Locality makes $\Theta_{q^{-1}}[\mathcal F]$ an
$\langle\Gamma_0,g\rangle$-fixed frame. Every $\Gamma_0$-fixed selector is a member of $\mathcal F$, since
its coefficients relative to that frame are invariant scalars and
therefore zeros or ones. The transported frame is consequently
the same frame, and $g$ fixes it pointwise. Finally its ground
cardinality is $|J|$, since its matrix is a Boolean bijection and
ccc preserves ground cardinals.
\end{proof}

\begin{lemma}\label{lem:scf-countable-reflection}
Every atom of $C^\Gamma$ is a selector. Its resulting frame is
fixed by every automorphism fixing the complement of a countable
coordinate set pointwise.
\end{lemma}
\begin{proof}
Let $U$ be an atom. Its scalar support is $1$. If it is not a
selector, choose $i\ne j$ with $U_i\wedge U_j>0$.
Write $m^\gamma_{jk}=(\Theta_\gamma\epsilon_j)_k$.
Each row and each column has countably many nonzero entries,
and every entry has countable coordinate support.
Build increasing countable $S_n\subseteq\Omega$ and
$J_n\subseteq I$, starting with the overlap and its supports.
For every $\gamma\in\Gamma_{S_n}$, include the labels occurring in
nonzero entries of rows and columns indexed by $J_n$. Include the
scalar supports of these entries and the supports of the newly
included $U_j$.
Also, for every pair of positive basic partial selectors below
$U$ with labels in $J_n$ and conditions on $S_n$, choose a finite
translation whose image of the first meets the second and add its
support. Such a translation exists because the orbit join of any
positive element below $U$ is $U$.

Put $S=\bigcup_nS_n$, $J=\bigcup_nJ_n$, and let $V$ agree with
$U$ on $J$ and be zero elsewhere. The recorded matrices define a
$\Gamma_S$-action on $B_S^J\restr V$. Its basic partial selectors
are order-dense, and the recorded intersection witnesses make the
action ergodic. Its original two-coordinate overlap remains nonzero.

Force first with $A=B_{\Omega\setminus S}$. The quotient is the
recomputed standard Cohen algebra $Q$ on $S$. The recorded inequalities persist, and the basic partial selectors
remain order dense. The intersection test therefore proves ergodicity
for new elements of $Q^J\restr V$ as well. By \cref{lem:full-quotient-action}, the
whole original cover descends to $Q^I$ with its full local
relative action, including all new automorphisms.
The complementary extension has at least $\kappa$ distinct reals,
preserves $\operatorname{cf}(\kappa)=\omega$, and therefore has continuum
strictly greater than $\kappa$. By \cref{lem:scf-standard-frame},
the full cover $Q^I$ has an invariant frame. Intersect it with
$V$. The intersections are invariant partial selectors covering
$V$; ergodicity makes any nonzero one equal to $V$. This contradicts
its overlapping coordinates. Hence $U$ is a selector.

Let $\mathcal F$ be the frame of these atoms. For any countably
infinite $S$, pass again to the complementary extension and choose
a full invariant frame $\mathcal E$ there. Each $f\in\mathcal F$
is still a $\Gamma_S$-fixed selector. Its coefficients relative
to $\mathcal E$ are $\Gamma_S$-invariant scalars, hence zeros or
ones, so $f$ is one row of $\mathcal E$. Every new relative automorphism fixes $f$. By the lifting in
\cref{lem:full-quotient-action}, the corresponding ground automorphisms
fix $f$ as well, including those whose action depends arbitrarily on
the complement. The frame has cardinality $|I|$ by ccc and
its Boolean bijection with $I$.
\end{proof}

\begin{lemma}[Small actions of the standard Cohen group]
\label{lem:scf-standard-small-index}
Every homomorphism $\Aut(Q)\to\Sym(J)$, where
$|J|<2^{\aleph_0}$ and $Q$ is the standard Cohen algebra,
is trivial.
\end{lemma}
\begin{proof}
We first prove that $G=\Aut(Q)$ is simple. Let $G_a$ consist of
the automorphisms supported on a principal region $a$, and use
$[x,y]=xyx^{-1}y^{-1}$. If $N\lhd G$ is nontrivial, choose
$g\in N$ and $a>0$ with $a\wedge g(a)=0$. For $u,v\in G_a$,
\[
 [[u,g],v]=[u,v],
\]
since $gu^{-1}g^{-1}$ is supported on the disjoint region $g(a)$.
Thus $[G_a,G_a]\subseteq N$. For $0<b<a$, every $h\in G_b$
is such a commutator. Partition $a$ into nonzero $b_n$, $n\in
\mathbb Z$, with $b_0=b$, and choose $q\in G_a$ shifting them.
Let $R$ equal $q^nhq^{-n}$ on $b_n$ for $n\geq0$ and the
identity on the negative banks. Then $[R,q]=h$. All proper
nonzero principal regions are conjugate, since they and their
complements are standard Cohen algebras. Hence $N$ contains every
$G_c$ for $0<c<1$.

These subgroups generate $G$. Given $f\ne1$, choose $a>0$ with
$a\wedge f(a)=0$ and $a\vee f(a)<1$. The involution $t$ equal
to $f$ on $a$, $f^{-1}$ on $f(a)$, and the identity elsewhere
belongs to $G_{a\vee f(a)}$, while $tf\in G_{\neg a}$.
Thus $f=t(tf)\in N$, proving simplicity.

Identify $Q$ with $\RO(\mathbb R)$. Increasing homeomorphisms
of $\mathbb R$ embed faithfully in $G$. Rosendal--Solecki's
small-index theorem \cite[Theorem~19]{RosendalSolecki} says that
$\operatorname{Homeo}_+(\mathbb R)$ has no nontrivial permutation
action on fewer than continuum many points. The kernel of the
given action of $G$ therefore contains this nontrivial subgroup,
and simplicity makes the entire action trivial.
\end{proof}

\begin{proof}[Proof of \cref{thm:scf-initial-frame}]
Only fixation by all coordinate permutations remains. Let
$\mathcal F$ be the frame from \cref{lem:scf-countable-reflection}.
Fix an infinite $E\subseteq\Omega$ with
$|\Omega\setminus E|=\kappa$, and force the complementary factor
$A=B_{\Omega\setminus E}$. In that extension, let $L_E$ be all
automorphisms of the recomputed $B_E$ fixing the complement of
some countable subset of $E$. Every such automorphism fixes $\mathcal F$. Ccc covering puts its
named countable support inside a ground countable set, and the lift
fixes that set's full complement in $B$. Thus \cref{lem:scf-countable-reflection} applies to its lift.
Since $L_E$
contains the finite translations, coefficientwise scalar ergodicity
gives
\[
 ((B_E)^I)^{L_E}=\mathcal P(\mathcal F).
\]

Partition $E$ into countably infinite blocks. The full recomputed
$\Aut(Q)$ acts diagonally by the same standard automorphism on
each block. To define this Boolean action, choose homeomorphism representatives
between comeager subspaces on each block. Their countable products
give compatible automorphisms on every countable collection of blocks. Every event has countable support,
so these actions and their inverses extend to $B_E$; ccc reduces
arbitrary joins to countable ones. This diagonal group normalizes
$L_E$, since the countably many blocks meeting a countable support
still have countable union. It therefore permutes the atoms
$\mathcal F$ of the displayed fixed algebra. The complementary
extension has continuum greater than $\kappa\geq|\mathcal F|$.
By \cref{lem:scf-standard-small-index}, this permutation action is
trivial. In particular every specified old diagonal map fixes
$\mathcal F$ in the ground.

Every coordinate permutation is a product of two involutions.
An involution with countably many pairs is covered by
\cref{lem:scf-countable-reflection}. If it has uncountably many
pairs and its support has a $\kappa$-sized complement, group its
pairs into countably infinite blocks. It is the diagonal action
of one fixed-point-free standard involution, so the preceding
argument applies. If its complement is smaller than $\kappa$,
split its $\kappa$ pairs into two collections of size $\kappa$.
Each restriction has a $\kappa$-sized complement and fixes the
frame. This treats every involution and completes the proof.
\end{proof}


\subsection{Proper factors and countable cofinality}

Let $N$ be generated by independently extended automorphisms of $B_E$
with $|E|=|\Omega\setminus E|=\kappa$. In particular, an independently
extended automorphism of a short raw factor belongs to $N$.

The proper-factor and short-adjustment lemmas,
\cref{lem:sr-proper-fixation,lem:sr-short-adjustment}, apply to this
group $N$. In particular, an $N$-fixed frame remains fixed by the
full recomputed relative proper-factor group after any short raw
coordinate extension.

\begin{lemma}\label{lem:scf-countable-pure}
Suppose $\operatorname{cf}(\kappa)=\omega$, $|I|\leq\kappa$,
and a full local action on $B^I$ has an $N$-fixed frame.
Any constant comparison on a preserved ground block is the identity.
\end{lemma}
\begin{proof}
Write $\gamma$ for the comparison on $J$. The Cohen case of
\cref{lem:product-countable-clocks} gives commuting involutions
$\tau_n$, bits $b_n$ with target comparisons
\eqref{eq:sr-bit-comparison}, and a fixed algebra $D$ such that
$B\cong\RO(D^+\times\operatorname{Fn}(\omega,2))$.

In a $B$-extension the continuum is greater than $\kappa$, since
there are at least $\kappa$ distinct new reals and the continuum
cannot have countable cofinality. One standard Cohen real
preserves its ground continuum, by counting nice names.
Consequently the $D$-extension in \eqref{eq:sr-clock-product}
already has continuum greater than $\kappa$. Descend the whole cover to the standard Cohen quotient there.
Every new quotient automorphism lifts to a $D$-fixing
automorphism of $B$, so the relative action is full.
By \cref{lem:scf-standard-frame}, it has a fully invariant frame.

The permutation exchanging clock bits $k,l$ and fixing $D$ is
the pasting of $\tau_k\tau_l$ where the bits differ and the
identity where they agree. The two exponents in
\eqref{eq:sr-bit-comparison} cancel, so it fixes each old $f_j$, $j\in J$.
Express $f_j$ in the new invariant frame in the $D$-extension.
Every scalar coefficient is fixed by all finite permutations
of the standard Cohen coordinates, hence is $0$ or $1$ by the
finite-condition zero--one argument. Thus $f_j$ is a member
of that frame and is fixed by all $\tau_k$. Equation
\eqref{eq:sr-bit-comparison} now gives $\gamma(j)=j$.
\end{proof}

\begin{theorem}[Cohen frames at countable cofinality]
\label{thm:scf-countable-frame}
Let $\kappa$ be uncountable of countable cofinality in a ZFC
ground. Every full local diagonally equivariant action of
$\Aut(\Coh_\kappa)$ on $\Coh_\kappa^I$, with
$0<|I|\leq\kappa$, has a frame fixed pointwise by the whole group.
\end{theorem}
\begin{proof}
Use \cref{thm:scf-initial-frame,lem:scf-proper-fixation} to obtain
an $N$-fixed frame. Given $h$ and $i\in I$, close $\{i\}$ under
the possible images and inverse images of $P_h$. The ccc makes
this a countable ground block $J$ preserved by $P_h$.
Its matrix entries have countable raw support. Close that support
under $h^{\pm1}$ to obtain a countable invariant factor $A$.
If $k\in N$ independently extends $h|A$, then $r=hk^{-1}$ fixes
$A$ and $P_r=P_h$. After forcing $A$, the relative action is
full and its frame remains proper-factor fixed by
\cref{lem:scf-proper-fixation}. The comparison on $J$ is now
a constant permutation. Apply \cref{lem:scf-countable-pure}
in that relative ground. It follows that $P_h$ fixes $i$.
Varying $h,i$ proves the theorem.
\end{proof}

\begin{corollary}\label{cor:scf-countable-od}
Let $W\models\mathsf{ZFC}+\mathsf{GA}+(V=\operatorname{HOD})$,
let $\operatorname{cf}^W(\kappa)=\omega<\kappa$, and let $G$
be $\Add(\omega,\kappa)^W$-generic. Every OD family of size at
most $\kappa$ in $W[G]$ has an OD bijective enumeration by a
ground cardinal at most $\kappa$. Every $\ODlt{\kappa}$ family
of that size has an $\ODlt{\kappa}$ bijective enumeration.
No GCH hypothesis is required.
\end{corollary}
\begin{proof}
For a nonempty OD family, choose an invariant name and a named
bijection indexed by a ground cardinal. The theorem supplies a fixed frame,
which relabels this bijection to an invariant function-name.
Apply \cref{lem:stable-ground-codes,lem:invariant-od}.
If the family is defined from a short parameter, choose a nice name
for that parameter supported on a short raw coordinate factor. Code that restricted generic by a subset
of a short ordinal, and apply the same argument to the residual
$\kappa$-Cohen extension. Stable ground codes retain this one
short parameter. The empty family has its empty enumeration.
\end{proof}

\subsection{Eliminating constant comparisons at higher cofinality}

The next lemma applies to frames of any size. We will use it after
one short capture has made the frame proper-factor fixed. The proof
uses an $\omega_1$-long chain of proper raw factors. These factors
may have size equal to the full width.

\begin{lemma}\label{lem:scf-proper-pure}
Let $\kappa>\omega_1$ in a ZFC ground. In a full local action
on $\Coh_\kappa^I$ with an $N$-fixed frame, a constant
comparison on any preserved ground block $J\subseteq I$
is the identity.
\end{lemma}
\begin{proof}
Let $u$ have constant comparison $\gamma$ on $J$.
First construct commuting automorphisms $J_i$, $i<\omega_1$,
on one raw bank $A$ of size $\omega_1$. Partition $A$ into
countable banks indexed by $\omega\leq\delta<\omega_1$.
In bank $\delta$, partition $1$ into positive events indexed
by the countable group
$E_\delta=\bigoplus_{i<\delta}\mathbb Z$, choose coherent
isomorphisms between their principal Cohen algebras, and let
$E_\delta$ translate the indices. Let $J_i$ perform its
$i$th translation in every bank with $i<\delta$, and act
trivially in the others. These products and their inverses
are Cohen automorphisms, obtained on finite products and
then extended by order completion. All $J_i$ belong to $N$.

Each countable prefix $E_\theta$ has a fundamental event
$p_\theta\in B_A$ satisfying
\begin{equation}\label{eq:scf-master-clocks}
 \{J_v(p_\theta):v\in E_\theta\}\text{ partitions }1,
 \qquad J_\theta(p_\theta)=p_\theta.
\end{equation}
For infinite $\theta$, use the identity cell in bank $\theta$;
for finite $\theta$, use the union of cells in bank $\omega$
whose first $\theta$ entries vanish. Also, each $b\in B$ is fixed by all sufficiently late $J_i$, since
the indices of the banks meeting its countable raw support are bounded
in $\omega_1$.

We next prepare an increasing chain $D_i=B_{S_i}$,
$i<\omega_1$, of $u$-invariant proper raw factors, all
containing $B_A$, such that
\begin{equation}\label{eq:scf-proper-exhaustion}
 |S_i|=|\Omega\setminus S_i|=\kappa,
 \qquad B=\bigcup_{i<\omega_1}D_i\quad\text{as a set}.
\end{equation}
For each raw coordinate $x$, let $C_x$ be its countable
closure under chosen supports for $u^{\pm1}$ of raw
literals. Thus $y\in C_x$ implies $C_y\subseteq C_x$.
The set $K=\bigcup_{x\in A}C_x$ has size at most $\omega_1$.
Hajnal's free-set theorem, in its arbitrary-cardinal form
\cite[p.~69]{ErdosHajnalMate}, supplies $Z\subseteq\Omega$
of size $\kappa$ with $C_z\cap Z=\{z\}$ for $z\in Z$.
The theorem requires the fixed bound $|C_x|<\omega_1<\kappa$;
it does not require $\kappa$ to be regular.
Remove $K$ from $Z$ and color $Z$ with $\omega_1$ colors,
each used $\kappa$ times. Put
\[
 \rho(x)=\sup\{\operatorname{col}(z):z\in C_x\cap Z\},
 \qquad S_i=\{x:\rho(x)\leq i\},
\]
where the empty supremum is $0$. Each $\rho(x)<\omega_1$,
and $y\in C_x$ implies $\rho(y)\leq\rho(x)$. Hence each $S_i$ is closed under the dependencies from both $u$ and
its inverse, and contains $K$. Since $\rho(z)=\operatorname{col}(z)$
on $Z$, both $S_i$ and its complement have size $\kappa$. Every countable support is contained
in some $S_i$, proving \eqref{eq:scf-proper-exhaustion}.

Call a map chain preserving if it preserves every $D_i$ setwise.
The maps $u,J_i$ have this property. So do their compositions
and inverses, independent extensions of their restrictions
to any $D_i$, and principal pastings on countable partitions
in $B_A$. For independent extensions, this follows by restricting to earlier
factors and using the product decomposition on later factors. For
pastings, compute images and inverse images as joins in each complete
$D_i$.

We construct permanent commuting maps $h_i$ and full maps
$F_\theta$, all chain preserving, with
\begin{align}
 h_i|B_A&=J_i|B_A,& P_{h_i}|J&=\gamma,\label{eq:scf-permanent}\\
 F_\theta h_iF_\theta^{-1}&=J_i\quad(i<\theta),&
 F_\theta|B_A&=\id,\nonumber\\
 F_\eta|D_\theta&=F_\theta|D_\theta\quad(\theta<\eta).
 &&\nonumber
\end{align}
Start with $F_0=\id$. The source prefix generated by the
$h_i$, $i<\theta$, is a copy of $E_\theta$, since it agrees
with the target action on $A$. Its comparisons on $J$ are
$\gamma^{\sum_i v_i}$ for $v\in E_\theta$.

We use two pasting constructions. First, if a countable action $h_v$
has fundamental event $p$, conjugate a map supported on $p$ by $h_v$
to define it on each $h_v(p)$. The pasted map centralizes the action.
If the action has comparisons which are powers of $\gamma$, this
construction preserves a constant comparison $\gamma$ or $1$. Second, suppose source and target
actions agree on $A$, share $p\in B_A$, and an
equivariant automorphism $T$ of an invariant raw factor
$D\supseteq B_A$ fixes $B_A$. Independently extend $T$
to $Q$. On the common partition $h_v(p)=J_v(p)$ put
\begin{equation}\label{eq:scf-common-pasting}
 F=J_vQh_v^{-1}.
\end{equation}
The inverses of these restrictions show that $F$ is a full
automorphism. Equivariance shows that it extends $T$, fixes $B_A$,
and conjugates the source action to the target action. These pastings
preserve the chain whenever their defining restrictions do.

At successor stage $\theta$, put $D=D_\theta$ and
$T=F_\theta|D$. The automorphism
$\sigma=T^{-1}(J_\theta|D)T$ has an independent extension
in $N$. Localize that extension to $p_\theta$, and mirror
it under the old source prefix. The resulting map $j$
centralizes the prefix, restricts to $\sigma$ on $D$,
and has identity comparison on $J$. This restriction
holds because $\sigma$ commutes on $D$ with the old
source action. It preserves the chain, as does each restriction used in its construction.
The action generated by the old maps and $j$ has fundamental event
$p_{\theta+1}$ in $B_A$.

Independently extend $(u|D)^{-1}$ to $q\in N$.
Then $w=qu$ fixes $D$ and retains comparison $\gamma$.
Localize $w$ to $p_{\theta+1}$ and apply the first pasting construction
to the enlarged action. This gives a chain-preserving map $r$
centralizing that action, fixing $D$, and having
comparison $\gamma$ on $J$. Set $h_\theta=rj$.
It is a permanent extension of the required source
restriction, commutes with the old maps, and satisfies
\eqref{eq:scf-permanent}. Apply
\eqref{eq:scf-common-pasting} with $p_{\theta+1}$ and
the independent extension of $T$ to obtain $F_{\theta+1}$.
This preserves all the inductive requirements, including preservation
of every $D_i$.

At a countable limit $\theta$, coherent restrictions and
inverse restrictions give an automorphism $T$ of
$D^-=B_{\bigcup_{i<\theta}S_i}$: the union of the earlier
raw factors is order dense in $D^-$. It is equivariant
with the permanent source prefix, fixes $B_A$, and
preserves each earlier $D_i$. Its independent extension
preserves the whole chain, since the raw coordinates of
$D^-$ are contained in every later $S_i$.
Use \eqref{eq:scf-common-pasting} with $p_\theta$ to
obtain the full $F_\theta$. No continuity of the chain
at $\theta$ is needed.

Finally, by \eqref{eq:scf-proper-exhaustion}, the coherent
restrictions define a full automorphism $F$ with
$Fh_i=J_iF$ for every $i<\omega_1$. Put $P=P_F$.
Since $P_{J_i}=1$, \eqref{eq:scf-comparison} gives, on
columns in $J$,
\[
 P\gamma=J_i(P)\qquad(i<\omega_1).
\]
If $\gamma(a)\ne a$, choose $b$ with $P(b,a)>0$.
A sufficiently late $J_i$ fixes this event exactly, so
$P(b,\gamma(a))=P(b,a)>0$, contradicting disjointness
within row $b$. Thus $\gamma=1$. At limits we took unions only of restrictions of
the conjugators. Every source map used in the final identity remained
unchanged after its construction.
\end{proof}

\begin{theorem}[Uniform strict-small Cohen families]
\label{thm:scf-all-singular}
Let $W\models\mathsf{ZFC}+\mathsf{GA}+(V=\operatorname{HOD})$
and let $\kappa$ be any singular cardinal of $W$.
In a $\Add(\omega,\kappa)^W$-extension, every
$A\in\ODlt{\kappa}$ of size less than $\kappa$ has
an $\ODlt{\kappa}$ bijective enumeration.
No GCH or strong-limit assumption is required.
\end{theorem}
\begin{proof}
Countable cofinality is \cref{cor:scf-countable-od}.
Suppose $\operatorname{cf}(\kappa)>\omega$.
Capture the original short parameter and represent the
family by a full local cover with $|I|<\kappa$ labels.
The uniform normalization lemma proved in the next
section, \cref{lem:ssf-uniform-normalization}, supplies,
after one further short capture, a frame fixed by all
raw permutations and all relative short-factor maps.
For Cohen algebras, that lemma uses only
\cref{thm:scf-countable-frame}, so the present argument is not circular. By \cref{lem:scf-proper-fixation} the
frame is proper-factor fixed, also in every subsequent
short relative ground.

For any scalar automorphism $g$, the entries of $P_g$
have a joint raw support of size at most
$\max(|I|,\aleph_0)<\kappa$. Close it under $g^{\pm1}$
to obtain a short invariant factor $B_T$. Independently
extend $g|B_T$ to $h$, which fixes the frame. Then
$r=h^{-1}g$ fixes $B_T$ and
$P_r=h^{-1}(P_g)$ has all entries in $B_T$.
After forcing $B_T$, it is a constant permutation of
the whole frame. The residual width is still $\kappa$
and $\kappa>\omega_1$, so
\cref{lem:scf-proper-pure} makes it the identity.
This proves $P_g=1$ for every $g$.

We used the factor on $T$ separately for each $g$ to prove invariance.
Its generic is not a parameter of the final definition. The invariant enumeration
exists by \cref{lem:stable-ground-codes,lem:invariant-od}
using just the original parameter capture and the single
normalization capture. Together these are one short code.
\end{proof}

\begin{remark}
The strict inequality is necessary for uniform enumeration. Under GCH
and uncountable cofinality, the Cohen extension has continuum $\kappa$. Its family of all reals cannot
have an $\ODlt{\kappa}$ enumeration: after capturing a
proposed short parameter, homogeneity would place every
real enumerated from it in that short coordinate
extension, contrary to the existence of new reals in the remaining Cohen extension. The next
section proves the pointwise endpoint for singular
strong-limit $\kappa$.
\end{remark}

\section{Small index and small families at singular strong limits}
\label{sec:ssf}

We prove the Cohen and random small-index theorems together. All
intermediate domains are coordinate factors, so we can amalgamate
maps by taking products over a common coordinate base. The two
constructions use different completions at limit stages.

Throughout the first subsection, $\Omega$ has singular strong-limit
cardinality $\kappa$.  Write $B_E$ for the complete coordinate factor on
$E\subseteq\Omega$, and let $B=B_\Omega$ be either the ordinary Cohen
algebra or the fair product probability algebra.  In the former case put
$G=\Aut(B)$, and in the latter put $G=\Aut(B,\mu)$.  A coordinate set is
\emph{short} if it has cardinality less than $\kappa$.  Subscripts in
parentheses denote pointwise stabilizers.

\subsection{A common small-index theorem}

\begin{theorem}\label{thm:ssf-small-index}
Let $D$ be a complete subalgebra of $B$ contained in a short coordinate
factor, and put $P=G_{(D)}$.  If $H\leq P$ and $[P:H]\leq\kappa$, then
there is a short $S\subseteq\Omega$ such that
\[
 D\subseteq B_S\qquad\text{and}\qquad G_{(B_S)}\subseteq H.
\]
This holds also when $\operatorname{cf}(\kappa)=\omega$.
\end{theorem}

The proof uses the tree method of Melles and Shelah~\cite{MellesShelah}.
We give the tree construction in detail and prove its extension step
using coordinate products.

Every element of either algebra has countable coordinate support.  Hence,
given a short set $T$ and fewer than $\kappa$ specified full maps, closing
$T$ under supports of the images and inverse images of its coordinate
bits, in countably many rounds, gives a common invariant coordinate
factor.  Its coordinate set has cardinality at most
\begin{equation}\label{eq:ssf-capture}
 \max(\aleph_0,|T|,\text{number of specified maps})<\kappa.
\end{equation}
For an increasing chain of coordinate sets, the union of their complete
factors is order dense in the factor on their union in the Cohen case,
and metrically dense in the random case.  Indeed, it contains every
finite-coordinate cylinder.  Compatible maps and compatible inverses
therefore extend uniquely to the appropriate completion.  In the random
case these extensions remain measure preserving.

We also use the following elementary product amalgamation.  Suppose
$S,Y,Z$ are disjoint, and automorphisms $u$ of $B_{S\cup Y}$ and $v$ of
$B_{S\cup Z}$ preserve $B_S$ and have the same restriction $\beta$ there.
Extend $\beta$ independently, remove this extension on the left of both
maps, and extend the two residual maps by the identity on the other
bank.  These residual extensions commute, since each fixes the entire factor
containing the images of the other's additional coordinate bits.
Their product, followed by the extension of $\beta$, amalgamates $u,v$.
All maps are complete automorphisms, with the corresponding product
inverses.  If $u,v$ preserve measure, every extension and product used
here preserves measure.  Dependence on the common base $B_S$ is allowed.

\begin{lemma}[One row of equations]\label{lem:ssf-row}
Let $N_0=B_S\subseteq N_1=B_T$ be short coordinate factors containing
$D$, and let $V$ be a set of fewer than $\kappa$ indices with a
distinguished index $\eta$.  Suppose full maps $h_\nu\in P$ preserve
$N_0,N_1$, with $h_\eta=\id$.  Suppose automorphisms
$g_0,m_\nu,l_\nu$ of $N_0$, fixing $D$, satisfy
\begin{equation}\label{eq:ssf-old-row}
 g_0m_\eta m_\nu^{-1}g_0^{-1}
   =l_\eta l_\nu^{-1}(h_\nu\restr N_0).
\end{equation}
Prescribe extensions $m_\nu^+,l_\nu^+$ to automorphisms of $N_1$ fixing
$D$.  There are full maps $g,m'_\nu,l'_\nu\in P$ retaining these
restrictions, with $g\restr N_0=g_0$, such that
\begin{equation}\label{eq:ssf-new-row}
 gm'_\eta(m'_\nu)^{-1}g^{-1}
   =l'_\eta(l'_\nu)^{-1}h_\nu
 \qquad(\nu\in V).
\end{equation}
In the random case all the prescribed and constructed maps are measure
preserving.
\end{lemma}

\begin{proof}
Put $Y=T\setminus S$, and choose a fresh coordinate copy $Y'$ of $Y$
outside $T$.  Let $g$ act as $g_0$ on $B_S$, swap $Y$ with $Y'$, and fix
all remaining coordinates.  Then
\[
 C'=g^{-1}[N_1]=B_{S\cup Y'}.
\]
Independently extend $m_\eta^+,l_\eta^+$ to full maps $a,b$.  Both
preserve $B_S$ and fix $Y'$, hence preserve $C'$.  For $\nu\ne\eta$
define an automorphism $v_\nu$ of $C'$ by
\begin{equation}\label{eq:ssf-copy-map}
 v_\nu(a^{-1}g^{-1}b(x))=g^{-1}h_\nu^{-1}l_\nu^+(x)
 \qquad(x\in N_1).
\end{equation}
Both parameterizations are isomorphisms of $N_1$ onto $C'$, and are
measure preserving in the random case.  Rearranging
\eqref{eq:ssf-old-row} gives
\[
 m_\nu m_\eta^{-1}g_0^{-1}l_\eta
   =g_0^{-1}(h_\nu\restr N_0)^{-1}l_\nu.
\]
Thus $v_\nu$ and $m_\nu^+$ agree on $B_S$.  The product amalgamation
above, followed by independent extension to $B$, gives $m'_\nu$
extending both.  Put $m'_\eta=a$, and define on all of $B$
\begin{equation}\label{eq:ssf-define-l}
 l'_\nu=h_\nu g m'_\nu a^{-1}g^{-1}b.
\end{equation}
For $\nu=\eta$ this is $b$; otherwise \eqref{eq:ssf-copy-map} says
exactly that it extends $l_\nu^+$.  Equation
\eqref{eq:ssf-new-row} is a rearrangement of
\eqref{eq:ssf-define-l}.  Every map fixes $D$.
\end{proof}

\begin{proof}[Proof of \cref{thm:ssf-small-index}]
Put $\theta=\operatorname{cf}(\kappa)$.  Choose increasing infinite
cardinals $\mu_i$, $i<\theta$, cofinal in $\kappa$, with
$\mu_i^+<\kappa$, and consider the tree
\[
 T_\alpha=\prod_{i<\alpha}\mu_i^+\qquad(\alpha\leq\theta).
\]
Every level below $\theta$ has size less than $\kappa$. Indeed, for
some infinite $\xi<\kappa$, its size is at most $\xi^\xi=2^\xi<\kappa$.
K\"onig's theorem gives $|T_\theta|>\kappa$.  Fix continuous increasing
short coordinate sets $X_\alpha$, $\alpha<\theta$, with union $\Omega$.

Suppose the conclusion fails.  We construct increasing short coordinate
factors $N_\alpha$ containing $D$ and $B_{X_\alpha}$.  At every node
$\eta\in T_\alpha$ we construct three automorphisms
$g_\eta,m_\eta,l_\eta$ of $N_\alpha$ fixing $D$, coherent along branches.
Once $N_\alpha$ is chosen, choose permanently
\[
 h_\alpha\in G_{(N_\alpha)}\setminus H.
\]
All later coordinate closures include $h_\alpha^{\pm1}$, so it
preserves every later domain and fixes every earlier one pointwise.
For nodes $\eta,\nu$ at the same level, let $h_{\eta\nu}=h_\gamma$
if their first difference is at $\gamma$ and
$\eta(\gamma)<\nu(\gamma)$; in the opposite order, and for equal
nodes, let $h_{\eta\nu}=\id$.  Maintain
\begin{equation}\label{eq:ssf-tree-row}
 g_\eta m_\eta m_\nu^{-1}g_\eta^{-1}
   =l_\eta l_\nu^{-1}(h_{\eta\nu}\restr N_\alpha).
\end{equation}
The initial maps can all be the identity.

At a successor stage, give each child its
parent's maps on $N_\alpha$.  The equations hold there also for new
siblings, since $h_\alpha$ fixes $N_\alpha$.  Independently extend all
inherited $m,l$ maps, and use \eqref{eq:ssf-capture} to obtain a common
short invariant coordinate factor containing $N_\alpha$,
$B_{X_{\alpha+1}}$, and invariant under all previously chosen $h$'s.  Initially
retain each $g$ only on $N_\alpha$.

Choose an infinite $\chi<\kappa$ bounding the size of this coordinate
set, the new level, and the previously chosen maps.  In an iteration of length
$\chi$, visit every row cofinally often.  Such a schedule exists even when $\chi$ is singular. Partition
$\chi$ into as many sets of cardinality $\chi$ as there are rows,
and visit each row at the stages in its assigned set.  When row $\eta$ is visited, let
$N_0$ be its last completed domain and let $N_1$ be the current common
domain.  Its old equation holds on $N_0$, all current $m,l$ maps extend
their restrictions there, and all $h_{\eta\nu}$ preserve both domains.
Apply \cref{lem:ssf-row}.  Capture the resulting full maps, the previously
chosen maps, and the current domain in a common invariant coordinate factor
of size at most $\chi$.  This extends all $m,l$ restrictions and the
selected $g$ restriction.  Every unvisited row keeps its old $g$ and
its old equation on its own domain; no equation is imposed for that row
on the larger domain until it is visited.

At an inner limit complete the increasing union of common coordinate
factors.  For a particular $g$, complete the union of its visit domains,
which is an earlier stage, possibly a limit stage, of the same chain.
The coordinate bound remains $\chi$.  At the end all rows have been
visited cofinally, so all three maps act on the same factor
$N_{\alpha+1}$ and satisfy \eqref{eq:ssf-tree-row} there.  Their
inverses are coherent as well.

At an outer limit $\alpha<\theta$, the union of the coordinate sets
is still short.  Complete the coherent maps along each branch.  At
height $\theta$, every branch similarly gives full maps
$g_\rho,m_\rho,l_\rho\in P$: the union contains every finite-coordinate
cylinder, so order completion or metric completion gives all of $B$.
This last assertion applies also when $\theta=\omega$.

For two distinct branches, ordered so that
$\rho(\gamma)<\nu(\gamma)$ at their first difference, the two
instances of \eqref{eq:ssf-tree-row} imply
\begin{equation}\label{eq:ssf-coset-contradiction}
 aLa^{-1}=Lh_\gamma,\qquad
 a=g_\rho g_\nu^{-1},\quad L=l_\rho l_\nu^{-1}.
\end{equation}
There are more than $\kappa$ branches and at most $\kappa$ pairs of
right cosets $(Hg_\rho,Hl_\rho)$.  Two branches therefore give
$a,L\in H$, and \eqref{eq:ssf-coset-contradiction} gives
$h_\gamma\in H$, a contradiction.  The argument does not assume that membership in $H$ is preserved
under limits.
\end{proof}

\subsection{A uniform normalization for fewer than \texorpdfstring{$\kappa$}{kappa} labels}

We now prove the frame normalization used in both kinds of extension.
This argument does not require $\kappa$ to be strong limit.  A full local diagonal action on
$B^I$ and its selector frames are understood as in
\eqref{eq:cohen-diagonal}--\eqref{eq:cohen-locality} and
\cref{subsec:sr-frame}.  In the random case ``full'' refers to all
Boolean automorphisms, including the nonsingular ones.  Relative actions
are always recomputed in the intermediate extension.

\begin{lemma}\label{lem:ssf-random-countable-frame}
In any ZFC ground, if $\nu$ is uncountable of countable cofinality and
$0<|I|\leq\nu$, every full local diagonal action on the random
$B_\nu^I$ has a frame fixed by the full Boolean automorphism group.
\end{lemma}

\begin{proof}
By \cref{lem:sr-initial-frame}, there is a frame fixed by raw
permutations and all maps fixing the complement of a countable
coordinate set.  \Cref{lem:sr-proper-fixation,thm:sr-frame-rigidity}
then give fixation by every measure-preserving automorphism.  For a
nonsingular $g$, the density $f=d(g_*\mu)/d\mu$ has countable coordinate
support.  On a countably infinite factor containing that support, the
isomorphism theorem for atomless standard probability spaces supplies
a nonsingular $c$ with $c_*\mu=f\mu$.  Extend it independently.  Then
$c^{-1}g$ is measure preserving, while $c$ acts only on a countable coordinate factor. Both therefore
fix the frame.
\end{proof}

\begin{lemma}[Uniform normalization]\label{lem:ssf-uniform-normalization}
Work in any ZFC ground.  Let $\kappa$ be singular with
$\operatorname{cf}(\kappa)>\omega$, let $B$ be Cohen or random of width
$\kappa$, and let a full local diagonal action on $B^I$ be given, where
$0<|I|<\kappa$.  Fix any original frame $(e_i:i\in I)$, and put
$\lambda=\max(\aleph_0,|I|)$.  There is a coordinate set $S$ of size at
most $\lambda$ such that, after forcing $B_S$, the descended original
frame is fixed by every raw permutation and by every full
automorphism fixing the complementary factor of some short coordinate
set pointwise.  The latter maps may depend on all complementary
parameters.
\end{lemma}

\begin{proof}
Use comparison matrices in the original frame, with convention
\begin{equation}\label{eq:ssf-comparison}
 P_{gh}=P_g\,g(P_h).
\end{equation}
Choose $\lambda<\nu<\kappa$ of countable cofinality, and a coordinate
bank $T$ of size $\nu$.  Force the complementary factor first.
The relative cover is the original recomputed $B_T^I$, with its full
action.  This is \cref{lem:sr-relative-descent} for random forcing;
the same argument for Cohen forcing uses names for operators and their
inverses in the product iteration, and descends their equality tests
and the locality identity.  In particular, the relative action includes every operator in the
intermediate extension.

By \cref{thm:scf-countable-frame,lem:ssf-random-countable-frame}, this
relative cover has a fully invariant frame.  Lift it to a frame $F$
over $B$.  It is fixed by every automorphism fixing the complement of
$T$ pointwise.  The change-of-frame matrix from $F$ to $(e_i)$ has at
most $\lambda$ entries, supported together by a set $U$ of size at
most $\lambda$.  Choose $Q\subseteq T\setminus U$ of size $\nu$.
Every automorphism fixing the complement of $Q$ fixes both $F$ and
the change-of-frame matrix, and hence fixes the original frame.

Let $t_\alpha$ flip coordinate $\alpha$, and put
\[
 S=\{\alpha\in\Omega:P_{t_\alpha}\ne\id\}.
\]
For a raw permutation $\pi$, choose a support $U_\pi$ of its comparison
matrix of size at most $\lambda$.  If $\alpha\in Q$ and
$\beta=\pi(\alpha)$, then $t_\beta\pi=\pi t_\alpha$, so
\[
 P_{t_\beta}\,t_\beta(P_\pi)=P_\pi.
\]
For $\beta\notin U_\pi$ this gives $P_{t_\beta}=\id$.  Thus $S\cap\pi[Q]\subseteq U_\pi$. If $|S|>\lambda$, choose a
permutation sending $Q$ to a set of size $\nu$ containing $\lambda^+$
elements of $S$. This contradicts the bound on $U_\pi$.  Consequently $|S|\leq\lambda$.

Force $B_S$.  The original frame in the recomputed tail is fixed by
every finite translation.  Let $E$ now be any short coordinate set in
this intermediate ground.  Choose a countable-cofinality cardinal
$\nu$ strictly between $\max(|E|,\lambda)$ and $\kappa$, and a bank
$T\supseteq E$ of size $\nu$.  After forcing its complement, choose a
full invariant frame $F$ as above.  The coefficients of every $e_i$
in the $F$ coordinates are fixed by finite translations of $T$.
The scalar fixed algebra of those translations is $\{0,1\}$, by
the finite-condition argument for Cohen forcing and product-measure
ergodicity for random forcing.  Hence each selector $e_i$ is a member of $F$ and is fixed by the
full relative group.  Every automorphism fixing the complement
of $E$ belongs to this relative $T$ group.  This proves the claimed
fixation for all such new $E$ and all new relative maps.

It remains to consider raw permutations.  The finite-translation fixed
cover algebra consists of unions of the original frame members.
Permutations normalize these translations, and therefore act ordinarily
on $I$.  Every action of $\Sym(\kappa)$ on fewer than $\kappa$ points
is trivial.  For completeness, the small-index support theorem
of Dixon--Neumann--Thomas~\cite[Theorem~$2^{\flat}$, p.~582]{DNT}
gives, for any point stabilizer
$L$ of index less than $\kappa$, a short $A$ with
$\Sym(\kappa)_{(A)}\subseteq L$.  Among $\kappa$ permutations sending
$A$ to pairwise disjoint sets, two have the same left $L$ coset.
Thus some $h\in L$ sends $A$ disjointly from itself.  The pointwise
stabilizers of two disjoint short sets $A,D$ generate $\Sym(\kappa)$:
given $f$, choose $v$ fixing $A$ with $vf[A]\cap D=\varnothing$,
extend $vf\restr A$ to $w$ fixing $D$, and note that
$w^{-1}vf$ fixes $A$.  Hence $L=\Sym(\kappa)$, as required.
\end{proof}

\subsection{Arbitrary-rank definable families}

\begin{theorem}[The Cohen endpoint]\label{thm:ssf-cohen-endpoint}
Suppose $W\models\mathrm{ZFC}+\mathrm{GA}+(V=\mathrm{HOD})$ and
$\kappa$ is a singular strong-limit cardinal of uncountable cofinality
in $W$.  In an extension $M=W[G]$ by $\kappa$ ordinary Cohen reals,
every $A\in\ODlt{\kappa}$ of cardinality at most $\kappa$ satisfies
$A\subseteq\ODlt{\kappa}$.  The members may have arbitrary rank.
\end{theorem}

\begin{proof}
Capture the short parameter defining $A$, and a condition deciding the
definition, nonemptiness, and cardinal bound, in a short coordinate
extension $V_0$ of $W$.  The empty case is immediate.  This gives a
fully invariant residual name $\dot A$, with a forced bijection
$(\tau_i:i<\mu)$ onto it, where $\mu\leq\kappa$.

The required arithmetic holds also in $V_0$.  In $W$, strong limitness
and uncountable cofinality give $\kappa^{\aleph_0}=\kappa$; a short
Cohen or random factor has size less than $\kappa$.  Forcing of size
$\delta<\kappa$ has at most $2^{\delta\cdot\xi}<\kappa$ names for
subsets of each infinite $\xi<\kappa$.  Since our short factors are
ccc, $\kappa$ remains strong limit singular of the same cofinality.
Thus in $V_0$ the residual Cohen algebra $B$ has cardinality $\kappa$.

Let $Z$ be the set of names forced to belong to $\dot A$, modulo forced
equality.  Each is a countable Boolean mixture of the $\tau_i$, so
\[
 |Z|\leq(\mu\cdot|B|)^{\aleph_0}\leq\kappa.
\]
The full automorphism group of $B$ acts on the set $Z$.  For each
$i$, \cref{thm:ssf-small-index} gives a short $S_i$ such that the full
pointwise fixer of $B_{S_i}$ fixes $\tau_i$.  Force this factor first.
By \cref{lem:hc-quotient-invariance}, the quotient name is fixed by
every automorphism recomputed in $V_0[G_{S_i}]$: names for an operator
and its inverse lift through the iteration to maps fixing $B_{S_i}$.
The stable-code and invariant-name lemmas
\ref{lem:stable-ground-codes} and~\ref{lem:invariant-od} give an
$\ODlt{\kappa}$ definition of its value.  The original parameter
capture and this further generic have one combined short code.
Only the original $W$ is required to satisfy the stable-ground
hypotheses.  Applying the argument separately to every $i$ proves the
claim; no union of the $S_i$ is taken.
\end{proof}

\begin{theorem}[Uniform enumeration for random families]
\label{thm:ssf-random-enumeration}
Under the same ground and cardinal assumptions, let $M=W[G]$ be the
extension by $\kappa$ random reals.  Every
$A\in\ODlt{\kappa}$ with $|A|<\kappa$ has a bijective enumeration
belonging to $\ODlt{\kappa}$.  There is no restriction on the ranks of
its members.
\end{theorem}

\begin{proof}
Capture the defining parameter and a condition deciding nonemptiness
and $|A|=\mu<\kappa$ in a short coordinate extension of $W$.  A
name for a bijection of $\mu$ onto the fully invariant family gives
the original frame $(e_i:i<\mu)$ of its cover.  Apply
\cref{lem:ssf-uniform-normalization} and make its one common short
capture.  In the resulting ground $V_*$ this original frame is fixed
by every short relative full automorphism.  The preceding arithmetic
argument shows that the residual probability algebra still has
strong-limit singular type $\kappa$, uncountable cofinality, and
cardinality $\kappa$.

Its selectors form an ordinary set of size at most
$(\mu\cdot|B|)^{\aleph_0}=\kappa$.  For each $i$, apply
\cref{thm:ssf-small-index} to the stabilizer of $e_i$ in
$\Aut(B,\mu)$, obtaining a short $S_i$ whose full measure-preserving
pointwise fixer fixes $e_i$.  Given an arbitrary $g\in\Aut(B,\mu)$,
close $S_i$ under coordinate supports of $g^{\pm1}$ to a short
invariant set $E$.  Let $n$ independently extend $g\restr B_E$.
The map $n$ is short and measure preserving, so fixes the entire
original frame.  The map $n^{-1}g$ fixes $B_E$ pointwise, so fixes
$e_i$.  Therefore $g$ fixes $e_i$.  This holds for all $g$ and $i$.

Every nonsingular automorphism factors into a nonsingular map acting
on a countable coordinate factor and a measure-preserving map, by the
density correction in
\cref{lem:ssf-random-countable-frame}.  Both fix the original frame.
Thus its single enumeration name is fully invariant.  Apply
\cref{lem:stable-ground-codes,lem:invariant-od}.  The parameters are the generics from the original short capture and
the common normalization capture. They have one short code for the
entire enumeration.  The
memberwise sets $S_i$ and the auxiliary sets $E$ were used only to
prove invariance and are not added to this parameter.
\end{proof}

\begin{remark}\label{rem:ssf-bounds}
The Cohen conclusion at size $\kappa$ is pointwise; it does not assert
a common short-parameter enumeration.  The random strict inequality
is sharp: at the present cardinals $\kappa^{\aleph_0}=\kappa$, and
\cref{prop:higher-random-cardinality,prop:no-od-less-kappa-random}
give a definable family of size $\kappa$ with no
$\ODlt{\kappa}$ member.
\end{remark}

\section{Definable models of arithmetic}\label{sec:models-arithmetic}

We first show that a definable structure whose domain is pointwise
definable has an isomorphic copy in the corresponding hereditary inner
model. We use this to exclude definable models with full binary standard
system at $\aleph_1$ in ordinary Cohen and random extensions. We then
construct definable saturated models at successors of singular strong
limits.

\subsection{Definition codes and hereditary classes}
\label{subsec:definition-codes}

Write $\ODR$ for definability from ordinals and one real, and
$\HODR$ for its hereditary part. We use the same construction with
parameters in $H_\lambda$, and occasionally with one additional fixed
set $a$. In this subsection let
\[
 Z=\mathbb R\ \text{ or }\ H_\lambda,\qquad
 O=\mathrm{OD}_{a,Z},\qquad
 S=\{x:\operatorname{tc}(\{x\})\subseteq O\},
\]
where $\lambda$ is regular uncountable. The real-parameter assertions
hold in ZF; for $H_\lambda$ we work in ZFC. The fixed parameter $a$
may be omitted and need not belong to $S$. Finite tuples of allowed
parameters can be coded by one allowed parameter.
The hereditary-model and copy lemmas also allow any nonempty
$Z\subseteq\mathbb R$ definable from $a$ and ordinals and closed
under coding finite tuples of its members.

There is a uniform definable partial evaluation map
\begin{equation}\label{eq:definition-evaluation}
 \operatorname{Val}_a:Z\times\Ord\rightharpoonup V
 \quad\text{with range }O.
\end{equation}
The ordinal input codes a formula, ordinal parameters, and a rank
segment containing $a$ and the other parameters in which uniqueness
is witnessed. Reflection supplies such a code for every definition;
conversely, evaluating a code gives an allowed definition. Only
satisfaction for set-sized ranks is used.

\begin{lemma}[Hereditary definability]\label{lem:definition-inner-model}
\label{lem:general-hereditary-model}
$S$ is a transitive inner model of ZF containing every member of $Z$.
For $Z=H_\lambda$ and every infinite $\kappa<\lambda$, it contains
$L(\mathcal P(\kappa)^V)$ and computes $\mathcal P(\kappa)$ correctly.
\end{lemma}
\begin{proof}
Substitution of definitions makes $O$ closed under definitions using
finitely many of its members. The class $S$ is definable from $a$ and ordinals,
transitive, contains all ordinals and all members of $Z$, and is closed
under pairing and union. For Separation and Replacement, relativize
the formula to $S$ and form the required set in $V$. It belongs to
$O$ by substitution and has all its elements in $S$, so belongs to
$S$. The same argument applies to $\mathcal P^V(x)\cap S$, giving
the internal power set. The other ZF axioms are inherited, with
$L\subseteq S$ supplying Infinity. In the $H_\lambda$ case, every
subset of $\kappa$ belongs to $S$, as does their ordinal-definable
collection $\mathcal P(\kappa)^V$. Minimality gives the stated
constructible inner model.
\end{proof}

\begin{lemma}[The all-codes copy lemma]\label{lem:rod-copy}
\label{lem:add-copy}
Every structure in $O$, in a language indexed by $\omega$, whose
domain is pointwise in $O$ has an externally isomorphic copy in $S$.
No bound on the domain's cardinality is required.
\end{lemma}
\begin{proof}
For each domain point $d$, take the least $\xi$ such that
$\operatorname{Val}_a(z,\xi)=d$ for some $z\in Z$. Replacement
bounds these ordinals by one $\delta$. The set
\[
 C=\{(z,\xi)\in Z\times\delta:
                     \operatorname{Val}_a(z,\xi)\in\operatorname{dom}(\mathcal M)\}
\]
maps onto the domain. Pull back equality, the relations, and the
function graphs through $\operatorname{Val}_a$. These sets, and their
language-indexed collection, are in $O$ and have hereditary entries
in $S$, so belong to $S$. Their quotient by the pulled-back equality
exists in $S$ by ZF; evaluation induces the external isomorphism.
The construction retains all codes below the ordinal bound and
requires no choice of representatives.
\end{proof}

\begin{lemma}[Closure from ordinal covers]\label{lem:definition-sequence-closure}
Work in ZFC. Let either $Z=\mathbb R$ and $\mu=\omega$, or
$Z=H_{\mu^+}$ for an infinite cardinal $\mu$. Suppose that the range
of every ordinal sequence of length at most $\mu$ is contained in
the range of an ordinal-definable map $e:\mu\to\Ord$. Then $S$ is
closed under ambient sequences of length at most $\mu$ of its
members, and consequently satisfies $\mathsf{DC}_\mu$.
\end{lemma}
\begin{proof}
Choose codes $x_i=\operatorname{Val}_a(z_i,\xi_i)$ for such a
sequence, and choose $e$ covering the $\xi_i$. Put
$j_i=\min\{j:e(j)=\xi_i\}$. The sequence of $(z_i,j_i)$ is coded
by one real in the first case and belongs to $H_{\mu^+}$ in the
second. Together with an ordinal definition of $e$, it defines the
whole sequence in $O$. All other sets in its transitive closure belong to $S$, so the
sequence belongs to $S$. Sequences witnessing ambient dependent choice
therefore also witness dependent choice in $S$.
\end{proof}

The cover hypothesis holds in a cardinal-preserving
$\mu^+$-cc extension of a ground with the stable ordinal codes of
\cref{lem:stable-ground-codes}: antichains deciding the ordinal
values give a ground cover of size at most $\mu$, and a ground
enumeration of that cover is ordinal-definable in the extension.


\begin{corollary}[A countable ground-model copy]\label{cor:cohen-countable-copy}
Let $a$ be a set of ordinals and let $c$ be Cohen-generic over $L[a]$.
In $V=L[a][c]$, every countable $\mathrm{OD}(a)$ structure in a fixed
countable language coded in $L[a]$ has an $\mathrm{OD}(a)$ isomorphic
copy on a finite ordinal or on $\omega$ whose code belongs to $L[a]$.
In particular, for a countable $\mathrm{OD}(a)$ model $\mathcal M$
of $I\Delta_0$,
\[
 \SSy(\mathcal M)\subseteq\mathcal P(\omega)^{L[a]}.
\]
\end{corollary}

\begin{proof}
The proof applies to any original ground $W_0$ with the stable
ordinal codes of \cref{lem:stable-ground-codes}, with $a\in W_0$
as a fixed parameter; $L[a]$ supplies the same codes from $a$.
Write $W$ for the ground. The empty-domain case is immediate.
Otherwise Cohen homogeneity
gives an invariant name for the domain.
Apply \cref{thm:cohen-invariant} to obtain an invariant bijective
enumeration. Applying \cref{lem:invariant-od} to the name for the
entire enumeration gives an $\mathrm{OD}(a)$ bijection from a finite ordinal
or $\omega$ onto the domain.  Pull back the structure along it.
The atomic diagram is an $\mathrm{OD}(a)$ real.  Every Boolean value
deciding a bit is fixed by the Cohen automorphism group, hence is zero
or one, so the diagram belongs to $W$. Satisfaction in a set-sized
first-order structure is absolute.  Every real coded by an element
of the pulled-back arithmetic model therefore belongs to $W$.
\end{proof}

\subsection{Canonical exponential cuts in bounded arithmetic}
\label{subsec:bounded-arithmetic-cuts}

We use $I\Delta_0$ in the ordinary language of arithmetic.  For each
standard $n$, let $\beta_n(a)$ assert that the $n$th binary digit of
$a$ is one, using the numeral for $2^n$:
\[
 \beta_n(a)\ \longleftrightarrow\
 \exists u\leq a\ \exists v<\overline{2^n}\,
       \bigl(a=(2u+1)\overline{2^n}+v\bigr).
\]
For a set-sized model $M\models I\Delta_0$, put
\[
 \SSy_{\rm bin}(M)=
 \bigl\{\{n<\omega:M\models\beta_n(a)\}:a\in M\bigr\}.
\]
This convention requires no total exponentiation in $M$.  For models
with exponentiation it agrees with the binary standard system used
below.  The metatheory of this subsection is ZF; standard natural
numbers and satisfaction refer to that ambient universe.

Fix the Paris $\Delta_0$ graph $E(x,y)$ for partial exponentiation
$y=2^x$.  In $I\Delta_0$ it is functional, its domain is initial,
and $E(x,y)$ implies $E(x+1,2y)$. The usual algebraic and order laws
hold wherever the values exist
\cite[Section~2, Lemma~2.1]{EnayatBounded}.
For standard $n$ define ordinary arithmetic formulas
\[
 T_0(x):=(x=x),\qquad
 T_{n+1}(x):=\exists y\,[E(x,y)\wedge T_n(y)],
\]
and define, externally,
\begin{equation}\label{eq:canonical-exponential-cut}
 C(M)=\{a\in M:M\models T_n(a)\text{ for every }n<\omega\}.
\end{equation}
The restricted structure is uniformly definable from $M$ alone,
by the satisfaction relation for set-sized structures.  Here an
initial cut is allowed to be all of $M$.

\begin{lemma}\label{lem:canonical-exponential-cut}
For every $M\models I\Delta_0$, the structure $C(M)$ is an initial
arithmetic substructure satisfying $I\Delta_0+\mathrm{Exp}$, or EFA.
Its operations and partial-exponential graph are inherited from $M$.
\end{lemma}
\begin{proof}
Every standard numeral lies in $C(M)$.  Downward closure and
monotonicity of partial exponentiation, iterated through each standard
finite number of steps, show that $C(M)$ is initial.  If $a\in C(M)$
and $E(a,b)$, functionality and $T_{n+1}(a)$ give $T_n(b)$ for every
standard $n$.  Thus exponentiation is total on $C(M)$ and has values
there.

The elementary bound
\[
 E(m,e)\ \wedge\ m\geq4\quad\Longrightarrow\quad m^2\leq e
\]
is provable in $I\Delta_0$.  Indeed, with $m,e$ fixed, all exponential
values through $m$ exist and are bounded by $e$.  Bounded induction
therefore applies to the assertion that $x^2\leq y$ whenever
$4\leq x\leq m$, $y\leq e$, and $E(x,y)$; the successor step uses
$(x+1)^2\leq2x^2$ for $x\geq3$.  For $a,b\in C(M)$, take
$m=\max(a,b,4)$ and $e=2^m\in C(M)$.  Both $a+b$ and $ab$ are at
most $e$, so initiality gives arithmetic closure, including successor.

An initial arithmetic substructure is $\Delta_0$-elementary.
Bounded induction holds in $C(M)$: a counterexample there would have
a least counterexample below it in $M$, and this element and its
predecessor would still lie in the cut.  The $\Delta_0$ graph $E$ is
absolute as well.  Hence $C(M)\models I\Delta_0+\mathrm{Exp}$.
\end{proof}

\begin{lemma}\label{lem:nonstandard-cut-standard-system}
If $D$ is a nonstandard initial arithmetic substructure of
$M\models I\Delta_0$ on which exponentiation is total with values
in $D$, then $\SSy_{\rm bin}(D)=\SSy_{\rm bin}(M)$.
\end{lemma}
\begin{proof}
Fix a nonstandard $c\in D$ and put $d=2^c\in D$.  For any $a\in M$,
division with remainder gives $a=qd+r$ with $r<d$, so $r\in D$.
For every standard $n$, the initial-domain property and $n+1$
applications of the exponential recurrence show that $2^{n+1}$
divides $d$.  Thus $a$ and $r$ have the same standard binary digits.
Bounded-formula absoluteness gives both inclusions of standard systems.
\end{proof}

\begin{theorem}\label{thm:full-system-canonical-cut}
There is a fixed recursive real $G_{\rm tow}$ such that, for every
$M\models I\Delta_0$, if $G_{\rm tow}\in\SSy_{\rm bin}(M)$, then
$C(M)$ is nonstandard and
\[
 \SSy_{\rm bin}(C(M))=\SSy_{\rm bin}(M).
\]
In particular this holds whenever $M$ has the full ambient binary
standard system.  No saturation assumption is required.
\end{theorem}
\begin{proof}
In the standard natural numbers let $e_0=1$, $e_{i+1}=2^{e_i}$, and
use the injective polynomial pairing $\pi(i,y)=(i+y)^2+i$.  Set
\[
 G_{\rm tow}=\{\pi(i,e_i):i<\omega\}.
\]
This set is recursive: for any input, test its finitely many possible
paired coordinates and compute the required finite tower.
Suppose that $a\in M$ codes this real.  Its infinite trace makes $a$
nonstandard.  Introduce the fixed bounded formulas
\begin{align*}
 B(k,a)&:\Longleftrightarrow
   \exists p\leq a\,[E(k,p)\wedge
     \exists u\leq a\ \exists v<p\ (a=(2u+1)p+v)],\\
 F(i,y,a)&:\Longleftrightarrow
   y\leq a\ \wedge\ B(\pi(i,y),a)
       \ \wedge\ \forall z<y\,\neg B(\pi(i,z),a).
\end{align*}
For standard $k$, functionality of $E$ shows that $B(k,a)$ is
equivalent to $\beta_k(a)$; if $a<2^k$, both are false.
For fixed $i$, at most one $y$ satisfies $F(i,y,a)$.
For standard $i$ it is $e_i$: the marked position is standard,
and every smaller candidate is standard and unmarked.

Let $\operatorname{Good}(n,a)$ be the following single $\Delta_0$
formula:
\begin{align*}
 &n\leq a\ \wedge\ F(0,1,a)\\
 &\quad\wedge\ \forall i\leq n\ \exists y\leq a\,
                 [F(i,y,a)\wedge y\geq i+1]\\
 &\quad\wedge\ \forall i<n\ \forall y\leq a\ \forall z\leq a\,
     [(F(i,y,a)\wedge F(i+1,z,a))\longrightarrow E(y,z)].
\end{align*}
Every standard $n$ satisfies it.  Bounded maximum in $M$ gives a
greatest $N\leq a$ satisfying it, and $N$ is nonstandard.
Put $j=\lfloor N/2\rfloor$ and let $b$ be the unique row value
with $F(j,b,a)$.  The explicit growth clause gives $b\geq j+1$,
so $b$ is nonstandard.  For every standard $k$, the rows
$j,j+1,\ldots,j+k$ exist below $N$ and satisfy successive
$E$-relations. This standard finite list of relations implies $T_k(b)$.
Therefore $b\in C(M)$.  Apply
\cref{lem:nonstandard-cut-standard-system}.
\end{proof}

The code for $G_{\rm tow}$ is used only to prove nonstandardness;
it is not a parameter in the definition of $C(M)$.  Thus this
construction preserves definability from any specified parameters
and does not increase the size of the domain.  It also takes a
definable family of such structures to a definable family of EFA
structures without selecting a member of the original family.
No saturation of $C(M)$ is asserted.

\subsection{A common category environment for Cohen extensions}
\label{subsec:common-cohen-category}

The same category argument works for ordinary and generalized Cohen
forcing. Fix a ground
$W_0\models\mathsf{ZFC}+\mathsf{GA}+(V=\mathrm{HOD})$, an infinite
regular $\kappa$ with $\kappa^{<\kappa}=\kappa$ in $W_0$, and a ground
cardinal $\Lambda>\kappa$. Let $G$ be generic and put
\begin{equation}\label{eq:common-cohen-category-setting}
 V=W_0[G],\quad G\subseteq\Add(\kappa,\Lambda)^{W_0},\quad
 \lambda=\kappa^+,\quad
 \mathcal S=\operatorname{HOD}_{H_\lambda^V}^{V}.
\end{equation}
Write $\mathcal D=\mathrm{OD}_{H_\lambda^V}^{V}$ for the corresponding
nonhereditary class.
Write $P_I^W=\Add(\kappa,I)^W$, and $P_1^W$ for one column.
The forcing is $<\kappa$-closed and $\kappa^+$-cc, the latter by
$\kappa^{<\kappa}=\kappa$ and the delta-system argument. Thus it
preserves cardinals and adds no ordinal sequences of length below
$\kappa$. These facts also hold over a one-column extension and
require no regularity of $\Lambda$. The support calculation itself
needs no stable-ground assumption.

\begin{lemma}\label{lem:generalized-support}
Let $W_0\models\mathsf{ZFC}$, let $\kappa$ be infinite regular with
$\kappa^{<\kappa}=\kappa$ in $W_0$, and let
$V=W_0[G]$ for $\Add(\kappa,\Lambda)^{W_0}$, where
$\Lambda>\kappa$ is any ground cardinal. Put $\lambda=\kappa^+$.
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item If $\delta\leq\kappa$ and $f:\delta\to\Ord$ belongs to $V$, its
range is contained in a set $B\in W_0$ with $|B|^{W_0}\leq\kappa$.
\item For every $z\in H_\lambda^V$ there is a ground set
$E\subseteq\Lambda$ of ground cardinality at most $\kappa$ such that
$z\in W_0[G\restr E]$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (i), maximal antichains deciding each $f(i)$ have size at most
$\kappa$. Their at most $\kappa$ sets of possible values give the
required ground cover.

For (ii), in $V$ code the transitive closure of $\{z\}$ by a well-founded
extensional relation on an ordinal at most $\kappa$, with a distinguished
point.  Ground pairing functions code this relation and the distinguished
point by a subset of $\kappa$.  A nice name for this subset uses at most
$\kappa$ antichains of size at most $\kappa$.  Each condition mentions fewer
than $\kappa$ columns.  Consequently the name is supported on a ground set
$E$ of at most $\kappa$ columns. The code belongs to $W_0[G\restr E]$.
Well-foundedness is downward absolute, and the transitive collapse in this
intermediate model is the same as in $V$.  Thus $z$ belongs to it as well.
\end{proof}

\begin{lemma}\label{lem:cohen-category-closure}
In \eqref{eq:common-cohen-category-setting}, $\mathcal S$ is a
transitive inner model of $\mathsf{ZF}+\mathsf{DC}_\kappa$,
closed under ambient sequences of length at most $\kappa$ of its
members, and
\[
 L(\mathcal P(\kappa)^V)\subseteq\mathcal S,\qquad
 \mathcal P(\kappa)^{\mathcal S}=\mathcal P(\kappa)^V.
\]
Every hereditary-small parameter belongs to the extension generated
by at most $\kappa$ original columns. For $\kappa=\omega$, $\mathcal S=\HODR^V$.
\end{lemma}
\begin{proof}
Apply \cref{lem:definition-inner-model,lem:definition-sequence-closure}
with $Z=H_{\kappa^+}$. The ordinal covers in
\cref{lem:generalized-support}(i) have ground enumerations with
stable ordinal codes, so the closure hypothesis holds.
Part (ii) gives the assertion about parameters. Finally, a pointed well-founded
relation on $\omega$ codes any hereditarily countable set by a real;
conversely every real is hereditarily countable. Hence the two
parameter conventions define the same class at $\kappa=\omega$.
\end{proof}

\subsubsection{Good bases and common extensions}
\label{subsec:gce-common-bases}

\begin{definition}\label{def:generalized-good-base}
With the ground and extension fixed as above, a \emph{good base}
is a subset
$h$ of $\kappa$ which is one-Cohen generic over $W_0$ and satisfies
\begin{equation}\label{eq:gi6-good}
 V=W_0[h][H]\quad\text{for some }P_\Lambda^{W_0[h]}\text{-generic }H.
\end{equation}
Let $D\subseteq\mathcal P(\kappa)^V$ be the set of good bases. Equality of universes
means that every set is the evaluation of a $W_0[h]$-name under $H$.
Define this relation in $V$ using its mantle predicate for $W_0$.
It is OD, and its hereditary-small entries put $D$ in $\mathcal S$.
In inner-model arguments,
$D$ always denotes this actual set. An original block on $\kappa$
columns recodes in $W_0$ as a good base, since removing those columns
leaves $\Lambda$ columns. Consequently any at most $\kappa$ members
of $H_\lambda^V$ can be captured in a good base by joining their codes.
\end{definition}

\begin{lemma}[The actual quotient over a real]\label{lem:cse-quotient}
Let $W_0$ be any ground model of $\mathsf{ZFC}$ and let
$V=W_0[G]$ for $\Add(\omega,\Lambda)^{W_0}$, where $\Lambda$ is
uncountable. For every
$z\in\mathbb R^V$, the actual extension $V/W_0[z]$ is an
extension by a forcing equivalent to
$\Add(\omega,\Lambda)^{W_0[z]}$.
\end{lemma}
\begin{proof}
A nice name for $z$ uses a countable set $T\subseteq\Lambda$ of
the original coordinates.  Enlarge $T$ to be countably infinite and let
$\mathbb A$ be the complete Cohen algebra on $T$.  Let
$\mathbb D\subseteq\mathbb A$ be the complete subalgebra generated
by the Boolean values of the bits of this name.  The intermediate-model
theorem identifies its generic extension with $W_0[z]$; see
\cite{Grigorieff,Jech}.  Factor first through $\mathbb D$ and then
through the quotient, using the quotient completion in the intermediate
extension.  The standard quotient factorization is also described in
\cite[556F--556G]{FremlinForcing}.

We verify the needed density bound.  Fix a countable dense subset
$D\subseteq\mathbb A^+$ in $W_0$, and let
\[
 \pi(b)=\bigwedge\{d\in\mathbb D:b\le d\}
 \quad(b\in\mathbb A)
\]
be the projection onto $\mathbb D$.  If $H$ is the induced
$\mathbb D$-generic filter, the quotient image of $b$ is positive
exactly when $\pi(b)\in H$.  Projection preserves arbitrary joins, so
\[
 \pi(b)=\bigvee\{\pi(d):d\in D,\ d\le b\}.
\]
When $\pi(b)\in H$, a ground maximal antichain refining the displayed
cover ensures that $\pi(d)\in H$ for some $d\in D$ below $b$.
Thus the positive images of $D$ are dense in the quotient and in its
completion, taken in the intermediate extension.

A complete Boolean algebra with a countable dense subset has at most
countably many atoms.  The complement of their join, if nonzero, is
atomless and has countable density; a countable dense Boolean subalgebra
there is the countable atomless Boolean algebra, so its completion is
the Cohen algebra.  The quotient is consequently a disjoint sum of
countably many atomic components and at most one Cohen component.
The untouched $\Lambda$ Cohen coordinates absorb each of these
components: one additional Cohen real can be reindexed into the tail.
A nonempty finite or countable disjoint sum of full $\Lambda$-Cohen
algebras is again that algebra, by partitioning one coordinate into a
maximal antichain of the required size and using homogeneity on its
cones.  This proves the assertion for the actual factorization of $V$.
All the forcings involved are ccc and preserve cardinals.
\end{proof}

\begin{lemma}\label{lem:gce-common-base}
Let $(h_i:i<\delta)$ be an ambient family of good bases, with
$\delta\leq\kappa$, and let $a\subseteq\kappa$.  There is an actual
original coordinate model $N=W_0[G\restr I]$, where $I\in W_0$ and
$|I|^{W_0}=\kappa$, which contains $a$ and all $h_i$, and satisfies
\[
 N=W_0[h_i][k_i]\qquad(i<\delta)
\]
for a one-Cohen generic $k_i$ over $W_0[h_i]$.
A ground recoding $h$ of $G\restr I$ is itself good and has $N=W_0[h]$.
The conclusion can capture any additional family of at most $\kappa$
members of $H_\lambda^V$.
\end{lemma}
\begin{proof}
First let $\kappa=\omega$. Capture the sequence of $h_i$ and all
additional parameters in an original countable block $I_0$, and
adjoin one fresh original coordinate $c$. Put
$N=W_0[G\restr I_0][c]$. For each $i$, the proof of
\cref{lem:cse-quotient} gives countable density for the quotient
of $W_0[G\restr I_0]$ over $W_0[h_i]$. Its product with the fresh
Cohen coordinate has countable density and is atomless, so its
completion is a Cohen algebra.
Thus $N$ is an actual one-Cohen extension of every $W_0[h_i]$.
The enlarged original block is countable and leaves $\Lambda$
columns, so its recoding is good. This proves the countable case.

Now let $\kappa>\omega$. Write $W_i=W_0[h_i]$. In the ambient
ZFC universe choose actual
presentations $V=W_i[K_i]$ by $P_\Lambda^{W_i}$.
The sequence $(h_i)$ and $a$ is coded by a subset of $\kappa$.
The support bound gives a ground set $I_0$ of size $\kappa$ capturing
this code.  The case $\delta=0$ ends here.

Recursively choose increasing $I_n\in W_0$ and, for each $i<\delta$,
increasing $J_{i,n}\in W_i$, all of size $\kappa$ in their grounds.
Put
\[
 N_n=W_0[G\restr I_n],\qquad
 N_{i,n}=W_i[K_i\restr J_{i,n}].
\]
Given $I_n$, capture a subset-of-$\kappa$ code for $G\restr I_n$
in the relative presentation over each $W_i$.  Include the previous
$J_{i,n-1}$ and pad to size $\kappa$.  This gives
$N_n\subseteq N_{i,n}$.  Recode each relative block as a subset
$k_{i,n}$ of $\kappa$.  These $\delta$ subsets have one code,
which can be captured in an original block $I_{n+1}$ containing
$I_n$.  Because $h_i\in N_0$, we obtain
\begin{equation}\label{eq:gce-sandwiches}
 N_n\subseteq N_{i,n}\subseteq N_{n+1}
 \qquad(i<\delta,\ n<\omega).
\end{equation}

For the limit step, if transitive ZFC models $W\subseteq V$
have the same ordinals and no new countable ordinal sequences,
every countable ambient sequence of members of $W$ belongs to $W$:
bound their ranks, well-order that rank segment in $W$, and code the
entries by a countable ordinal sequence.  Consequently
$(I_n)_{n<\omega}\in W_0$ and $(J_{i,n})_{n<\omega}\in W_i$.
Put $I=\bigcup_n I_n$ and $J_i=\bigcup_nJ_{i,n}$, and form
\[
 N=W_0[G\restr I],\qquad N_i=W_i[K_i\restr J_i].
\]
Their complementary coordinate sets have size $\Lambda$.
Their remaining forcing is $<\kappa$-closed, so both $N$ and $N_i$
are closed under countable ambient sequences of their own elements.

By \eqref{eq:gce-sandwiches}, $N_i$ contains each partial generic
function $\bigcup(G\restr I_n)$.  It contains their countable
sequence and its union, which reconstructs the full generic on
$I\times\kappa$.  Hence $N\subseteq N_i$.  Conversely, $N$ contains
each relative function $\bigcup(K_i\restr J_{i,n})$, its sequence,
and its union; it also contains $W_i$.  Thus $N_i\subseteq N$.
We have $N=N_i$ for every $i$.

Recode the $J_i$-generic in $W_i$ as one Cohen generic $k_i$, and
recode the original $I$-generic in $W_0$ as $h$.  The remaining $\Lambda$ columns show that $h$ is good. To include
a family of hereditary-small parameters, first combine their relation
codes into one subset of $\kappa$.
\end{proof}

\subsubsection{Good points on a ground Cohen tree}
\label{subsec:gce-tree-points}

For $\kappa=\omega$ take $\mathcal T=2^{<\omega}$ and $X=2^\omega$.
For $\kappa>\omega$, fix a ground tree isomorphism
$\kappa^{<\kappa}\cong\mathcal T$ and put $X=[\mathcal T]^V$.
In either case $X$ is the full ambient branch space.  Its basic cylinders are $[s]$, for
$s\in\mathcal T$.  A fixed ground indexing codes its nodes and
branches by subsets of $\kappa$; if the original nodes have higher
rank, retain the ground decoding map as an ordinal-definable
parameter.  In particular $X\in\mathcal S$ and every ambient branch
is present there.

The tree forcing is equivalent to $P_1$. For $\kappa=\omega$ this is immediate. For uncountable $\kappa$,
concatenate the words
$1^\alpha0$, $\alpha<\kappa$, to identify the successor-length
nodes of $\kappa^{<\kappa}$ with a dense set of binary strings.
Regularity keeps each short concatenation below length $\kappa$.
This identifies the forcing presentations and their generic extensions.

For a good $W=W_0[h]$ and positive finite $n$, let
\begin{equation}\label{eq:gce-good-points}
\begin{split}
 \Gamma_n(W)=\{\vec x\in X^n:\;&
 \vec x\text{ is }\mathcal T^n\text{-generic over }W,\ \text{and}\\
 &V=W[\vec x][H]\text{ for a }
 P_\Lambda^{W[\vec x]}\text{-generic }H\}.
\end{split}
\end{equation}
All short nodes and partial-function conditions are unchanged in
the relevant extensions.  The relation $(h,\vec x)$ in
\eqref{eq:gce-good-points} is an ambient OD set with hereditary
entries in $\mathcal S$, so it too belongs to $\mathcal S$.
Inside $\mathcal S$ we use the relation formed in $V$, so no
computation of the mantle inside $\mathcal S$ is required. Whenever forcing
relations are taken, expand the original ambient definitions;
\cref{lem:stable-ground-codes} keeps their ground predicate equal
to $W_0$ in each relevant ZFC extension.

\begin{lemma}\label{lem:gce-good-laws}
The good points have the following properties.
\begin{enumerate}[label=(\roman*),leftmargin=*,beginpenalty=10000]
\item If $W'=W[k]$ is a one-Cohen extension and both are good bases,
then $\Gamma_n(W')\subseteq\Gamma_n(W)$.
\item For positive finite $r,s$,
\[
 (\vec x,\vec y)\in\Gamma_{r+s}(W)
 \quad\Longleftrightarrow\quad
 \vec x\in\Gamma_r(W)\text{ and }
 \vec y\in\Gamma_s(W[\vec x]).
\]
The model $W[\vec x]$ on the right is a good base after a ground
recoding as one Cohen extension of $W_0$.
\item Every nonempty basic open subset of $X^n$ meets $\Gamma_n(W)$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (i), the pair $(k,\vec x)$ is product generic over $W$.
Its ground forcing factors are unchanged, so we may reverse their
order. Over $W[\vec x]$, adding $k$ and then forcing with the full tail
is equivalent, by reindexing, to forcing with a full
$\Lambda$-column tail.  This gives (i).
For (ii), use the product forcing theorem.  In the forward
direction the finite $\vec y$-factor can be absorbed into the
remaining full tail.  In the reverse direction the asserted joint
genericity and actual tail are precisely
\eqref{eq:gce-good-points}.  The forcing adding $h$ and $\vec x$
over $W_0$ is equivalent to one Cohen column.

For (iii), factor an actual full-tail presentation over $W$ into
$\mathcal T^n$ and a full tail.  A tree cone is isomorphic in the
ground to the entire tree, so the distinguished generic can be
placed in any prescribed product of cylinders while preserving
the intermediate extension and its remaining tail.
\end{proof}

\begin{definition}\label{def:gce-ideals}
For each positive finite $n$, define in $\mathcal S$
\[
 \mathcal I_n=\{E\in\mathcal P(X^n)^{\mathcal S}:
       E\cap\Gamma_n(W_0[h])=\varnothing
       \text{ for some }h\in D\}.
\]
We abbreviate $\mathcal I_1$ to $\mathcal I$.
\end{definition}

\begin{proposition}\label{prop:gce-ideals}
The $\mathcal I_n$ are proper ideals in $\mathcal S$, closed under
unions of at most $\kappa$ members, even when the family is initially
given in $V$.  They contain every $\mathcal S$-set covered by at most
$\kappa$ nowhere dense sets and contain no nonempty open set.
They are invariant under
permutations of the finite coordinates and under ground tree
automorphisms.
\end{proposition}
\begin{proof}
For a family $(E_i:i<\delta)$, $\delta\leq\kappa$, choose witnessing
good bases in $V$.  A common base from \cref{lem:gce-common-base},
together with \cref{lem:gce-good-laws}(i), has good points avoiding
every $E_i$.  The union belongs to $\mathcal S$ by
\cref{lem:cohen-category-closure}.  This proves closure under these unions. Downward closure is immediate.  Part (iii) of
\cref{lem:gce-good-laws} proves positivity of nonempty basic open
sets, hence properness and positivity of every nonempty open set.

A closed set is coded by a subset of the ground basis, which has
size $\kappa$.  Capture the codes of $\kappa$ closed nowhere dense
covers in one good base.  Their nowhere denseness is absolute:
every basic condition has a stronger one avoiding the coded set,
and all these conditions are unchanged.  Generic tuples over that
base avoid the covers, proving the assertion for nowhere dense sets.
Ground tree automorphisms and coordinate permutations preserve
genericity and the generated intermediate extension.  They
therefore preserve the good points over each ground containing
their codes, and hence preserve the ideals.
\end{proof}

The same argument applies to any tree automorphism coded by a subset
of $\kappa$: first capture its code in a common good base. In particular
it applies to the ground XOR translations used below.
Also $X^n\setminus\Gamma_n(W)\in\mathcal I_n$.
Consequently, for every nonempty basic open $U$,
\begin{equation}\label{eq:gce-positive-good}
 U\cap\Gamma_n(W)\notin\mathcal I_n.
\end{equation}
Otherwise, taking the union with that complement would put $U$ in the ideal.

\subsubsection{Regular-open traces and the product theorem}
\label{subsec:gce-traces}

\begin{theorem}\label{thm:gce-traces}
For every $E\subseteq X^n$ in $\mathcal S$ there are a good base $W$
and a regular open set $O$, coded in $W$, such that
\[
 E\cap\Gamma_n(W)=O\cap\Gamma_n(W).
\]
In particular $E\mathbin\triangle O\in\mathcal I_n$.
\end{theorem}
\begin{proof}
Capture the hereditary-small parameter defining $E$, and retain its
ordinal parameters, in a good $W$.  Expand every ambient class
definition before taking forcing relations.  The actual full tail
over $W$ is homogeneous.  Since the definition is unique in $V$,
its uniqueness and the relevant type of its value are forced by
the greatest condition.

Factor the tail as $\mathcal T^n$ followed by a full tail.
In the first forcing, take the Boolean value of the assertion that
the top of the remaining tail forces the distinguished tuple to
belong to the defined set.  Represent it by the regular open set
$O$ in the completion of $\mathcal T^n$.
Its code is a subset of a basis of size $\kappa$.
Its interpretation in $V$ is still regular open, because closure
and interior of this coded open set are computed by compatibility
and refinement among the unchanged basic conditions.

For each $\vec x\in\Gamma_n(W)$, the ambient universe $V$ is a full
tail extension of $W[\vec x]$. Homogeneity and the two forcing theorems
therefore identify membership in the actual $E$ with membership
of $\vec x$ in $O$.  The set where the two memberships differ misses $\Gamma_n(W)$
and therefore belongs to the ideal.
\end{proof}

\begin{corollary}\label{cor:gce-local-decision}
If $E\subseteq X^n$ belongs to $\mathcal S$ and $E\cap U$ is positive
on a nonempty basic open set $U$, there is a nonempty basic
$U'\subseteq U$ such that $U'\setminus E\in\mathcal I_n$.
Every $\mathcal S$-function with range of ambient size at most $\kappa$
is constant modulo $\mathcal I_n$ on a smaller basic open subset
of any given nonempty basic open domain.
\end{corollary}
\begin{proof}
A regular-open trace of $E$ must meet $U$, since otherwise
$E\cap U$ would be small.  Choose $U'$ inside their intersection.
For a function, its at most $\kappa$ fibers cannot all be small
on $U$, by \cref{prop:gce-ideals}; apply the first assertion to
a positive fiber.  An ambient enumeration of its small range
belongs to $\mathcal S$ by \cref{lem:cohen-category-closure}.
\end{proof}

\begin{theorem}[Finite-product category theorem]\label{thm:gce-ku}
For $E\subseteq X^{r+s}$ in $\mathcal S$, where $r,s$ are positive finite,
\[
 E\in\mathcal I_{r+s}
 \quad\Longleftrightarrow\quad
 \{\vec x\in X^r:E_{\vec x}\notin\mathcal I_s\}\in\mathcal I_r.
\]
\end{theorem}
\begin{proof}
If $E$ misses $\Gamma_{r+s}(W)$, then for every
$\vec x\in\Gamma_r(W)$ its section misses $\Gamma_s(W[\vec x])$,
by the product law.  This is a good base, so the section is small.
The set of positive sections consequently misses $\Gamma_r(W)$.

Conversely, a positive $E$ has a nonempty regular-open trace.
Choose a nonempty basic rectangle $U\times U'$ inside this trace.
For $\vec x\in U\cap\Gamma_r(W)$, the product law gives
\[
 U'\cap\Gamma_s(W[\vec x])\subseteq E_{\vec x}.
\]
The left side is positive by \eqref{eq:gce-positive-good}.
Thus the set of positive sections contains
$U\cap\Gamma_r(W)$, itself positive.  This proves the converse
by contraposition.  All sets of sections used here belong to
$\mathcal S$ by separation from the ideals already constructed there.
\end{proof}

\subsubsection{Uniformization with arbitrary codomain}
\label{subsec:gce-uniformization}

\begin{theorem}\label{thm:gce-uniformization}
Suppose $Y,R\in\mathcal S$, $R\subseteq X\times Y$, and every section
of $R$ is nonempty.  There is a partial function $F\in\mathcal S$
uniformizing $R$ such that
$X\setminus\operatorname{dom}(F)\in\mathcal I$.
More precisely, its domain contains $\Gamma_1(W')$ for some good base $W'$.
\end{theorem}
\begin{proof}
Capture the definition parameters of $Y,R$ in a
good $W=W_0[h]$, retaining their ordinal parameters. Expand ambient
class definitions before forcing. By homogeneity, the greatest condition forces their membership
in $\mathcal S$ and the nonemptiness of all sections.

Present the tail as $\mathcal T\times P_\Lambda^W$.
The evaluation \eqref{eq:definition-evaluation}, after coding its
hereditary-small parameter by a subset of $\kappa$, has the form
$\operatorname{Val}:\mathcal P(\kappa)\times\Ord\rightharpoonup
\mathcal D$. The maximum principle gives names $\dot r\subseteq\kappa$
and $\dot\xi\in\Ord$ such that the top forces
\begin{equation}\label{eq:gce-witness-name}
 \operatorname{Val}(\dot r,\dot\xi)\in Y,\qquad
 R\bigl(\dot x,\operatorname{Val}(\dot r,\dot\xi)\bigr).
\end{equation}
The $\kappa^+$-cc bounds the possible values of $\dot\xi$ by a ground
set of size at most $\kappa$. Enumerate it by $b:\kappa\to B$ and
replace $\dot\xi$ by $b(\dot j)$, taking the least suitable index.
Nice names for $\dot r,\dot j$ use at most $\kappa$ tail columns.
Pass to the extension by their actual generic $a$, keeping the
distinguished tree factor in the remaining forcing. The new base
$W'=W[a]$ is good, since a nonempty captured block recodes as one Cohen
column and $\Lambda$ columns remain. This also holds for singular
$\Lambda$.

Over $W'$, the retained names are $\mathcal T$-names. Indexing the
unchanged tree conditions by $\kappa$ makes their nice-name reading
tables hereditary of size at most $\kappa$, hence members of
$H_\lambda^V$. For trees with higher-rank nodes use the fixed ground
decoding map. Evaluate the tables at the initial-segment filter of
$x$, leaving invalid readings undefined, to obtain $r(x),j(x)$.
These readings are correct at every $W'$-generic branch.
The ordinal table $b\in W_0[h]$ is definable from $h$ and ordinals
by \cref{lem:stable-ground-codes}. Consequently
\[
 F(x)=\operatorname{Val}^V(r(x),b(j(x)))
\]
defines an $\mathrm{OD}_{H_\lambda}^V$ partial function when its value
is retained precisely if it exists, belongs to the actual $Y$, and
witnesses the actual $R$. Its arguments and values are hereditary
members of that class, so $F\in\mathcal S$.

The Boolean value of \eqref{eq:gce-witness-name} remains one after
passing to the new base. For $x\in\Gamma_1(W')$, the first forcing theorem says
that the residual full tail forces the retained reading to witness
$R$. The ambient universe $V$ is such a tail extension of $W'[x]$.
Thus $x\in\operatorname{dom}(F)$, proving the claim.
\end{proof}


\subsection{The ordinary Cohen case}\label{subsec:cse-environment}

Specialize the preceding construction to
$V=W_0[G]$, $G\subseteq\Add(\omega,\omega_1)^{W_0}$, and write
$S=\HODR^V$ and $C=2^\omega$. Taking $W_0=L$ gives the stated
version below. Addition on $C$ is coordinatewise modulo two.
For $M=W_0[z]$, let $\operatorname{NC}_n(M)$ be the tuples in
$C^n$ that are not jointly Cohen generic over $M$; equivalently,
they lie in a meager Borel set coded in $M$. Define in $S$
\begin{equation}\label{eq:cse-ideal}
 \mathcal I_n=\{E\in\mathcal P(C^n)^S:
     E\subseteq\operatorname{NC}_n(W_0[z])\text{ for some real }z\},
 \qquad\mathcal I=\mathcal I_1.
\end{equation}
The actual non-genericity relation is formed in $V$ using the stable
ground predicate and belongs to $S$; no internal computation of the
mantle is intended.

\begin{theorem}[The Cohen category environment]
\label{thm:cohen-category-environment}
In the stated $L$-extension, the following hold, with (ii)--(v)
interpreted in $S$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item $S$ is a transitive model of $\mathrm{ZF}+\mathrm{DC}$ and contains
all the reals of $V$.  In fact, it is closed under countable $V$-sequences
of its elements.
\item For each positive integer $k$, $\mathcal I_k$ is a proper
$\sigma$-ideal containing every ordinary meager set in $S$ and containing
no nonmeager Borel set.  It is invariant under translations of $C^k$ and
permutations of the coordinates.
\item Every $X\subseteq C^k$ in $S$ differs from a Borel set by a member
of $\mathcal I_k$.
\item For positive integers $k,l$ and $E\subseteq C^{k+l}$ in $S$,
\begin{equation}\label{eq:cse-ku}
 E\in\mathcal I_{k+l}
 \quad\Longleftrightarrow\quad
 \{x\in C^k:E_x\notin\mathcal I_l\}\in\mathcal I_k.
\end{equation}
In particular, if every section is in $\mathcal I_l$, then
$E\in\mathcal I_{k+l}$.
\item If $Y,R\in S$, $R\subseteq C\times Y$, and every section of
$R$ is nonempty, there is a function $F\in S$ uniformizing $R$ on a
domain $D\subseteq C$ with $C\setminus D\in\mathcal I$.
\end{enumerate}
\end{theorem}

\begin{proof}
For a good base $W_0[h]$, its good points are exactly its actual
Cohen-generic tuples in $V$: apply \cref{lem:cse-quotient} to the
full tail over that base and the real coding the tuple. Moreover,
any real $z$ is captured by an original countable block, whose
recoding $h$ is good. Thus
$\operatorname{NC}_n(W_0[z])\subseteq\operatorname{NC}_n(W_0[h])$.
Conversely, every good $h$ is a real. Consequently \eqref{eq:cse-ideal}
is precisely the ideal of \cref{def:gce-ideals}.

Now (i) is \cref{lem:cohen-category-closure};
(ii)--(v) follow from
\cref{prop:gce-ideals,thm:gce-traces,thm:gce-ku,thm:gce-uniformization}.
A nonmeager Borel set cannot lie in $\mathcal I_n$: modulo a
meager Borel set it contains a nonempty open set, and the ideals
contain all meager sets and no nonempty open set. Translation by
any real tuple is a coded homeomorphism preserving genericity over
a base capturing that tuple, so the symmetry assertion applies.
\end{proof}

We will also use the following trace description. If $E\subseteq C^n$
is defined from $z$ and ordinals, then some regular open $B$ coded
in $W_0[z]$ satisfies
\begin{equation}\label{eq:cse-all-generic-trace}
 \vec x\text{ Cohen generic over }W_0[z]
 \quad\Longrightarrow\quad(\vec x\in E\ \longleftrightarrow\ \vec x\in B).
\end{equation}
Indeed, the proof of \cref{thm:gce-traces} only needs a full
homogeneous tail over the base, which \cref{lem:cse-quotient}
supplies over $W_0[z]$ and over $W_0[z,\vec x]$. In particular
$E\mathbin\triangle B\subseteq\operatorname{NC}_n(W_0[z])$.

\begin{corollary}[Local decisions]\label{cor:cse-local-decision}
Let $A\subseteq C$ belong to $S$, and let $U$ be a nonempty basic
cylinder.  If $A\cap U\notin\mathcal I$, there is a nonempty basic
cylinder $W\subseteq U$ such that $W\setminus A\in\mathcal I$.
Consequently every function in $S$ with standard finite range is
constant modulo $\mathcal I$ on some subcylinder of any prescribed
nonempty basic cylinder.
\end{corollary}
\begin{proof}
Apply \cref{cor:gce-local-decision} at $\kappa=\omega$.
\end{proof}

\subsection{The category Scott obstruction}\label{subsec:cso-obstruction}

Work in $S$ with the notation of
\cref{thm:cohen-category-environment}.
A statement holds \emph{on $U$ modulo $\mathcal I$}
when its exceptions in $U$ belong to $\mathcal I$; a set is
\emph{conull} when its complement belongs to $\mathcal I$.

We use $\mathrm{EFA}=Q+I\Delta_0+\mathrm{Exp}$ in its conservative
definitional expansion by $x\mapsto2^x$, with induction for bounded
formulas in the exponential language; see
\cite[Section~1.1]{Beklemishev}. Binary digits are boundedly definable,
for example by
\[
 \operatorname{Bit}(a,m)\quad\Longleftrightarrow\quad
 \exists u\le a\;\exists v<2^m\;
       a=(2u+1)2^m+v.
\]
For EFA, $\SSy(\mathcal M)$ means the family of binary-coded traces
$\{n<\omega:\mathcal M\models\operatorname{Bit}(a,\bar n)\}$,
as $a$ ranges over $M$. No equivalence with traces of arbitrary
unbounded formulas is required.

\begin{theorem}[The Cohen Scott obstruction]\label{thm:cohen-scott-obstruction}
There is no model $\mathcal M\in S$ of $I\Delta_0$ such that
\[
 \SSy(\mathcal M)=\mathcal P(\omega)^S.
\]
\end{theorem}

By \cref{thm:full-system-canonical-cut}, any such model has a
canonically definable EFA initial substructure with the same binary
standard system. It suffices to exclude that substructure. We therefore
work with EFA models throughout the proof below; the metatheory is ZF.
The proof uses only the preceding category environment, so we may
work over its fixed stable ground $W_0$.
The arithmetic strategy follows \cite{GlazerFullStandard}.
Local uniformization and Kuratowski--Ulam turn pointwise arithmetic
overspill into a bound on a nonempty open set.  Periodic translations
then give a countable coinitial sequence in the nonstandard part.
A repeated schedule of basic open sets produces infinitely many
arithmetic blocks, from which one formula defines the standard cut.

\subsubsection{Local overspill}

A basic cylinder always means a nonempty set of the form
$[s]=\{x\in2^\omega:s\subseteq x\}$, where $s\in2^{<\omega}$.
We use \cref{cor:cse-local-decision}: a positive subset of a cylinder
is conull on a smaller cylinder, and a function with standard finite
range is constant modulo $\mathcal I$ on some smaller cylinder.

\begin{lemma}[Local overspill]\label{lem:cso-local-overspill}
Let $\mathcal M\in S$ be a nonstandard model of EFA, and identify
$\omega$ with its standard copy in $\mathcal M$.
Suppose $P\subseteq C\times M$ belongs to $S$ and, outside a set
in $\mathcal I$, every $x$ admits a nonstandard $h\in M$ such that
\[
 \forall N<h\;P(x,N).
\]
For every basic cylinder $U$ there are a nonstandard $H\in M$
and a basic cylinder $W\subseteq U$ such that
\begin{equation}
 \forall N<H\;P(x,N)
 \quad\text{holds on $W$ modulo $\mathcal I$}.
 \label{eq:cso-local-bound}
\end{equation}
The same assertion holds for a space $X$ with a basis of positive
open sets and ideals $\mathcal I,\mathcal I_2$, provided conull
uniformization, local decision, Kuratowski--Ulam, and invariance under
transposition hold and nonempty
basic rectangles are positive.  One may replace $M\setminus\omega$
by any final segment $A$ of positive arithmetic elements closed under
$h\mapsto\lfloor h/2\rfloor$.
\end{lemma}

\begin{proof}
Conull uniformization gives a function $h:C\to M\setminus\omega$
whose bound works outside a set in $\mathcal I$. To apply
uniformization to a relation with all sections nonempty, use a fixed
nonstandard default value where no bound exists; use the same default
off the resulting conull domain.

For nonstandard $a\in M$, put $E_a=\{x:h(x)>a\}$. Suppose that
$E_a\cap U\in\mathcal I$ for every such $a$, and consider
\[
 Q=\{(x,y)\in U^2:h(y)>\lfloor h(x)/2\rfloor\}.
\]
For each $x\in U$, the element $\lfloor h(x)/2\rfloor$ is
nonstandard. Thus every vertical section of $Q$ belongs to
$\mathcal I$, and Kuratowski--Ulam gives $Q\in\mathcal I_2$.
Coordinate-permutation invariance gives
$Q^{\mathsf T}\in\mathcal I_2$ as well. But
$Q\cup Q^{\mathsf T}=U^2$: of two positive integers in $\mathcal M$,
at least one exceeds half the other. This contradicts the fact that
the nonempty open rectangle $U^2$ does not belong to $\mathcal I_2$.

Consequently $E_a\cap U$ is positive for some nonstandard $a$.
Local decision gives a basic $W\subseteq U$ on which $h(x)>a$
modulo $\mathcal I$. Discard also the exception to the chosen
bound property, and take $H=a$.
The proof uses only the stated properties and therefore proves the
general form as well. If a suitable function $h$ is already given, no
uniformization assumption is needed.
\end{proof}

To prove \cref{thm:cohen-scott-obstruction}, suppose that
$\mathcal M\in S$ is an EFA model with full standard system.
It is necessarily nonstandard. Uniformize the coding relation to
obtain a conull $X\subseteq C$ and a function $F:X\to M$ in $S$
such that $F(x)$ codes $x$. Extend $F$ to $C$ by a fixed default
value. Write $\operatorname{Bit}(a,m)$ for the assertion that
the $m$th binary digit of $a$ is one, and put
\[
 \operatorname{Ext}(x)(m)=1
 \quad\Longleftrightarrow\quad
 \mathcal M\models\operatorname{Bit}(F(x),m).
\]
For every $x\in X$, all its standard bits are correct simultaneously:
\begin{equation}
 \operatorname{Ext}(x)(n)=x(n)\qquad(n<\omega).
 \label{eq:cso-standard-bits}
\end{equation}
Here and below arithmetic positions such as $n$ denote their images
in $\mathcal M$.

For finite binary words $s,t$, with $t$ nonempty,
$s{}^\frown t^\omega$ is the ordinary eventually periodic real.
The notation $s{}^\frown t^\infty$ denotes the corresponding
arithmetically defined pattern on all positions of $\mathcal M$;
it is not an assertion that an integer codes an infinite periodic
string.

\begin{lemma}[Local periodic overspill]\label{lem:cso-periodic}
For every such $s,t$ and every basic cylinder $U$, there are a
nonstandard $H\in M$ and a basic $W\subseteq U$ such that
\begin{equation}
 \operatorname{Ext}(x+s{}^\frown t^\omega)(m)
 =
 \operatorname{Ext}(x)(m)\mathbin\oplus
        (s{}^\frown t^\infty)(m)
 \qquad(m<H)
 \label{eq:cso-periodic}
\end{equation}
holds on $W$ modulo $\mathcal I$.
Addition of reals is Cantor-group addition, and $\oplus$ on the
right is addition of bits modulo two.
\end{lemma}

\begin{proof}
Discard the two coding exceptions
$C\setminus X$ and its translate by $s{}^\frown t^\omega$.
They belong to $\mathcal I$ by translation invariance.
For each remaining $x$, the displayed identity holds at every
standard position by \cref{eq:cso-standard-bits}. For this fixed $x$,
the identity is a bounded formula of the exponential language, using
only the two integer codes and the finite periodic pattern as
parameters. Division by the positive standard length of $t$ defines
the periodic part by a bounded formula. Choose any nonstandard
cutoff. Either the identity holds below it, or bounded minimization
gives its least failure below that cutoff. This least failure is
nonstandard, and the identity holds below it. Thus bounded induction
alone gives the required pointwise nonstandard bound.
Apply \cref{lem:cso-local-overspill} to make one bound work on a
subcylinder modulo $\mathcal I$.
\end{proof}

\subsubsection{A countable coinitial sequence}

Let
\[
 a_n=0{}^\frown1^n{}^\frown0^\omega,\qquad
 a_\infty=0{}^\frown1^\omega.
\]
For $n<\omega$, define $u_n(x)$ to be the least $m>n+1$
such that
$\operatorname{Ext}(x)(m)\ne\operatorname{Ext}(x+a_n)(m)$,
if such an $m$ exists, and put $u_n(x)=\infty$ otherwise.
The least witness is obtained by bounded minimization: both bits
vanish above $F(x)+F(x+a_n)$, so it suffices to search below
$F(x)+F(x+a_n)+1$.
The symbol $\infty$ is a formal value larger than every element of
$M$. For each basic cylinder $U$, set
\begin{equation}
 I_{n,U}=
 \bigl\{b\in M\setminus\omega:
          \{x\in U:u_n(x)\ge b\}\in\mathcal I\bigr\}.
 \label{eq:cso-final-segments}
\end{equation}
These are final segments of the nonstandard part.

\begin{lemma}\label{lem:cso-coinitiality}
Every $I_{n,U}$ is proper, and
\begin{equation}
 \bigcup_{n<\omega,\ U\text{ basic}}I_{n,U}=M\setminus\omega.
 \label{eq:cso-cover}
\end{equation}
Consequently $S$ contains a strictly decreasing coinitial sequence
$N_0>N_1>\cdots$ in $M\setminus\omega$.
\end{lemma}

\begin{proof}
Apply \cref{lem:cso-periodic} for $a_n$ inside $U$.
On a basic $W\subseteq U$, modulo $\mathcal I$, the two strings
agree throughout $(n+1,H)$ for some nonstandard $H$.
Thus $u_n\ge H$ there. Since $W$ is positive,
$\{x\in U:u_n(x)\ge H\}\notin\mathcal I$.
This proves $H\notin I_{n,U}$ and hence properness.

Suppose \cref{eq:cso-cover} fails, and take a nonstandard $d$
outside the union. Apply \cref{lem:cso-periodic} for $a_\infty$
to obtain a basic $U$ and nonstandard $H$ on which
\[
 \operatorname{Ext}(x+a_\infty)(m)\ne\operatorname{Ext}(x)(m)
 \qquad(0<m<H)
\]
modulo $\mathcal I$. Choose a nonstandard $m$ with
$m+1<\min(d,H)$.
By local decision, refine $U$ to a basic cylinder $W$ on which
$f(x)=\operatorname{Ext}(x+a_\infty)(m)$ is constant modulo
$\mathcal I$.

The translations $t_n=a_n+a_\infty$ tend to zero. For every
sufficiently large standard $n$, $t_n$ vanishes on the prefix
defining $W$, so $W+t_n=W$. Outside the constancy exception and
its translate, both of which belong to $\mathcal I$, we have
\[
 \operatorname{Ext}(x+a_n)(m)
 =f(x+t_n)=f(x)
 =\operatorname{Ext}(x+a_\infty)(m)
 \qquad(x\in W).
\]
This bit differs from $\operatorname{Ext}(x)(m)$ modulo
$\mathcal I$ on $W$. Since $m$ is nonstandard, $m>n+1$,
and therefore $u_n(x)\le m$ on $W$ modulo $\mathcal I$.
It follows that $m+1\in I_{n,W}$, whence
$d\in I_{n,W}$ by finality, a contradiction.

There are only countably many pairs $(n,U)$. Enumerate their
proper final segments as $\langle J_i:i<\omega\rangle$, and use
countable choice in $S$ to choose $c_i\in(M\setminus\omega)
\setminus J_i$. The sequence $\langle c_i:i<\omega\rangle$ is
coinitial: if $d\in J_i$, finality and $c_i\notin J_i$ give
$c_i<d$. Define
\begin{equation}
 N_0=\lfloor c_0/2\rfloor,\qquad
 N_{i+1}=\left\lfloor\frac{\min(N_i,c_{i+1})}{2}\right\rfloor.
 \label{eq:cso-coinitial-sequence}
\end{equation}
All these elements are nonstandard and the sequence strictly
decreases. It is coinitial because $N_i\le c_i$. The use of countable choice
is justified by $\mathsf{DC}$ in $S$.
\end{proof}

\subsubsection{Infinitely many successful blocks}

The proof below requires only a real-coded nonincreasing coinitial
sequence. We use a strictly decreasing one for convenience.
Fix the sequence from \cref{lem:cso-coinitiality} and a sequence
$\langle U_n:n<\omega\rangle$ of basic cylinders in which every
basic cylinder occurs infinitely often. Put
\begin{equation}
 w_n=0{}^\frown1^n,\qquad p_n=n+1,\qquad L_n=2p_n.
 \label{eq:cso-block-lengths}
\end{equation}
The internal periodic pattern $w_n^\infty$ has exactly two zeros
in each interval of length $L_n$, separated by $p_n$.

\begin{lemma}\label{lem:cso-blocks}
There are a real $s_\omega\in S$, strictly increasing indices
$r_n\ge n$, and $x\in C$ with $x,x+s_\omega\in X$ such that,
writing $M_n=N_{r_n}$, the following hold.
The successive gaps in the ordinary zero set of $s_\omega$ are
nondecreasing and unbounded. For infinitely many standard $n$,
the strings $\operatorname{Ext}(x)$ and
$\operatorname{Ext}(x+s_\omega)$ agree at exactly two positions
in $[M_n,M_n+L_n)$, separated by $n+1$.
\end{lemma}

\begin{proof}
Construct finite words $s_n$, beginning with $s_0=\varnothing$.
Given $s_n$, let $z_n=s_n{}^\frown w_n^\omega$.
Local periodic overspill inside $U_n$ gives a basic
$W\subseteq U_n$ and a nonstandard $H_n$ on which the
comparison \cref{eq:cso-periodic} for $z_n$ holds below $H_n$
modulo $\mathcal I$. By coinitiality choose $r_n\ge n$,
larger than all previous indices, such that
\begin{equation}
 M_n=N_{r_n},\qquad M_n+L_n<H_n.
 \label{eq:cso-block-position}
\end{equation}
This is possible because $H_n-L_n$ is nonstandard.

The map
\[
 G_n(y)=
 \langle\operatorname{Ext}(y+z_n)(M_n+j):j<L_n\rangle
\]
has standard finite range. Refine $W$ to a basic cylinder
$V_n$ on which $G_n$ has one constant value modulo
$\mathcal I$. Let $\ell_n$ be the length of the prefix
defining $V_n$. Choose an integer $q_n\ge1$ with
$|s_n|+q_np_n\ge\ell_n$, and set
\begin{equation}
 s_{n+1}=s_n{}^\frown w_n^{q_n},\qquad
 s_\omega=\bigcup_{n<\omega}s_n.
 \label{eq:cso-translation}
\end{equation}
Dependent choice carries out this recursion in $S$. Since
$q_n\ge1$, the union is a real, and
\begin{equation}
 s_\omega\restr\ell_n=z_n\restr\ell_n.
 \label{eq:cso-matching-prefix}
\end{equation}

Put $t_n=s_\omega+z_n$. By
\cref{eq:cso-matching-prefix}, $t_n$ fixes $V_n$ under
translation. Let $E_n\subseteq V_n$ be the exception to the
chosen constant value of $G_n$. Then $E_n\in\mathcal I$,
and for $y\in V_n\setminus(E_n\cup(E_n+t_n))$,
\begin{equation}
 \left\langle
   \operatorname{Ext}(y+s_\omega)(M_n+j):j<L_n
 \right\rangle
 =G_n(y+t_n)=G_n(y).
 \label{eq:cso-local-comparison}
\end{equation}
Combining this equality with the periodic-overspill comparison
on $W$, we obtain, on $V_n$ modulo $\mathcal I$,
\begin{equation}
 \operatorname{Ext}(y+s_\omega)(M_n+j)
 =
 \operatorname{Ext}(y)(M_n+j)\mathbin\oplus
 (s_n{}^\frown w_n^\infty)(M_n+j)
 \quad(j<L_n).
 \label{eq:cso-good-block}
\end{equation}
The position $M_n$ is nonstandard and hence lies beyond the
standard finite prefix $s_n$. Thus the agreement positions
in this interval are exactly the two zeros of the periodic
pattern, at distance $p_n=n+1$.

For every standard $k$, the set
\[
 D_k=\bigcup_{n\ge k}V_n
\]
is dense open. Indeed, every basic cylinder occurs as $U_n$
for some $n\ge k$, and contains that stage's nonempty $V_n$.
Consequently $\bigcap_kD_k$ is ordinary comeager.
For each $n$, the failure set of \cref{eq:cso-good-block}
inside $V_n$ belongs to $\mathcal I$: it is contained in
the two translated constancy exceptions and the periodic
comparison exception. Remove their countable union, together
with the two coding exceptions for $y$ and $y+s_\omega$.
The complement of $\bigcap_kD_k$ is meager and also belongs
to $\mathcal I$. Since $\mathcal I$ is proper and closed under countable unions,
there is some $x$ in $\bigcap_kD_k$ outside all these exceptional sets.

This $x$ belongs to infinitely many $V_n$, so infinitely many
blocks satisfy the required comparison.
Finally, \cref{eq:cso-translation} shows that
the gaps between successive zeros of $s_\omega$ are
$1,2,3,\ldots$, each repeated a positive finite number of
times. Within stage $n$ the gaps are $n+1$, and the gap
to the first zero of the next stage is also $n+1$.
\end{proof}

\subsubsection{Defining the standard cut}

\begin{proof}[Proof of \cref{thm:cohen-scott-obstruction}]
Assume that $\mathcal M\in S$ has full standard system,
and carry out the preceding constructions.
Take $s_\omega,x$ from \cref{lem:cso-blocks}, and let
$p=F(x)$ and $q=F(x+s_\omega)$.
Define the arithmetic set
\begin{equation}
 B=\{b\in M:
     \mathcal M\models
       \operatorname{Bit}(p,b)\leftrightarrow
       \operatorname{Bit}(q,b)\}.
 \label{eq:cso-agreement}
\end{equation}
Define the bounded formula
\[
 \operatorname{Next}_B(u,v)\quad\Longleftrightarrow\quad
 u<v\ \wedge\ B(u)\ \wedge\ B(v)\ \wedge
 \forall z<v\,(u<z\to\neg B(z)).
\]
It says that $u,v$ are consecutive agreement positions.
Consider
\begin{equation}
 \begin{split}
 C(d)\quad\Longleftrightarrow\quad
 \forall u,v,r,s<d\;\bigl[&\operatorname{Next}_B(u,v)
       \wedge\operatorname{Next}_B(r,s)\\
       &{}\wedge v\le r\ \longrightarrow\ v+r\le s+u\bigr].
 \end{split}
 \label{eq:cso-cut}
\end{equation}
This compares every earlier consecutive gap with every later one,
using addition instead of subtraction. After expanding $B$ and $\operatorname{Next}_B$, we obtain a bounded
formula with parameters
$p,q$. At standard positions, \cref{eq:cso-standard-bits} identifies
$B$ with the ordinary zero set of $s_\omega$. Its gaps are
nondecreasing, so $C(d)$ holds for every standard $d$.

Let $d$ be nonstandard. Since $M_n=N_{r_n}$ is decreasing and
coinitial, eventually $M_n<\lfloor d/2\rfloor$. Every standard
$L_n$ is also less than $\lfloor d/2\rfloor$. Choose a sufficiently
late successful block; then $M_n+L_n<d$. Its two agreement
positions $r<s$ are nonstandard, consecutive in $B$, and satisfy
$s-r=n+1$. The ordinary zero gaps of $s_\omega$ are unbounded,
so choose standard consecutive agreement positions $u<v$ with
$v-u>n+1$. All four positions lie below $d$, and $v<r$.
They violate the inequality in \cref{eq:cso-cut}. Hence $C(d)$
fails for every nonstandard $d$.

We have proved externally that $C^{\mathcal M}=\omega$.
In particular, $\mathcal M\models C(0)$ and
$\mathcal M\models\forall d\,(C(d)\to C(d+1))$.
Bounded induction for $C$, with parameters $p,q$, yields
$\mathcal M\models\forall d\,C(d)$,
contradicting nonstandardness.
\end{proof}


\subsection{A Borel obstruction for a finite fragment}
\label{subsec:borel-finite-fragment}

A \emph{coded Borel quotient presentation} consists of a Borel
domain $P\subseteq\omega^\omega$, a Borel equivalence relation
$\mathord\approx$ on $P$, and Borel operations and relations on
representatives which respect $\mathord\approx$ and induce a
first-order structure on $P/{\mathord\approx}$. Constants are given
by distinguished representatives. Ordinary Borel presentations are
the special case in which $\mathord\approx$ is equality. All Borel
sets and maps in this subsection are given by real codes.

The proof uses the following facts about real codes. We state their
metatheoretic strength because it is part of the theorem.

\begin{lemma}[Coded category toolkit]
\label{lem:borel-coded-category-toolkit}
The following uniform statements are provable in
$\Pi^1_1\text{-}\mathsf{CA}_0$.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item If a coded Borel relation $R\subseteq2^\omega\times
       \omega^\omega$ has nonempty sections, there are codes for a
       comeager Borel set $X\subseteq2^\omega$ and a Borel map on $X$
       which uniformizes $R$.
 \item From a sequence of coanalytic codes one can uniformly obtain
       Baire-property data. In particular, if their union contains a
       comeager set, then in every nonempty basic cylinder one member
       of the sequence is comeager on a smaller basic cylinder.
 \item For every uniformly nonempty sequence of $\Sigma^1_1$
       relations, one real codes a sequence of witnesses. Ordinary
       countable recursions using uniform code operations and least
       natural-number choices likewise have single real codes.
\end{enumerate}
\end{lemma}

\begin{proof}
Borel evaluation and the usual operations on Borel codes are available
already in $\mathsf{ATR}_0$. For (i), use the coded
Jankov--von Neumann construction. For each basic open set in the
range, replace the corresponding selector preimage by an open set
modulo a coded meager set. Deleting the union of these countably many
errors leaves a coded comeager set on which the selector is Borel. It
may in fact be taken continuous relative to its domain. The same
uniform Baire-property construction for coanalytic codes proves (ii).
If every member were meager in a given cylinder, the coded union of
their meager hulls would make that cylinder meager. The first assertion
of (iii) is $\Sigma^1_1$ choice, which is provable already in
$\mathsf{ATR}_0$ \cite[Theorem~V.8.3]{Simpson}; the recursion assertion
is the usual arithmetical recursion on the joined codes. These are
the coded forms of separation, uniformization, and the Baire-property
theorems developed in \cite[Sections~V.3 and VI.2]{Simpson}.
\end{proof}

\begin{theorem}[Finite-fragment Borel obstruction]
\label{thm:borel-finite-fragment}
There are a fixed bounded formula $\operatorname{Bin}(a,i)$ and a
fixed finite subtheory $F_{\mathrm{Bor}}$ of the usual axiomatization
of $I\Delta_0$ such that $\Pi^1_1\text{-}\mathsf{CA}_0$ proves that no
Borel quotient presentation of an $F_{\mathrm{Bor}}$-model has every
real as a $\operatorname{Bin}$-trace. In every $I\Delta_0$-model the
$\operatorname{Bin}$-traces are exactly the binary standard system.
Consequently the same theory proves that no Borel quotient presentation
of an $I\Delta_0$-model has full binary standard system.
\end{theorem}

\begin{proof}
We first prove the obstruction, identifying each arithmetic axiom as
it is used. We then collect these axioms into a finite subtheory.
Fix the partial-exponentiation graph $E$ from
\cref{subsec:bounded-arithmetic-cuts}. For definiteness, put
\[
 \pi(u,v)=(u+v)^2+u,
 \qquad \rho(i,c,d)=1+\pi(i,\pi(c,d)).
\]
The discretely ordered semiring axioms prove that $\pi$, and hence
$\rho$, is injective and that $\rho(i,c,d)>i,c,d$. Put
\begin{align}
 \operatorname{Rem}(c,d,i,r)
 &\;\Longleftrightarrow\;
 r<1+(i+1)d\ \wedge\
 \exists q\leq c\,[c=q(1+(i+1)d)+r],
 \label{eq:borel-rem}\tag{Rem}\\
 \operatorname{BBit}(c,d,i)
 &\;\Longleftrightarrow\;
 d>0\ \wedge\ \operatorname{Rem}(c,d,i,1).
 \label{eq:borel-bbit}\tag{BBit}
\end{align}
Division with unique remainder makes this a two-valued decoder. The
bounded formula
\begin{equation}
 \delta(a,m)\quad\Longleftrightarrow\quad
 m=0\ \vee\ \exists q\leq a\ \exists r<m\,[a=qm+r]
 \label{eq:borel-division}
\end{equation}
has an induction instance which proves existence of quotient and
remainder; uniqueness follows from the discretely ordered semiring
axioms. We first convert the binary standard-system hypothesis to this
decoder without assuming total exponentiation.

For a finite binary word $\sigma$ of length $n$, put $d_\sigma=n!$
(with $0!=1$). The moduli
$1+(i+1)d_\sigma$, $i<n$, are pairwise coprime. Indeed, a common
divisor for the $i$th and $j$th moduli divides $(j-i)d_\sigma$,
is coprime to $d_\sigma$, and therefore divides $j-i$; since
$j-i$ divides $d_\sigma$, the $i$th modulus is congruent to $1$
modulo $j-i$. Thus the common divisor is $1$.
The Chinese remainder theorem supplies $c_\sigma$ whose remainder
at the $i$th modulus is $\sigma(i)$. Choose the least such $c_\sigma$
below the product of the moduli. This is a bounded search, so
$\sigma\mapsto(c_\sigma,d_\sigma)$ is primitive recursive.

Define the fixed bounded formula for binary coding by
\begin{equation}
 \begin{split}
 \operatorname{Bin}(a,i)\quad\Longleftrightarrow\quad
 i<a\ \wedge\ \exists p\leq a\,[E(i,p)\ \wedge
 \exists u\leq a\ \exists v<p\;a=(2u+1)p+v].
 \end{split}
 \label{eq:borel-bin}
\end{equation}
Functionality, the base value, and the successor law for $E$ imply
in $I\Delta_0$ that, for every standard $i$, this is equivalent to
$\beta_i(a)$. The weaker fragment below requires no property of $E$,
since its fullness hypothesis is stated directly in terms of
$\operatorname{Bin}$-traces.
For $x\in2^\omega$ let
\[
 Y_x=\{\rho(n,c_{x\restr n},d_{x\restr n}):n<\omega\}.
\]
Fullness for $\operatorname{Bin}$ gives $y$ whose standard
$\operatorname{Bin}$-trace is $Y_x$. Since the displayed support is
unbounded and $\operatorname{Bin}(y,i)$ includes $i<y$, the element $y$ is
nonstandard.

Write
\begin{align*}
 \operatorname{Mark}_y(i,c,d)
 &\;\Longleftrightarrow\;
 d>0\ \wedge\ \rho(i,c,d)<y\ \wedge
       \operatorname{Bin}(y,\rho(i,c,d)),\\
 \operatorname{Row}_y(i,c,d)
 &\;\Longleftrightarrow\;
 \operatorname{Mark}_y(i,c,d)\ \wedge
 \forall c',d'<y\,
   [\rho(i,c',d')<\rho(i,c,d)
        \longrightarrow\neg\operatorname{Mark}_y(i,c',d')].
\end{align*}
Thus $\operatorname{Row}$ chooses the least marked record in a row.
Let $\operatorname{GoodRow}(n,y)$ be the bounded formula asserting
that $n<y$, that every row $i\leq n$ has a row pair $c,d<y$, and that
whenever $i<j\leq n$, their selected pairs have the same
$\operatorname{BBit}$ values at every $k<i$. Explicitly, its last
clause is
\[
 \begin{split}
 \forall i<j\leq n\ \forall c,d,c',d'<y\;
 [&\operatorname{Row}_y(i,c,d)\wedge
   \operatorname{Row}_y(j,c',d')\longrightarrow{}\\[-2mm]
 &\forall k<i\,
 (\operatorname{BBit}(c,d,k)\leftrightarrow
  \operatorname{BBit}(c',d',k))].
 \end{split}
\]
This predicate is downward closed and holds for every standard $n$.
For a standard row the intended record is standard, and every smaller
record is standard, so the $\operatorname{Bin}$-trace of $y$ identifies
the intended pair. Put
\[
 \Theta_{\rm row}(z,y)\quad\Longleftrightarrow\quad
 \forall n<z\;\operatorname{GoodRow}(n,y).
\]
But $\operatorname{GoodRow}(y,y)$ fails. The induction
instance for $\Theta_{\rm row}$ gives a least $b\leq y$ at which
$\operatorname{GoodRow}$ fails. The element $b$ is nonstandard,
and its predecessor $N$ is a nonstandard good row. If $(c,d)$ is the
pair in row $N$, comparison with the standard row $k+1$ gives
\begin{equation}
 \operatorname{BBit}(c,d,\bar k)\quad\Longleftrightarrow\quad x(k)=1
 \qquad(k<\omega).
 \label{eq:borel-beta-code}
\end{equation}
Thus fullness for $\operatorname{Bin}$ implies that every real has a
remainder code $(c,d)$. In particular the model is nonstandard.
Fix one nonstandard element for the least-failure
arguments below.

Now let $P/{\mathord\approx}$ be the presented model. On
representatives, $\operatorname{BBit}$ is analytic, since the
quotient witness $q$ in \eqref{eq:borel-rem} is existentially
quantified. Its complement is analytic as well. The part with $d=0$
is Borel, and on $d>0$ division supplies $q,r$ with
\[
 r<1+(i+1)d,\qquad
 c\mathrel\approx q(1+(i+1)d)+r,\qquad r\not\mathrel\approx\bar1,
\]
and uniqueness makes this equivalent to failure of
$\operatorname{BBit}$. Analytic separation therefore gives a Borel
code for $\operatorname{BBit}$, uniformly from the presentation.
Consequently
\[
 R(x,c,d)\quad\Longleftrightarrow\quad
 d>0\ \wedge\ \forall n<\omega\,
 [\operatorname{BBit}(c,d,\bar n)\leftrightarrow x(n)=1]
\]
is Borel and has nonempty sections by
\eqref{eq:borel-beta-code}. Jankov--von Neumann uniformization and
Baire regularization give a comeager Borel set $X\subseteq2^\omega$
and a Borel map $F:X\to P^2$ uniformizing $R$. Extend $F$ by a fixed
value off $X$, and set
\[
 \operatorname{Ext}(x)(m)=1
 \quad\Longleftrightarrow\quad
 \operatorname{BBit}(F(x),m).
\]
This relation is Borel in $(x,m)$, and
$\operatorname{Ext}(x)(n)=x(n)$ for $x\in X$ and standard $n$.

The set of representatives of nonstandard elements,
\[
 P_{\rm ns}=\bigcap_{n<\omega}\{a\in P:\bar n<a\},
\]
is Borel, and the reverse of the arithmetic order is a Borel linear
quasiorder on it. We use the following coded consequence of the
Harrington--Marker--Shelah theorem. In
$\Pi^1_1\text{-}\mathsf{CA}_0$, every real-coded Borel linear
quasiorder has a real-coded countable cofinal sequence. Indeed,
Marcone's formalization of the representation theorem gives, already
in $\mathsf{ATR}_0$, a coded countable wellorder $W$ and a Borel
strong order-preserving map
\[
 f:P_{\rm ns}\longrightarrow(2^W,\leq_{\rm lex})
\]
\cite[Theorem~4.2 and Corollary~4.9]{MarconeBorel}; see also the
original theorem~\cite[Theorem~3.1]{HMSBorel}. The coded lexicographic
argument is as follows. Follow the lexicographically highest
extendible branch $b$. If $b=f(x)$ for some $x$, then $x$ is cofinal.
Otherwise let $\lambda$ be the first limit initial segment at which
extendibility fails, and fix a coded increasing sequence
$(\lambda_n)$ cofinal in $\lambda$. For every $n$, the Borel set
\[
 E_n=\{x\in P_{\rm ns}:f(x)\restr\lambda_n
                  =b\restr\lambda_n\}
\]
is nonempty. By $\Sigma^1_1$ choice
\cite[Theorem~V.8.3]{Simpson}, one real codes a sequence $x_n\in E_n$.
The first-difference argument shows that $(f(x_n))$ is cofinal in the
image, and strong order preservation shows that $(x_n)$ is cofinal in
$P_{\rm ns}$. Applying this lemma to the reverse order gives
a real-coded coinitial family $(A_i)_{i<\omega}$ in the nonstandard
part. Put
\[
 Q_n=\min(A_0,\ldots,A_n),
\]
retaining the first representative which attains the minimum. Then
$(Q_n)$ is a nonincreasing coinitial sequence, obtained by finite
comparisons and without choosing new representatives.

We next prove a Borel version of \cref{lem:cso-local-overspill}.
If $a=s{}^\frown t^\omega$ is eventually periodic, with
$t\ne\varnothing$, choose standard remainder codes
$(c_s,d_s)$ and $(c_t,d_t)$ for the prefix and period, and put
$\alpha=(c_s,d_s,|s|,c_t,d_t,|t|)$. The fixed bounded formula
\begin{align*}
 \operatorname{Per}(m;\alpha)\quad\Longleftrightarrow\quad{}
 &[m<|s|\wedge\operatorname{BBit}(c_s,d_s,m)]\\
 &{}\vee[|s|\leq m\wedge
   \exists q\leq m\ \exists r<|t|\,
   (m=|s|+q|t|+r\wedge
       \operatorname{BBit}(c_t,d_t,r))]
\end{align*}
defines the corresponding internal periodic predicate $a^*(m)$.
Division supplies a unique residue modulo the fixed positive standard
number $|t|$. Thus $\operatorname{Per}$ is Borel on representatives,
since this predicate and its complement are analytic by the argument used for
$\operatorname{BBit}$. If $P,Q$ are two remainder-code
pairs, let
\[
 \Psi(m;P,Q,\alpha)\quad\Longleftrightarrow\quad
 [\operatorname{BBit}(P,m)\leftrightarrow
  \operatorname{BBit}(Q,m)]
 \leftrightarrow\neg\operatorname{Per}(m;\alpha),
\]
where $\operatorname{BBit}((c,d),m)$ abbreviates
$\operatorname{BBit}(c,d,m)$, and set
\[
 \Theta_{\rm per}(z;P,Q,\alpha)
 \quad\Longleftrightarrow\quad
 \forall m<z\;\Psi(m;P,Q,\alpha).
\]
For
$x\in X\cap(X+a)$, standard correctness gives
\begin{equation}
 \operatorname{Ext}(x+a)(m)=
 \operatorname{Ext}(x)(m)\mathbin\oplus a^*(m)
 \qquad(m<\omega).
 \label{eq:borel-periodic-standard}
\end{equation}
The induction instance for $\Theta_{\rm per}$ gives, for each $x$, a
nonstandard bound below which the identity holds. Indeed, below a fixed
nonstandard cutoff, either there is no failure or there is a least one,
which must be nonstandard. Hence the
coanalytic sets
\[
 D_i^a=\{x\in X\cap(X+a):
       \eqref{eq:borel-periodic-standard}\text{ holds for all }m<Q_i\}
\]
cover the comeager set $X\cap(X+a)$. In every nonempty basic cylinder
$U$, some $D_i^a\cap U$ is nonmeager. The Baire property for
coanalytic sets therefore gives a nonempty basic
$W\subseteq U$ on which the identity below $Q_i$ holds comeagerly.
This gives the required local common bound without selecting overspill
bounds.

We can now repeat the block construction in the proof of
\cref{lem:cso-blocks}, with the meager ideal in place of $\mathcal I$
and $(Q_i)$ in place of $(N_i)$. At stage $n$ take
\[
 w_n=0{}^\frown1^n,\qquad p_n=n+1,\qquad L_n=2p_n,
\]
and apply the preceding paragraph to
$z_n=s_n{}^\frown w_n^\omega$ inside the $n$th cylinder of a schedule
which repeats every basic cylinder infinitely often. It gives a
subcylinder $W_n$ and a nonstandard $H_n=Q_i$ on which the periodic
comparison holds comeagerly. Choose an increasing $r_n\geq n$ such
that, for $M_n=Q_{r_n}$,
\[
 M_n+L_n<H_n.
\]
The finite-valued Borel map
\[
 x\longmapsto
 \langle\operatorname{Ext}(x+z_n)(M_n+j):j<L_n\rangle
\]
is constant on a comeager part of a smaller basic cylinder $V_n$.
Extend $s_n$ by a positive finite number of copies of $w_n$ far
enough to cover the prefix defining $V_n$. Exactly as in
\eqref{eq:cso-translation}--\eqref{eq:cso-good-block}, the union
$s_\omega$ of these prefixes and the Baire category theorem supply
an $x$ such that $x,x+s_\omega\in X$ and unboundedly many stages are
successful. At every successful stage the two internal strings agree
at exactly two positions in $[M_n,M_n+L_n)$, and these positions are
consecutive and separated by $n+1$. The successful block positions
are coinitial. Once $Q_k<\lfloor e/2\rfloor$, every sufficiently late
$M_n$ is below $Q_k$, while the standard $L_n$ is below the
nonstandard element $\lfloor e/2\rfloor$.

Write $F(x)=(c,d)$ and $F(x+s_\omega)=(c',d')$, and define the fixed
bounded predicate
\[
 B(m)\quad\Longleftrightarrow\quad
 \bigl(\operatorname{BBit}(c,d,m)\leftrightarrow
       \operatorname{BBit}(c',d',m)\bigr).
\]
Use the formula $\operatorname{Next}_B$ and the four-endpoint gap
formula $C(e)$ from \eqref{eq:cso-cut}. At standard positions $B$ is
the zero set of $s_\omega$, whose successive gaps are nondecreasing
and unbounded, so $C(e)$ holds for every standard $e$. If $e$ is
nonstandard, choose a sufficiently late successful block below $e$.
Its consecutive agreement points have gap $n+1$. A standard
construction stage $k>n$ supplies an earlier consecutive standard
gap $k+1>n+1$, and all four endpoints are below $e$. Thus $C(e)$
fails. The extension of $C$ is exactly the standard cut, contradicting
the one induction instance for this fixed bounded formula.

The proof uses only finitely many first-order axioms. In the standard
presentation
\[
 I\Delta_0=\mathsf{PA}^{-}+
 \{\operatorname{Ind}(\varphi):\varphi\text{ is bounded}\},
\]
where $\mathsf{PA}^{-}$ is the fixed finite theory of discretely
ordered commutative semirings, take $F_{\mathrm{Bor}}$ to be
$\mathsf{PA}^{-}$ together with exactly four universally closed
induction instances: induction in $a$ for $\delta(a,m)$ from
\eqref{eq:borel-division}, induction in $z$ for
$\Theta_{\rm row}(z,y)$, induction in $z$ for
$\Theta_{\rm per}(z;P,Q,\alpha)$, and induction in $e$ for
$C(e;P,Q)$. Division uniqueness, the polynomial-record facts, and
the elementary arithmetic of standard finite intervals are already
provable in $\mathsf{PA}^{-}$. Thus $F_{\mathrm{Bor}}$ is a finite
subset of the axioms of $I\Delta_0$. For any other conventional finite base,
take the union of the finitely many $I\Delta_0$ axioms occurring in
fixed proofs of the required $\mathsf{PA}^{-}$ facts together with the
four displayed induction instances. Hence the conclusion that a finite
fragment suffices is independent of the choice of finite base.

All the descriptive-set-theoretic objects in the proof have uniform
real codes. Analytic separation gives the Borel code for
$\operatorname{BBit}$; parts~(i) and (ii) of
\cref{lem:borel-coded-category-toolkit} yield the codes for $X,F$ and
the local common bounds; the Harrington--Marker--Shelah representation
followed by the coded lexicographic lemma yields the coinitial
sequence; and part~(iii) codes the countable block recursion and all
of its choices and error sets. The representation theorem is
formalizable in $\mathsf{ATR}_0$ by Marcone~\cite{MarconeBorel}.
Since satisfaction of the fixed
finite theory $F_{\mathrm{Bor}}$ expands as a finite formula about
the presentation code, the whole assertion is one sentence of
second-order arithmetic, with no third-order satisfaction predicate.
\end{proof}

\begin{remark}
The forcing-theoretic transfer arguments above use the canonical
exponential cut. In \cref{thm:borel-finite-fragment}, we instead obtain the
coinitial sequence from the Borel order itself. Partial exponentiation
occurs only in the fixed bounded trace formula used to produce
remainder codes.
\end{remark}


\subsection{The obstruction at \texorpdfstring{$\aleph_1$}{aleph one}}

Continue with $V=L[G]$, where $G$ adds $\omega_1$ Cohen reals,
and $S=\HODR^V$.

\begin{theorem}[No small definable model with full standard system]
\label{thm:arithmetic-negative}
In $V$ there is no $\ODR$ model
$\mathcal M$ of $I\Delta_0$ with $|\mathcal M|\leq\aleph_1$ and
$\SSy(\mathcal M)=\mathcal P(\omega)^V$.
\end{theorem}

\begin{proof}
The domain is an $\ODR$ set of size at most $\aleph_1$.
A sequence of fewer than $\kappa=\omega_1$ reals is coded by one real,
so \cref{thm:regular-cohen} implies that every domain element is $\ODR$.
By \cref{lem:rod-copy}, an isomorphic structure
$\widehat{\mathcal M}$ belongs to $S$.  Absoluteness of satisfaction
makes it a model of $I\Delta_0$ there.

The models $S$ and $V$ have the same reals, because each real is
hereditarily definable from itself.  Every such real is coded in
$\mathcal M$, hence by the corresponding element of
$\widehat{\mathcal M}$.  This element belongs to $S$, and the bit
coding relation is absolute.  Thus $S$ regards
$\widehat{\mathcal M}$ as having full standard system, contrary to
\cref{thm:cohen-scott-obstruction}.
\end{proof}

\begin{corollary}[Absence of definable saturated copies]
\label{cor:saturated-no-od}
For every complete consistent
$T\supseteq I\Delta_0$, its saturated model of cardinality
$\aleph_1$ exists in $V$ and has no $\ODR$ copy there.
In particular it has no OD copy.
\end{corollary}

\begin{proof}
The extension satisfies CH, so $(\aleph_1)^{\aleph_0}=\aleph_1$.
Standard saturation theory gives a saturated model of $T$ of
cardinality $\aleph_1$, unique up to isomorphism.
For every real $x\subseteq\omega$, the type
\[
 \{\beta_n(v):n\in x\}
 \cup\{\neg\beta_n(v):n\notin x\},
\]
where $\beta_n(v)$ tests the residue modulo the standard integer
$2^{n+1}$, is finitely satisfiable by standard integers. Saturation realizes
it, so the model has full standard system.  Apply
\cref{thm:arithmetic-negative}.
\end{proof}

\subsection{The arithmetic obstruction after \texorpdfstring{$\omega_1$}{omega one} random reals}
\label{subsec:random-arithmetic}

For the random analogue we use an additional parameter. The members of
a small definable family are definable from reals and one fixed orbit of
generic filters, although they need not be individually
real-ordinal-definable. The hereditary class defined using this orbit
has the probability and uniformization properties needed below.

\begin{theorem}\label{thm:random-arithmetic-negative}
Let $V=L[F]$, where $F$ adds $\omega_1$ random reals.  There is no
$\ODR$ model $\mathcal M$ of $I\Delta_0$, on any domain, such that
\[
 |M|\leq\aleph_1,
 \qquad \SSy(\mathcal M)=\mathcal P(\omega)^V.
\]
Here the standard system is defined by binary coding.  The same conclusion
holds with $L$ replaced by any
$W\models\mathsf{ZFC}+\mathsf{CH}+\mathsf{GA}+(V=\mathrm{HOD})$.
\end{theorem}

We prove the stated generalization.  Write $\B=\mathbb B_{\omega_1}^W$
for the probability algebra, $\mu$ for its probability, and
$P=\Aut^W(\B,\mu)$.  Fix a ground ordinal coding of $\B$ and, in
$V=W[F]$, put
\begin{equation}\label{eq:ra-anchor}
 a=\{\operatorname{code}(h[F]):h\in P\},\qquad
 O_a=\mathrm{OD}_{a,\mathbb R}^{V},\qquad
 S_a=\{x:\operatorname{tc}(\{x\})\subseteq O_a\}.
\end{equation}
The parameter $a$ is fixed throughout; it need not belong to $S_a$.
Its canonical name is fixed by every member of $P$.

\subsubsection{Small-index supports and descent to the orbit}

\begin{lemma}[Local patching]\label{lem:ra-local-patching}
Let $D\subseteq\B$ be a separable complete probability subalgebra.
If $h\in P$ and $d>0$ satisfy
\begin{equation}\label{eq:ra-local-patching}
 h(d\wedge c)=h(d)\wedge c\qquad(c\in D),
\end{equation}
there is $k\in P_{(D)}$ agreeing with $h$ on the entire principal
algebra $\B\restr d$.
\end{lemma}

\begin{proof}
For every $c\in D$, the two sides of \cref{eq:ra-local-patching}
have equal measure.  Consequently
\[
 c\wedge\neg d\longmapsto c\wedge\neg h(d)
\]
is a well-defined measure-preserving isomorphism of the restricted
$D$-algebras.  Unless $d=1$, their ambient complementary principal
algebras have the same positive total measure and relative type
$\omega_1$ over these separable bases.  After normalizing the measures,
\cite[333C(b)]{Fremlin} extends the displayed map to the whole
complements.  Paste this extension with $h\restr(\B\restr d)$.
The result fixes $D$.  If $d=1$, the hypothesis already says that $h$
fixes $D$.
\end{proof}

\begin{lemma}[Descent to one orbit]\label{lem:ra-anchor-descent}
Every member of every $\ODR$ family of size at most $\omega_1$ in $V$
belongs to $O_a$.
\end{lemma}

\begin{proof}
First suppose that a ground name $\tau$ is fixed modulo forced equality
by every member of $P$ fixing a countable coordinate factor $\B_T$.
We recover $\tau^F$ from $a$, the real coding $F\cap\B_T$, and ground
ordinal codes.  Evaluate $\tau$ at all filters coded by $a$ whose
restriction to $\B_T$ is this same filter.  The value at $F$ occurs.
For another such filter $h[F]$, equality of the two restricted filters
holds below some $d\in F$; thus
\[
 d\wedge h^{-1}(c)=d\wedge c\qquad(c\in\B_T).
\]
After applying $h$, this is \cref{eq:ra-local-patching}.
Choose $k\in P_{(\B_T)}$ agreeing with $h$ on $\B\restr d$.
Then $k[F]=h[F]$, whereas invariance of $\tau$ gives
$\tau^{k[F]}=\tau^F$.  The proposed evaluation therefore has one value.
Stable ground codes, supplied by \cref{lem:stable-ground-codes}, make
it a definition in $O_a$. Local patching is needed because agreement
of two actual generic traces does not imply that $h$ fixes $\B_T$
globally.

Now let $A$ be the given family.  Capture its real definition parameter
and a condition forcing the definition, nonemptiness, and the size bound
on a countable ground coordinate set $T_0$.  Below this condition use
the named family, and elsewhere use a fixed singleton.  The resulting
everywhere nonempty family name is invariant under $P_{(\B_{T_0})}$.
The set of classes of its member names modulo equality forced by $1$
has ground cardinality at most $\omega_1$. To see this, choose a forced
enumeration by $\omega_1$. Every member name is a countable Boolean
mixture of entries of this enumeration, and there are at most
\[
 (\omega_1\cdot|\B|)^{\aleph_0}=\omega_1
\]
such mixtures.  The indicated group acts on this ordinary ground set.
Each point stabilizer has index at most $\omega_1$, so
\cref{lem:ra-small-index}, with constants for the finite-cylinder
algebra on $T_0$, gives a countable factor $\B_T$ whose fixer fixes
that member name.  Apply the first paragraph to each member.
\end{proof}

\subsubsection{The hereditary model and its probability}

Put $C=2^\omega$, and use addition modulo two on $C$.
All definitions in \cref{eq:ra-anchor} are made in the ambient model $V$.

\begin{lemma}\label{lem:ra-hereditary}
$S_a$ is a transitive inner model of $\mathsf{ZF}+\mathsf{DC}$,
contains every real of $V$, and is closed under countable $V$-sequences
of its elements.  Also $W[z]\subseteq S_a$ for every real $z\in V$.
An $O_a$ structure with pointwise $O_a$ domain has an isomorphic copy
belonging to $S_a$.
\end{lemma}

\begin{proof}
Apply \cref{lem:definition-inner-model,lem:definition-sequence-closure}
with $Z=\mathbb R$ and the fixed anchor $a$. The closure hypothesis
follows from ccc and stable ordinal codes for ground enumerations.
For every real $z$, the intermediate model $W[z]$ is a set-forcing
extension of $W$; its objects have definitions from $z$ and ground
ordinal codes, by \cref{lem:stable-ground-codes}. Its transitivity
therefore gives $W[z]\subseteq\HODR^V\subseteq S_a$.
The copy assertion is \cref{lem:rod-copy}, which does not require
$a\in S_a$.
\end{proof}

The next lemma allows us to obtain traces at every random real from
the traces at coordinate generics.

\begin{lemma}[Realizing a random real locally]\label{lem:ra-local-realization}
In a homogeneous random extension of uncountable type over any ZFC ground
$U$, let $r_\beta$ be a fixed coordinate real.  Every real $x$ in the
extension that is random over $U$ is the actual value of
$h\dot r_\beta$ for some measure-preserving ground automorphism $h$
supported on countably many coordinates.  The same holds for finite
tuples of jointly random reals.
\end{lemma}

\begin{proof}
Read $x$ as a ground measurable function $f$ of countably many coordinates,
and let $\rho$ be its distribution.  Write the absolutely continuous part
as $g\,dm$, and choose a ground $m$-null Borel set $Z$ carrying the
singular part.  Since $x$ is random over $U$, it avoids $Z$ and the null
exception to finiteness of $g$.  For some positive integer $N$ the event
$c_0=\{f\notin Z,\ g(f)\leq N\}$ belongs to the actual generic.
The marginal of $f$ restricted to $c_0$ is bounded by $Nm$.
Split $c_0$ into $N$ equal slices using a fresh independent uniform
variable, and select the slice $c$ in the generic.  Increasing $N$ first
if necessary ensures $\mu(c)<1$.  Then
\[
 \eta=f_*(\mu\restr c)\leq m.
\]
On $\neg c$, use a second independent uniform variable to sample
$(m-\eta)/(1-\mu(c))$.  Pasting this sample with $f$ on $c$ gives a
ground random variable $y$ of distribution exactly $m$ whose actual
value is $x$.

Take a countable coordinate factor containing all coordinates used,
$\beta$, and a further independent atomless variable. Conditional on
$y$, this factor is atomless because the last variable remains
independent. The standard relative product theorem identifies the
factor with the product of the $y$-factor and a standard atomless
factor, and also with the product of the $r_\beta$-factor and its
complement; see
\cref{lem:relative-fixed} and \cite{Delbaen}.
The resulting isomorphism sends $r_\beta$ to $y$.  Extend it by the
identity on the other coordinates. The slice containing the actual
generic was selected externally, but every candidate map belongs to
the ground. For finite tuples, use the standard finite product probability
as the target marginal in the same construction.
\end{proof}

\begin{lemma}[Universal Borel traces]\label{lem:ra-borel-trace}
For each $X\subseteq C^n$ belonging to $O_a$, there are a countable
coordinate intermediate ground $U=W[F_T]$ and a Borel set $B$ coded
in $U$ such that
\[
 \vec x\text{ jointly random over }U
 \quad\Longrightarrow\quad
 (\vec x\in X\ \longleftrightarrow\ \vec x\in B).
\]
The ground $U$ can be enlarged to include any prescribed real parameter.
\end{lemma}

\begin{proof}
Capture the real definition parameters and a suitable forcing condition
on a countable coordinate set $T$. Every new measure-preserving tail
automorphism over $U$ fixes the interpreted anchor. To prove this, take
a name for the map, extend it by the identity off a condition, and lift
it to the ground iteration while fixing $\B_T$.
Preservation of conditional probability,
followed by integration over $\B_T$, makes the lift measure-preserving.
It therefore fixes the canonical name of $a$.  This is also the scalar
case of \cref{lem:sr-relative-descent}.

Choose $n$ fresh coordinate reals $\vec r$.  The Boolean value of
$\vec r\in X$ is fixed by every measure-preserving map on the remaining
coordinates, so belongs to the factor generated by $\vec r$.
Indeed, conditional on that factor, the remaining product has no
nonconstant event invariant under all its finite flips.  Represent
this Boolean value by a Borel set $B$ over $U$.
For any actual jointly random tuple $\vec x$ over $U$,
\cref{lem:ra-local-realization} supplies a relative measure-preserving
map taking $\vec r$ to $\vec x$ at the generic.  This map fixes $a$
and the Borel code.  Applying it to the Boolean identity proves the
asserted equivalence.  The argument works after any further countable
capture.
\end{proof}

\begin{proposition}[The probability environment]\label{prop:ra-environment}
Inside $S_a$, every subset of $C$ has a countably additive,
translation-invariant probability extending the usual Borel probability.
There are compatible finite-product probabilities satisfying Fubini,
and every set differs from a Borel set by a null set.
Every relation $R\subseteq C\times Y$ in $S_a$ with nonempty sections
has, in $S_a$, a uniformizing function on a conull subset of $C$.
\end{proposition}

\begin{proof}
Let $\operatorname{NR}_n(W[z])$ denote the tuples of reals in $V$ that
are not jointly random over $W[z]$.  Define in $V$
\[
 \mathcal I_n=
 \{X\in\mathcal P(C^n)^{S_a}:\exists z\in C\;
                         X\subseteq\operatorname{NR}_n(W[z])\}.
\]
Joining a countable sequence of parameters into one real, using
\cref{lem:ra-hereditary}, shows that this is a $\sigma$-ideal.
It contains the usual null Borel sets and contains no positive Borel
set. For the latter assertion,
capture a Borel code and $z$ in a countable coordinate model.  Among
countably many fresh independent tuples, some belongs to the positive
Borel set, and every such tuple is jointly random over that model.
By \cref{lem:ra-borel-trace}, every $X\in\mathcal P(C^n)^{S_a}$
differs from a Borel set $B$ by a member of $\mathcal I_n$.
Thus
\begin{equation}\label{eq:ra-extension-measure}
 \nu_n(X)=m^n(B)\quad\text{whenever }X\mathbin\triangle B\in\mathcal I_n
\end{equation}
is well defined.  Its null ideal is exactly $\mathcal I_n$, since a null
Borel trace can be included in the exceptional ideal by adjoining its code.
Simultaneous traces over a joined real parameter give countable
additivity.  Including a translation parameter in the trace ground
gives translation invariance.

We must check that the ideals and measure graphs belong to $S_a$.
Interpret $W[z]$ in $V$ using stable ground codes. Membership in $W[z]$
is uniform in $z$. Range over ground names for reals and the complete
subalgebras generated by their bit values. Ground Borel operations
recover from $z$ the generic trace that gives it as a value. Each
resulting intermediate model is $W[z]$ by the
intermediate model theorem.  The ground names are quantified over, so
no further parameter depending on $z$ is introduced.
Consequently $\mathcal I_n$ and the graph in
\cref{eq:ra-extension-measure} belong to $O_a$.  Their entries and all
their transitive constituents belong to $O_a$, since $X\in S_a$ and
the values of $\nu_n$ are reals.  Hence both graphs belong to $S_a$.

For Fubini, choose a joint trace $B$ of $X\subseteq C^{k+l}$ over a
countable coordinate ground $U$.  If $x$ is random over $U$ and $y$
is random over $U[x]$, the pair is jointly random over $U$.
The section $X_x$ therefore differs from $B_x$ only on
$\operatorname{NR}_l(U[x])$.  Its section measure agrees with the
Borel section measure for $\nu_k$-almost every $x$.  Ordinary Borel
Fubini gives Fubini for the probabilities $\nu_n$.
Here $U[x]=W[z,x]$ for a real code $z$ of the captured generic.
All these assertions hold internally in $S_a$, since its countable
sequences are actual sequences and it has all the ambient reals.
The same trace argument shows that finite-valued functions are
approximable in measure by functions of finitely many bits, and that
\[
 t_j\longrightarrow0\quad\Longrightarrow\quad
 \nu_1(X\mathbin\triangle(X+t_j))\longrightarrow0.
\]

For uniformization use the witness-code construction in
\cref{thm:gce-uniformization}, now with
$\operatorname{Val}_a(z,\xi)$ from \eqref{eq:definition-evaluation}.
Capture the definitions and a condition
forcing nonempty sections on countably many coordinates, and choose
a fresh coordinate $\beta$. The maximum principle selects a real name
and an ordinal name coding a witness for $r_\beta$.
The real name has countable support; ccc replaces the ordinal name
by an integer index in a countable ground ordinal table. Pass to the
intermediate model containing their coordinates other than $\beta$.
Over this countable coordinate ground $U_1$, the remaining names have
Borel readings.
Evaluate those readings and keep a value precisely when it belongs
to the actual $Y$ and witnesses the actual $R$.
The reading codes and the stable ordinal-table code make this an
$O_a$ partial function, and its graph is hereditary, hence in $S_a$.
Its failure set has zero Boolean value at $r_\beta$.
Apply the proof of \cref{lem:ra-borel-trace} over this same $U_1$,
with $\beta$ as the distinguished coordinate. Local realization
transports the zero failure identity to every real random over $U_1$,
since the relative measure-preserving maps fix the anchor and the reading codes.
The failure set is therefore contained in $\operatorname{NR}_1(U_1)$,
hence belongs to $\mathcal I_1$. Thus the domain is conull.
\end{proof}

The measures just constructed are extensions on all the subsets
belonging to $S_a$.  Their null sets need not be null for ordinary
ambient outer measure.

\subsubsection{The probability Scott argument}

We now prove the probability counterpart of
\cref{thm:cohen-scott-obstruction}. The probability argument supplies
arithmetic blocks to which the same final bounded formula applies.

\begin{lemma}[The probability Scott obstruction]\label{lem:ra-scott}
Suppose a transitive model of $\mathsf{ZF}+\mathsf{DC}$ has the
probability and uniformization properties of
\cref{prop:ra-environment}.  It has no $I\Delta_0$ model with full binary
standard system.
\end{lemma}

\begin{proof}
Work in this model and write $\nu=\nu_1$.
If a counterexample exists, \cref{thm:full-system-canonical-cut}
gives a canonical nonstandard EFA cut with the same binary standard
system.  Write $M$ for this cut.  Uniformization gives a function
$F:C\to M$ whose value codes $x$ on a conull set $X$. Extend it by a
fixed value off its original domain. Put
\[
 \operatorname{Ext}(x)(m)=1
 \quad\Longleftrightarrow\quad M\models\operatorname{Bit}(F(x),m).
\]
For $x\in X$, all standard bits are correct simultaneously.
The arithmetic operations and bounded overspill used below are exactly
those justified in the proof of \cref{lem:cso-periodic}.

We first need a measure form of overspill.  Suppose that for almost
every $x$ there is a nonstandard $h$ with $P(x,N)$ for all $N<h$.
Uniformize these bounds, taking nonstandard defaults on the exceptional
set, to obtain $h(x)$.  Choose $k$ with $1/(k+1)<\varepsilon$.
Among $k+1$ independent samples, the events that a specified sample
has the unique strict minimum $h$-value are disjoint and equimeasurable.
Hence
\[
 \nu_{k+1}\{h(y)<\min_{i<k}h(x_i)\}\leq\frac1{k+1}.
\]
Fubini supplies a fixed tuple for which the corresponding section
has measure less than $\varepsilon$.
For $H=\lfloor\min_{i<k}h(x_i)/2\rfloor$, which is nonstandard,
the bound $\forall N<H\;P(y,N)$ therefore holds with probability
greater than $1-\varepsilon$.

Apply this observation to bounded overspill of the correct standard-bit
identities.  For finite words $s,t$ with $t\ne\varnothing$, there is
a nonstandard $H$ such that, with probability greater than
$1-\varepsilon$,
\begin{equation}\label{eq:ra-periodic}
 \operatorname{Ext}(x+s{}^\frown t^\omega)(m)
 =\operatorname{Ext}(x)(m)\mathbin\oplus
                         (s{}^\frown t^\infty)(m)
 \qquad(m<H).
\end{equation}
The internal periodic pattern on the right is boundedly defined;
no integer coding an infinite internal string is asserted.

We next obtain a countable coinitial sequence in $M\setminus\omega$.
Put $a_n=0{}^\frown1^n{}^\frown0^\omega$ and
$a_\infty=0{}^\frown1^\omega$.  Let $u_n(x)$ be the least
$m>n+1$ at which the codes of $x$ and $x+a_n$ disagree, or a formal
$\infty$ if no such position exists.  A least witness, when present,
is found by bounded minimization below $F(x)+F(x+a_n)+1$.
For fixed $0<p<1$, put
\[
 I_{n,p}=\{d\in M\setminus\omega:\nu\{x:u_n(x)<d\}>p\}.
\]
These are final segments. Each is proper. Apply
\cref{eq:ra-periodic} to $a_n$ with error less than $p$ to find a
nonstandard $H$ with $\nu(u_n<H)<p$.
They cover the nonstandard part as $n$ varies.  Otherwise take
nonstandard $d$ outside their union.  Apply
\cref{eq:ra-periodic} to $a_\infty$ with
$\delta<(1-p)/3$, and choose nonstandard
$m+1<\min(d,H)$.  With probability greater than $1-\delta$,
the codes of $x$ and $x+a_\infty$ disagree at $m$.
Translation continuity, applied to the set where the latter bit is one,
gives, for large standard $n$,
\[
 \nu\{x:\operatorname{Ext}(x+a_n)(m)
             =\operatorname{Ext}(x+a_\infty)(m)\}>1-\delta.
\]
Thus $\nu(u_n<m+1)>1-2\delta>p$, contradicting the choice of $d$.

Take $p=1/2$.  By countable choice choose a nonstandard
$c_n\notin I_{n,1/2}$. The sequence is coinitial, since
$d\in I_{n,1/2}$ implies $c_n<d$ by finality.
Successively halving finite minima, as in
\cref{eq:cso-coinitial-sequence}, gives a strictly decreasing
coinitial sequence $N_0>N_1>\cdots$ of nonstandard elements.

We now construct successful blocks with summable error.  Put
$w_n=0{}^\frown1^n$, $p_n=n+1$, $L_n=2p_n$, and
$\varepsilon_n=2^{-n-5}$.  Starting with the empty word $s_0$,
apply \cref{eq:ra-periodic} to
$z_n=s_n{}^\frown w_n^\omega$ with error $\varepsilon_n$.
Choose increasing indices $r_n\geq n$ so that
\[
 M_n=N_{r_n},\qquad M_n+L_n<H_n.
\]
Finite-coordinate approximation gives $\ell_n<\omega$ and a function
$f_n:2^{\ell_n}\to2^{L_n}$ such that
\[
 \nu\left\{x:
   \langle\operatorname{Ext}(x)(M_n+j):j<L_n\rangle
                       =f_n(x\restr\ell_n)\right\}>1-\varepsilon_n.
\]
Choose $q_n\geq1$ with $|s_n|+q_np_n\geq\ell_n$, and let
$s_{n+1}=s_n{}^\frown w_n^{q_n}$.
Dependent choice carries out the recursion; its union $s_\omega$ is a
real, with $s_\omega\restr\ell_n=z_n\restr\ell_n$.
Translation invariance and the two approximation errors imply, outside
a set of measure at most $3\varepsilon_n$,
\begin{equation}\label{eq:ra-successful-block}
 \operatorname{Ext}(x+s_\omega)(M_n+j)
 =\operatorname{Ext}(x)(M_n+j)\mathbin\oplus
                      (s_n{}^\frown w_n^\infty)(M_n+j)
 \qquad(j<L_n).
\end{equation}
Indeed the block read from $x+s_\omega$ equals the block read from
$x+z_n$ whenever both finite-coordinate approximations succeed, and
\cref{eq:ra-periodic} compares the latter block with that of $x$.
Since $\sum_n3\varepsilon_n<\infty$, countable additivity makes
\cref{eq:ra-successful-block} hold eventually for almost every $x$.
Choose such an $x$ with $x,x+s_\omega\in X$.

The standard agreement positions of $F(x)$ and $F(x+s_\omega)$ are
the zeros of $s_\omega$.  Their consecutive gaps are nondecreasing
and unbounded. The gap $n+1$ occurs a positive finite number of times
at stage $n$. For all sufficiently large $n$,
\cref{eq:ra-successful-block} gives exactly two agreement positions
in $[M_n,M_n+L_n)$, separated by $n+1$.
The bounded-cut argument at the end of the proof of
\cref{thm:cohen-scott-obstruction} now applies. Formula
\cref{eq:cso-cut}, with parameters $F(x),F(x+s_\omega)$, holds
exactly at standard $d$, contradicting bounded induction.
\end{proof}

\begin{proof}[Proof of \cref{thm:random-arithmetic-negative}]
Suppose $\mathcal M$ is a counterexample.  By
\cref{lem:ra-anchor-descent}, every element of its domain belongs to
$O_a$.  The structure itself is $O_a$, so
\cref{lem:ra-hereditary} gives an isomorphic copy
$\widehat{\mathcal M}\in S_a$.  Satisfaction for this structure and
its standard-index binary coding are absolute.  Since $S_a$ and $V$
have the same reals, $S_a$ regards the copy as an $I\Delta_0$ model
with full binary standard system.
This contradicts \cref{prop:ra-environment,lem:ra-scott}.
\end{proof}

\begin{corollary}\label{cor:random-saturated-no-od}
In the extension of \cref{thm:random-arithmetic-negative}, every
complete consistent extension of $I\Delta_0$ has a saturated model of
cardinality $\aleph_1$, and none of these models has an $\ODR$ copy.
In particular the saturated elementary extension of $\mathbb N$ of
cardinality $\aleph_1$ has no $\ODR$ presentation on any domain.
\end{corollary}

\begin{proof}
CH is preserved, so the usual construction gives a saturated model
of each such theory of cardinality $\aleph_1$.  Saturation realizes
the countable type prescribing the standard binary digits of any real,
since each finite part is realized by a standard integer. The binary
standard system is therefore full, and
\cref{thm:random-arithmetic-negative} applies.
\end{proof}


\subsection{Definable saturation at successors of singular strong limits}

At a singular strong limit $\kappa$, a regular ultrafilter can be
defined from data of hereditary size $\kappa$. Apply the finite-support
ultrapower construction of Kanovei and Shelah \cite{KanoveiShelah} to
all codes for these data. This gives the following theorem in ZFC.

\begin{theorem}\label{thm:definable-saturation}
For every singular strong limit cardinal $\kappa$, there is an
OD $\kappa^+$-saturated elementary extension of $\mathbb N$
of cardinality exactly $2^\kappa$.  The construction is uniform
from $\kappa$.
\end{theorem}

\begin{proof}
Put $\mu=\operatorname{cf}(\kappa)$ and $J=[\kappa]^{<\omega}$.
Choose an increasing cofinal sequence of infinite cardinals
$(\kappa_i:i<\mu)$ below $\kappa$, a uniform ultrafilter $D_0$
on $\mu$, and, for each $i$, an ultrafilter $U_i$ on
$J_i=[\kappa_i]^{<\omega}$ containing every cone
$\{s\in J_i:\xi\in s\}$, $\xi<\kappa_i$.
These ultrafilters exist by the finite-intersection property.
Call a tuple $q$ of this form \emph{admissible}, and define
\[
 X\in U_q
 \quad\Longleftrightarrow\quad
 \{i<\mu:X\cap J_i\in U_i\}\in D_0
 \qquad(X\subseteq J).
\]
Restriction to the $J_i$'s respects intersections and complements,
so $U_q$ is an ultrafilter on the full power set of $J$.
For every $\xi<\kappa$, the cone
$R_\xi=\{s\in J:\xi\in s\}$ belongs to $U_q$, since
$\xi<\kappa_i$ eventually and $D_0$ contains all final segments.
Each $s\in J$ belongs to only finitely many $R_\xi$'s, so these
cones regularize every $U_q$.

Since $\kappa$ is strong limit, each admissible tuple has hereditary size
at most $\kappa$:
\[
 |\operatorname{tc}(\{q\})|
 \leq\kappa+\sum_{i<\mu}2^{\kappa_i}+2^\mu=\kappa.
\]
Fix an ordinal coding of pointed well-founded extensional relations
on ordinals at most $\kappa$ by subsets of $\kappa$.
Let $C\subseteq\mathcal P(\kappa)$ be the set of codes whose
distinguished points collapse to admissible tuples. Admissibility is tested in the
ambient universe, including the ultrafilter assertions on the
full power sets of the $J_i$'s.  Thus $C$ is nonempty and OD from
$\kappa$, and $|C|\leq2^\kappa$.  Decoding assigns a regular
ultrafilter $U_c$ to each $c\in C$.  The lexicographic order of
characteristic functions linearly orders $C$ definably; no choice
of a particular tuple is made.

For finite $u=\{c_0<\cdots<c_{n-1}\}\subseteq C$, take the
Fubini ultrafilter $D_u$ on $J^u$ in this order:
\[
 X\in D_u
 \quad\Longleftrightarrow\quad
 U_{c_{n-1}}s_{n-1}\cdots U_{c_0}s_0\,
       ((s_0,\ldots,s_{n-1})\in X).
\]
Here $U_cs\,\varphi(s)$ means $\{s:\varphi(s)\}\in U_c$.
These products cohere when unused coordinates are added, and
therefore define an ultrafilter $D$ on the algebra of finitely
supported subsets of $J^C$.  For a structure $N$, let $F_N$ be
the set of all finitely supported functions $J^C\to N$ and put
$T(N)=F_N/D$.

The truth set of a formula on finitely supported functions
has finite support. For the existential step of \L o\'s's
theorem, choose witnesses as a function of the coordinates in the union
of the parameter supports. The witness function still has finite
support. Thus constant functions give an elementary
embedding $N\to T(N)$.

Moreover $T(N)$ realizes every finitely satisfiable type
over the diagonal $N$ with at most $\kappa$ formulas.
Write such a type as
\[
 p(y)=\{\varphi_\xi(y,\bar m_\xi):\xi<\nu\},
 \qquad \nu\leq\kappa.
\]
For each $s\in J$, choose $n_s\in N$ realizing the finite part
indexed by $s\cap\nu$. Fix a coordinate $c\in C$ and put
$f(z)=n_{z(c)}$. Its class realizes $p$, because the truth set for the
$\xi$-th formula contains the $D$-large cylinder $z(c)\in R_\xi$.
The choice of this coordinate and these witnesses is needed only to
verify realization. The operator $T$ uses all finitely supported functions.

Starting with $N_0=\mathbb N$, iterate this fixed operation
for $\kappa^+$ stages, taking canonical direct limits at limit
stages.  Quotients, embeddings, and direct limits are uniformly
definable, so the final model $N_{\kappa^+}$ is OD from $\kappa$.
Identify stages with their images.  Every parameter set of size
below $\kappa^+$ is contained in some $N_\alpha$,
$\alpha<\kappa^+$, by regularity.  A type over it has at most
$\kappa$ formulas.  By elementarity each finite part is
realized in $N_\alpha$, and the one-step property realizes
the whole type in $N_{\alpha+1}$.  Thus $N_{\kappa^+}$ is
$\kappa^+$-saturated.

Put $\Lambda=2^\kappa$.  If $|N|\leq\Lambda$, then
\[
 |F_N|\leq|[C]^{<\omega}|\cdot|N|^\kappa
       \leq\Lambda\cdot\Lambda^\kappa=\Lambda.
\]
The direct limits through $\kappa^+\leq\Lambda$ stages satisfy
the same bound. For the reverse inequality, fix a coordinate $c$.
For $X\subseteq\kappa$ and
$s=\{\xi_0<\cdots<\xi_{n-1}\}\in J$, put
\[
 f_X(s)=\sum_{\substack{j<n\\\xi_j\in X}}2^j.
\]
If $X\ne Y$ and $\xi\in X\mathbin\triangle Y$, then
$f_X(s)\ne f_Y(s)$ for every $s\in R_\xi\in U_c$.
The functions $z\mapsto f_X(z(c))$, each supported on one coordinate, give
$2^\kappa$ distinct elements already in $T(\mathbb N)$.
Elementary embeddings preserve their distinctness, proving
$|N_{\kappa^+}|=2^\kappa$. Definably relabel the diagonal images of
standard integers by those integers and tag all other elements. The
result is an elementary extension that contains $\mathbb N$ itself.
\end{proof}

\begin{corollary}\label{cor:gch-saturation}
If $\kappa$ is singular strong limit and $2^\kappa=\kappa^+$,
there is an OD saturated elementary extension of $\mathbb N$
of cardinality $\kappa^+$.
In particular, GCH gives such a model at the successor of every
singular cardinal, including $\aleph_{\omega+1}$.
\end{corollary}

\begin{proof}
Apply \cref{thm:definable-saturation}.  Under GCH every singular
cardinal is strong limit and satisfies $2^\kappa=\kappa^+$.
\end{proof}

The model in \cref{cor:gch-saturation} is unique up to isomorphism
among saturated models of $\operatorname{Th}(\mathbb N)$ of that
cardinality. The structure is ordinal-definable, but its domain need
not be pointwise OD.

\section{The relative Feferman--Levy model}\label{sec:feferman-levy}

We use the finite-stage filtration from \cite{GlazerMO} and include the
original ground in the relative construction. We identify the resulting
model with the class defined by ambient hereditary definability. Finite
collapse descent then proves countable choice for finite sets and shows
that every infinite set is Dedekind-infinite.

\subsection{The model and its finite stages}

Begin with $W_0=L$, and put
\[
 \kappa_i=\aleph_i^{W_0},\qquad
 P_i=\operatorname{Fn}(\omega,\kappa_i,{<}\omega),\qquad
 P=\prod_{i<\omega}^{\mathrm{fin}}P_i.
\]
Let $G$ be $P$-generic and work in the ambient ZFC universe $V=W_0[G]$.
Define
\[
 N_k=W_0[G\mathbin{\upharpoonright}k],\quad
 R_k=\mathcal P(\omega)^{N_k},\quad
 \mathcal A=\langle R_k:k<\omega\rangle,\quad
 \mathcal R=\bigcup_{k<\omega}R_k.
\]
Write
\begin{equation}\label{eq:fl-relative-model}
 M_0=W_0(\mathcal R,\mathcal A)
\end{equation}
for the least transitive inner model of ZF containing $W_0$ and the
sets $\mathcal R,\mathcal A$. We will identify it with
\begin{equation}\label{eq:fl-hereditary-model}
 \begin{split}
 \mathcal D_{\mathrm{FL}}
   &=\{x:\exists r\in\mathcal R\quad
                    x\in\mathrm{OD}^{V}_{\mathcal A,r}\},\\
 S_{\mathrm{FL}}
   &=\{x:\operatorname{tc}(\{x\})\subseteq\mathcal D_{\mathrm{FL}}\}.
 \end{split}
\end{equation}
The term \emph{relative Feferman--Levy model} refers throughout to $M_0$.
All definability in \eqref{eq:fl-hereditary-model} is computed in the
ambient $V$, with $\mathcal A$ as an allowed parameter.
All conclusions below also hold when the original ground $W_0$ satisfies
$\mathsf{ZFC}+\mathsf{GCH}+\mathsf{GA}+(V=\operatorname{HOD})$.
We give the proof in this form, using \cref{lem:stable-ground-codes}.
Intermediate stages are not assumed to satisfy GA or $V=\operatorname{HOD}$.

Use the ambient evaluation map of \cref{eq:definition-evaluation},
with anchor $\mathcal A$ and allowed reals $\mathcal R$:
\[
 \operatorname{Eval}_{\mathcal A}:
       \Ord\times\mathcal R\rightharpoonup V,
 \qquad \operatorname{Eval}_{\mathcal A}(\alpha,r)
       =\operatorname{Val}_{\mathcal A}(r,\alpha),
\]
whose values at $r$ are exactly $\mathrm{OD}^{V}_{\mathcal A,r}$.

\begin{lemma}[Stage facts]\label{lem:fl-stage-facts}
\begin{enumerate}[label=(\roman*)]
 \item $S_{\mathrm{FL}}$ is a transitive inner model of ZF containing
 $\mathcal A,\mathcal R$, and every member of $\mathcal R$.
 \item $\mathcal P(\omega)^{S_{\mathrm{FL}}}=\mathcal R$, and each
 $R_k$ is countable in $S_{\mathrm{FL}}$.
 \item Every $N_k\subseteq S_{\mathrm{FL}}$ has a real presentation
 $N_k=W_0[u]$, with $u\in R_k$. In $V$ the class $N_k$ is uniformly
 definable from $R_k$ and the ambiently definable original ground $W_0$.
 \item For $k\geq1$,
 \[
  \omega_1^{N_k}=\kappa_k,\qquad
  N_{k+1}\text{ is a }
  \Coll(\omega,\omega_1^{N_k})\text{-extension of }N_k.
 \]
\end{enumerate}
\end{lemma}

\begin{proof}
Finitely many reals in $\mathcal R$ lie in one finite stage, which
contains their tuple code. Since $\mathcal R$ is definable from
$\mathcal A$, \cref{lem:definition-inner-model} gives ZF and all
members of $\mathcal R$. The sets $\mathcal A,\mathcal R,R_k$
are definable from $\mathcal A$; their other transitive constituents
are built from allowed reals and natural numbers. Hence they also
belong to $S_{\mathrm{FL}}$, proving (i).

Every ground set is ambient OD by \cref{lem:stable-ground-codes}.
A finite tuple of collapse generics has a real code. For a generic
surjection $f:\omega\to\kappa_i$, code the relation
$n\preccurlyeq m$ iff $f(n)\leq f(m)$. The ordinal ranks of its quotient
well-order recover $f$. Joining finitely many such codes gives
$N_k=W_0[u]$ with $u\in R_k$; for $k=0$ use the zero real.
The stable ground codes make every object of $N_k$ ambiently definable
from $u$ and ordinals. Since $N_k$ is transitive, $N_k\subseteq
S_{\mathrm{FL}}$.

More precisely, relative constructibility over the definable class
$W_0$, including the transitive closure of the added parameter, gives
\begin{equation}\label{eq:fl-stage-definition}
 N_k=W_0(R_k)=\bigcup_{g\in R_k}W_0[g].
\end{equation}
Each class on the right is contained in $N_k$, and one of them is
$W_0[u]=N_k$. This also supplies a uniform ambient definition of $N_k$
from $R_k$ and $W_0$, proving (iii).

If a real $x\in S_{\mathrm{FL}}$ is ambiently defined from
$\mathcal A,r$, take $r\in N_k$. The rowwise automorphisms of the
remaining product preserve the canonical filtration name and $r$.
Tail weak homogeneity decides every bit of $x$ over $N_k$, so
$x\in R_k$. Conversely every real in $\mathcal R$ belongs to the
hereditary model. For $k\geq1$, the first $k$ rows have ground size
$\kappa_{k-1}$, and their real names number at most
$2^{\kappa_{k-1}}=\kappa_k$ in $W_0$. Thus $R_k$ has a well-order of
size at most $\kappa_k$ in $N_k$. Row $k$ makes this well-order countable,
giving an enumeration in $N_{k+1}\subseteq S_{\mathrm{FL}}$.
Similarly $R_0$ is countable
in $N_2$, since its ground cardinality is $\kappa_1$. This proves (ii).

Finally, the first $k$ rows preserve $\kappa_k$ by their
$\kappa_k$ chain condition, and their last row makes
$\kappa_{k-1}$ countable. Hence $\omega_1^{N_k}=\kappa_k$. The next row is
precisely $\operatorname{Fn}(\omega,\kappa_k,{<}\omega)$, proving (iv).
\end{proof}

\subsection{Recovering the relative construction}

We reconstruct ambiently defined relations from sets of forcing data
in the original ground. The argument requires no definition of the
ground or of ambient truth inside an intermediate ZF model.

\begin{lemma}[Relations on definition codes]\label{lem:fl-code-relations}
Let $T$ be any transitive ZF inner model of $V$ containing $W_0$ and
$\mathcal A$. Let $\delta$ be an ordinal and $d$ a positive integer.
If $B\subseteq(\delta\times\mathcal R)^d$ is definable in $V$ from
$\mathcal A$, ordinals, and one real $z\in\mathcal R$, then $B\in T$.
\end{lemma}

\begin{proof}
First $\mathcal R\in T$ and every $N_k\subseteq T$. Indeed a real
in $R_k$ codes the actual finite generic $G\restr k$, and evaluation
of ground names from this generic takes place in any such $T$.
Choose $m$ with $z\in N_m$, retain the actual filter $g_m=G\restr m$
as a parameter in $T$, and fix a ground $P_{<m}$-name $\dot z$ for $z$.
Here $P_{<k}=\prod_{i<k}P_i$.

Fix a formula $\varphi$ defining membership in $B$, with its ordinal
parameters understood. In $W_0$, choose for each $k$ the set
$\mathcal Q_k$ of all nice $P_{<k}$-names forced to be reals, and the
set $\mathcal E_k$ of all dense subsets of $P_{<k}$ in $W_0$.
For $k\geq m$, $\vec\alpha\in\delta^d$, and
$\vec\sigma\in\mathcal Q_k^d$, compute in $\RO(P)^{W_0}$ the value
\[
 b_{k,\vec\alpha,\vec\sigma}
   =\bigl\|\varphi(\check{\vec\alpha},\vec\sigma,
                          \dot{\mathcal A},\dot z)\bigr\|.
\]
This value belongs to the initial complete subalgebra
$\RO(P_{<k})^{W_0}$. To see this, the rowwise tail automorphisms fix
all the displayed parameters, including the filtration name. Their
weak homogeneity implies that whenever $(p,q)$ decides the formula,
its initial component $(p,1)$ decides it the same way. Thus the
Boolean value is a join of initial conditions.

The entire sequence of these tables, together with the sequences
$(\mathcal Q_k)$ and $(\mathcal E_k)$ and the forcing presentations,
is one set in $W_0$, hence a parameter available in $T$. Represent
the initial Boolean values by regular open subsets of $P_{<k}$.
Their evaluation under a generic filter $H$ is then the test
$H\cap b_{k,\vec\alpha,\vec\sigma}\ne\varnothing$.

Inside $T$, consider filters $H\subseteq P_{<k}$ such that
\begin{enumerate}[label=(\roman*),beginpenalty=10000]
 \item $H$ meets every member of $\mathcal E_k$;
 \item $H\restr m=g_m$;
 \item for every $j\leq k$, the values of $\mathcal Q_j$ under
 $H\restr j$ form exactly $\mathcal A(j)$.
\end{enumerate}
All these are tests on sets in $T$. In particular, (i) expresses
actual $W_0$-genericity without an internal predicate for $W_0$.
The actual filter $G\restr k\in T$ satisfies all three conditions.

Every filter satisfying these conditions has $W_0[H]=N_k$.
Both extensions have the real set $R_k$, and each is generated over
$W_0$ by a real coding its finite collapse generic. Each generating
real therefore belongs to the other extension, proving equality.
Consequently the actual tail $G_{\geq k}$ is generic over $W_0[H]$,
and
\[
 W_0[H][G_{\geq k}]=V.
\]
This new full generic presentation gives exactly the same filtration
$\mathcal A$. Condition (iii) gives equality at stages through $k$,
and equality of the $k$th models gives equality at all later stages.
The presentation also gives the same $z$, by (ii).

For a tuple $((\alpha_i,r_i):i<d)\in(\delta\times\mathcal R)^d$,
membership in $B$ is therefore equivalent to the following test in $T$:
there exist $k\geq m$, a filter $H$ satisfying (i)--(iii), and
$\vec\sigma\in\mathcal Q_k^d$ such that
\[
 \sigma_i^H=r_i\quad(i<d),\qquad
 H\cap b_{k,\vec\alpha,\vec\sigma}\ne\varnothing.
\]
The preceding comparison shows that every witness certifies membership
in $B$.
Conversely, the actual finite generics and their real names supply
witnesses for every tuple in $B$. Separation in $T$ proves $B\in T$.
No sequence of choices of the filters $H$ is used.
\end{proof}

\begin{theorem}[Relative model identification]\label{thm:fl-relative-identification}
In the stated ground-model setting,
\[
 W_0(\mathcal R,\mathcal A)=S_{\mathrm{FL}}.
\]
GCH is not needed for this identification; GA and $V=\operatorname{HOD}$
are used only to supply the stable ground codes of
\cref{lem:stable-ground-codes}.
\end{theorem}

\begin{proof}
By \cref{lem:fl-stage-facts}, $S_{\mathrm{FL}}$ is a transitive ZF
inner model containing $W_0$ and $\mathcal A$. We show that it is
contained in every such inner model $T$.

Take $x\in S_{\mathrm{FL}}$, put $Y=\operatorname{tc}(\{x\})$,
and fix an ambient definition of $x$ from $\mathcal A,z$ and ordinals.
Every $y\in Y$ has an evaluation code $(\alpha,r)$ with
$r\in\mathcal R$. The least possible ordinal codes, as $y$ ranges
over $Y$, are bounded by some ordinal $\delta$. In $V$ define
\[
 C=\{(\alpha,r)\in\delta\times\mathcal R:
             \operatorname{Eval}_{\mathcal A}(\alpha,r)\in Y\},
\]
where membership requires that the evaluation be defined.
On $C$ let $E$ mean equality of evaluation values and let $F$ mean
membership between them. These three relations are ambiently
definable from $\mathcal A,z$ and ordinals, by expanding $Y$ using the
chosen definition of $x$. By \cref{lem:fl-code-relations},
$C,E,F\in T$.

Form $C/E$ and its induced membership relation in $T$. External
evaluation identifies this quotient with the transitive set $Y$,
so its relation is well-founded and extensional, also as computed
in $T$. Its Mostowski collapse in $T$ is therefore the actual $Y$.
In particular $x\in T$. This proves the required minimality and
the asserted equality. The reconstruction uses neither GCH nor
an internal definition of $W_0$; stable ground codes ensure that
$S_{\mathrm{FL}}$ itself contains the whole ground.
\end{proof}

\subsection{Localization and stage descent}

The next lemmas transfer definitions between the ambient universe and
finite ZFC stages. They do not require the hereditary model to compute
the mantle.

\begin{lemma}[Tail localization]\label{lem:fl-tail-localization}
Let $z,S\in N_k$. Suppose $B\subseteq S$ is defined in $V$ by a fixed
formula from $\mathcal A,z$, ordinals, and specified parameters in
$N_k$. Then $B\in N_k$, with a definition there from those parameters
and $R_0,\ldots,R_{k-1}$. If $S$ is definable from the other parameters
in $N_k$, it need not be retained as an additional parameter.
\end{lemma}

\begin{proof}
Factor the product after its first $k$ rows. In $N_k$ the original
ground is its mantle, by \cref{lem:stable-ground-codes}; the tail forcing
is therefore definable from $k$ and its ground cardinal sequence.
The filtration name uses check names for $R_0,\ldots,R_{k-1}$,
the intrinsic set of reals of $N_k$ at position $k$, and the canonical
real-set names for later finite tail extensions. Every rowwise tail
automorphism preserves this name and the specified ground parameters.
Consequently weak homogeneity decides the formula at the top condition
for each $s\in S$. Otherwise, conditions forcing opposite answers
could be moved to compatible conditions without changing the parameters.
Separation in $N_k$, using the forcing relation for the fixed formula,
gives $B$ with the asserted definition. Any abbreviations involving
ambient definability or evaluation are expanded before taking this
forcing relation.
\end{proof}

\begin{lemma}[Lifting a stage definition]\label{lem:fl-lift}
A set ordinal-definable in $N_k$ from a real $u\in R_k$ and
$R_0,\ldots,R_{k-1}$ belongs to $\mathrm{OD}^{V}_{\mathcal A,u}$.
\end{lemma}

\begin{proof}
In $V$, replace $N_k$ by the uniformly definable class in
\eqref{eq:fl-stage-definition}, put $R_i=\mathcal A(i)$, and
relativize the fixed defining formula to that class. The original
ground is ambiently definable and $k$ is an ordinal parameter.
\end{proof}

For $k\geq1$ set
\[
 D_k=\{g\in R_k:R_k\subseteq W_0[g]\}.
\]
This nonempty set is intrinsic to $N_k$, using its definable original
ground. Its members are exactly the maximal presentations
$N_k=W_0[g]$. Indeed, a real generating $N_k$ belongs to $R_k$, and the defining
inclusion for $D_k$ puts that real in $W_0[g]$.

\begin{lemma}[Stage lowering]\label{lem:fl-stage-lowering}
Fix $m\geq1$ and $z$ with $N_m=W_0[z]$. Let
$X\in\mathrm{OD}^{V}_{\mathcal A,z}$ be finite and nonempty.
Suppose $k>m$ and every member of $X$ belongs to
$\mathrm{OD}^{V}_{\mathcal A,r}$ for some $r\in R_k$.
Then, for every maximal presentation $N_{k-1}=W_0[u]$,
\[
 X\subseteq\mathrm{OD}^{V}_{\mathcal A,u}.
\]
\end{lemma}

\begin{proof}
Choose $g_0\in D_k$. Stable ground codes make every real of $R_k$
ordinal-definable in $N_k=W_0[g_0]$ from $g_0$; the definition lifts
by \cref{lem:fl-lift}. Thus every member of $X$ is ambiently definable
from $\mathcal A,g_0$ and ordinals. Choose a finite ordinal list
$\alpha_0,\ldots,\alpha_{d-1}$ such that
\[
 \{\operatorname{Eval}_{\mathcal A}(\alpha_i,g_0):i<d\}=X.
\]
In $V$, form
\[
 B=\{(i,g)\in d\times D_k:
       \operatorname{Eval}_{\mathcal A}(\alpha_i,g)
                         \text{ is defined and belongs to }X\},
\]
and, for $(i,g),(j,h)\in B$, put
\[
 (i,g)\mathrel E(j,h)\quad\Longleftrightarrow\quad
 \operatorname{Eval}_{\mathcal A}(\alpha_i,g)
       =\operatorname{Eval}_{\mathcal A}(\alpha_j,h).
\]
Expand $X$ by its original definition from $\mathcal A,z$ and
ordinals, without using $X$ as a ground parameter. Expand the evaluation
and $D_k$ definitions as well. Applying
\cref{lem:fl-tail-localization} to subsets of $d\times R_k$ and its
square puts $B,E$ in $N_k$, ordinal-definably from
$z,R_0,\ldots,R_{k-1}$. Here $R_k$ and $D_k$ are intrinsic to the stage.

The quotient partition $\mathcal P=B/E$ belongs to $N_k$ and has
exactly $|X|$ nonempty cells. Each cell has one evaluation value, and
the values at $g_0$ show that every member of $X$ occurs.
The quotient cells are the same in $N_k$ and $V$, because both use
the same sets $B,E$ and all their elements. In particular this is a
finite family of sets of coded reals in $N_k$, even if its evaluation
values lie outside $N_k$.

All the parameters $z,R_0,\ldots,R_{k-1}$ belong to
$N_{k-1}=W_0[u]$. Its stable ordinal codes make $\mathcal P$ an
$\mathrm{OD}(u)$ family in $N_k$. The latter is a collapse extension
of $N_{k-1}$ at its $\omega_1$, by \cref{lem:fl-stage-facts}.
Homogeneity gives an invariant name for $\mathcal P$, and
\cref{thm:finite-collapse-invariant} gives invariant names enumerating
its cells. In $N_k$, the preceding ground $W_0[u]$ and its names are
definable from $u$ and ordinals by \cref{lem:stable-ground-codes}.
Thus \cref{lem:invariant-od} makes each cell $\mathrm{OD}(u)$ in $N_k$.
This uses no GA or HOD assumption on $N_{k-1}$.

Apply \cref{lem:fl-lift} to the definition of a cell. In $V$ the cell
defines its unique common evaluation value, using the fixed ordinal list
$(\alpha_i)_{i<d}$. No real representative of the cell is selected.
These values exhaust $X$, proving the conclusion.
\end{proof}

\subsection{Uniform parameters, finite choice, and infinite sets}

\begin{theorem}[Uniform finite-family definability]\label{thm:fl-uniform-real}
If $\langle F_n:n<\omega\rangle\in S_{\mathrm{FL}}$ is a sequence
of finite sets, there is one real $z\in\mathcal R$ such that
\[
 \bigcup_{n<\omega}F_n\subseteq\mathrm{OD}^{V}_{\mathcal A,z}.
\]
\end{theorem}

\begin{proof}
The whole sequence has an ambient definition from $\mathcal A$,
ordinals, and one real. Choose $m\geq1$ containing that real and a
maximal presentation $N_m=W_0[z]$. By stable ground codes and
\cref{lem:fl-lift}, the original real is ambiently definable from
$\mathcal A,z$ and ordinals. The sequence therefore has such a
definition too.

Fix $n$ with $F_n\ne\varnothing$. Its finitely many members belong
to the hereditary model; choose their real definition parameters
and one stage $k\geq m$ containing them. If $k>m$, apply
\cref{lem:fl-stage-lowering} repeatedly, keeping the original
definition of $F_n$ from $\mathcal A,z$ at every step. This finite
descent reaches $R_m$. Every real there is ordinal-definable from
$z$ in $N_m$, and its definition lifts. Thus
$F_n\subseteq\mathrm{OD}^{V}_{\mathcal A,z}$. For each $n$ only
finitely many choices and descent steps are used; no sequence of
stage bounds is chosen.
\end{proof}

\begin{corollary}\label{cor:fl-finite-choice}
$S_{\mathrm{FL}}\models\mathsf{AC}^{\mathrm{fin}}_\omega$.
\end{corollary}

\begin{proof}
For a sequence of nonempty finite sets take $z$ from
\cref{thm:fl-uniform-real}. In $V$, select from each set the value
with least ordinal evaluation code from $(\mathcal A,z)$.
The sequence itself has an allowed definition, so the resulting
choice function belongs to $\mathcal D_{\mathrm{FL}}$.
Its entries and their transitive closures are already in
$S_{\mathrm{FL}}$; hence the function belongs to $S_{\mathrm{FL}}$.
Thus the ambient selection defines a function in $S_{\mathrm{FL}}$
without an internal application of Replacement to ambient evaluation.
\end{proof}

\begin{theorem}\label{thm:fl-dedekind-infinite}
Every infinite set in $S_{\mathrm{FL}}$ has a countably infinite
subset in $S_{\mathrm{FL}}$.
\end{theorem}

\begin{proof}
Take $x\in S_{\mathrm{FL}}$ infinite and $z\in\mathcal R$ such that
$x\in\mathrm{OD}^{V}_{\mathcal A,z}$. Define in $V$
\[
 X_k=\{a\in x:\exists r\in R_k\quad
                       a\in\mathrm{OD}^{V}_{\mathcal A,z,r}\}.
\]
These sets cover $x$. Their definitions are uniform from
$\mathcal A,z$ and the definition of $x$, and all their elements
belong to $S_{\mathrm{FL}}$. Thus both the sets $X_k$ and their sequence
belong to $S_{\mathrm{FL}}$.

If some $X_k$ is infinite, choose one enumeration
$e:\omega\to R_k$ in $S_{\mathrm{FL}}$. For each $a\in X_k$, take
the least $i$ for which $a$ is ambiently definable from
$(\mathcal A,z,e(i))$, followed by its least ordinal definition code
from these parameters. Lexicographic comparison of these pairs
well-orders $X_k$, since it is an ordinal sum of well-orders of codes.
The relation is ambiently definable from $\mathcal A,z,e$; substitute
an allowed definition of $e$ and join its real parameter with $z$.
Its elements are built from members of $x$, so the well-order belongs
to $S_{\mathrm{FL}}$. Its first $\omega$ elements, formed inside that
ZF model, give the required injection into $x$.

Otherwise every $X_k$ is finite. Inside $S_{\mathrm{FL}}$, the set
of linear orders on $X_k$ is nonempty and finite, with its unique
empty order when $X_k=\varnothing$. Apply
\cref{cor:fl-finite-choice} to choose one order for each $k$.
Listing the sets $X_k$ in these orders gives a partial map on
$\omega\times\omega$ with range $x$. Retaining first occurrences
in a fixed enumeration gives an injection $\omega\to x$, since
$x$ is infinite.
\end{proof}

By \cref{thm:fl-relative-identification}, both choice conclusions
hold in the relative Feferman--Levy model
$M_0=W_0(\mathcal R,\mathcal A)$.


\section{Hereditary definability and choice}
\label{sec:add-kappa-lambda}

We now consider hereditary definability in extensions obtained by
adding generalized Cohen subsets. The parameter bound is the successor
of the closure cardinal, while the number of columns may be larger.

\providecommand{\Sstar}{S^*}

\begin{definition}\label{def:od-less-lambda}
For regular $\lambda$, write
$\mathcal D_\lambda=\mathrm{OD}_{H_\lambda}^V$, also denoted
$\mathrm{OD}_{<\lambda}$ in this setting. Its hereditary part is
$\operatorname{HOD}_{H_\lambda}^V$, with the parameter convention
of \cref{subsec:definition-codes}.
\end{definition}

Throughout the generalized Cohen results, unless another width is
specified, let $\kappa$ be regular uncountable in $L$, let
$\Lambda>\kappa$ be a cardinal of $L$, and put
\begin{equation}\label{eq:generalized-direct-setting}
\begin{gathered}
 P=\Add(\kappa,\Lambda)^L,\qquad V=L[G],\qquad
 \lambda=(\kappa^+)^L=\kappa^+,\\
 A=\mathcal P(\kappa)^V,\qquad
 \Sstar=\operatorname{HOD}_{H_\lambda^V}^{V},\qquad
 \mathcal D=\mathrm{OD}_{H_\lambda^V}^{V}.
\end{gathered}
\end{equation}
This is the setting of \cref{subsec:common-cohen-category}, since
$L\models\kappa^{<\kappa}=\kappa$; we retain its forcing notation.

As in \cref{lem:stable-ground-codes}, the proofs of the hereditary-model
and arithmetic results work over a fixed
$W_0\models\mathsf{ZFC}+\mathsf{GCH}+\mathsf{GA}+(V=\mathrm{HOD})$,
using its stable ordinal codes. Taking $W_0=L$ gives the stated results.
The support and ultrafilter theorems require only a GCH ground.

\subsection{The direct hereditary-definability theorem}

\begin{theorem}\label{thm:generalized-direct-inner-model}\label{lem:gce-small-codes}
In the setting \eqref{eq:generalized-direct-setting}, the class
$\Sstar$ is an inner model of $\mathsf{ZF}$. Every member
of $H_\lambda^V$ belongs to an original coordinate extension on at
most $\kappa$ columns. Moreover:
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item $L(\mathcal P(\kappa)^V)\subseteq\Sstar$ and
$\mathcal P(\kappa)^{\Sstar}=\mathcal P(\kappa)^V$.
\item Every function in $V$ with domain an ordinal at most $\kappa$ and
range contained in $\Sstar$ belongs to $\Sstar$.
\item No uniform ultrafilter on $\kappa$ belongs to $\mathcal D$.
In particular, $\Sstar$ has no uniform ultrafilter on $\kappa$.
\end{enumerate}
\end{theorem}
\begin{proof}
\Cref{lem:cohen-category-closure} gives ZF, parameter capture, (i),
and (ii). Assertion (iii) is the following theorem.
\end{proof}

\begin{theorem}\label{thm:generalized-no-uniform-ultrafilter}
Let $W_0\models\mathsf{ZFC}+\mathsf{GCH}$, let $\kappa$ be regular
uncountable in $W_0$, and let $\Lambda>\kappa$ be any ground cardinal.
In $V=W_0[G]$, where $G$ is $\Add(\kappa,\Lambda)^{W_0}$-generic,
no uniform ultrafilter on $\kappa$ belongs to
$\mathrm{OD}_{H_{\kappa^+}}^V$.
\end{theorem}
\begin{proof}
Put $\lambda=\kappa^+$. Suppose $U$ is a uniform ultrafilter on $\kappa$ defined from
$z\in H_\lambda$ and ordinal parameters.  Choose a name for $z$ supported
on a ground set $E$ of at most $\kappa$ columns, as in
\cref{lem:generalized-support}(ii).  Choose $p\in G$ forcing the definition
to have a unique value that is a uniform ultrafilter, and enlarge $E$ to
include the columns of $p$.  Fix $\beta\in\Lambda\setminus E$ and let
$c_\beta$ be the generic subset of $\kappa$ on that column.

Choose $q\in G$, $q\leq p$, deciding whether $c_\beta$ or its complement
belongs to the named ultrafilter; call the selected name $\dot A$.
Put
\[
 D_q=\{\xi<\kappa:(\beta,\xi)\in\operatorname{dom}(q)\}.
\]
This ground set has size less than $\kappa$.  The ground forcing
automorphism $\tau$ that flips precisely the bits
$(\beta,\xi)$ with $\xi\notin D_q$ fixes $q$, $p$, and the parameter
name.  By uniqueness of the definition, it fixes the ultrafilter name
modulo forced equality below $p$.  Hence
\[
 q\Vdash \dot A\in\dot U\ \wedge\ \tau\dot A\in\dot U.
\]
Outside $D_q$ these two sets are complementary.  Their intersection is
therefore forced to have size less than $\kappa$, contrary to uniformity.

\end{proof}

\begin{corollary}\label{cor:generalized-small-closure}
In the setting \eqref{eq:generalized-direct-setting}, every ambient
set of at most $\kappa$ elements of $\Sstar$ belongs to
$\Sstar$ and has an enumeration in $\Sstar$.  The model $\Sstar$ satisfies
$\mathsf{DC}_\kappa$.  Every nontrivial Boolean algebra in $\Sstar$ of
ambient cardinality at most $\kappa$ has an ultrafilter in $\Sstar$,
although $\Sstar$ fails the Boolean prime ideal theorem.
\end{corollary}
\begin{proof}
Choose an ambient enumeration and apply
\cref{thm:generalized-direct-inner-model}(ii); take its range inside
$\Sstar$.  The same argument puts any ambient dependent-choice sequence
of length $\kappa$ into $\Sstar$. This also proves the version in which
each choice depends on the entire preceding sequence, since all
shorter sequences of members of $\Sstar$ are available there.

For a small Boolean algebra choose an ultrafilter in $V$ and enumerate it.
Its members lie in $\Sstar$, so the enumeration and its range do too.
Finally, subsets of $\kappa$ of size less than $\kappa$ are the same in
$V$ and $\Sstar$. One direction is absolute,
and the other follows by putting an ambient short enumeration into
$\Sstar$.  The co-$<\kappa$ filter is consequently a proper filter in
$\Sstar$.  If the Boolean prime ideal theorem held there, this filter
would extend to a uniform ultrafilter, contradicting
\cref{thm:generalized-direct-inner-model}(iii).
\end{proof}

\subsection{Sharpness at the next cardinal}

The next theorem shows that the small-family bound cannot be increased.
It holds over every GCH ground, without a ground definability assumption.

\begin{theorem}\label{thm:generalized-sharpness}
Let $W_0\models\mathsf{ZFC}+\mathsf{GCH}$, let $\kappa$ be regular
uncountable in $W_0$, and put $\lambda=\kappa^+$. In
$V=W_0[G]$, where $G$ is $\Add(\kappa,\lambda)^{W_0}$-generic,
the family $\mathscr U_\kappa$ of all uniform ultrafilters on $\kappa$ is
ordinal-definable and satisfies
\[
 |\mathscr U_\kappa|^V=\lambda^+,
 \qquad
 \mathscr U_\kappa\cap\OD_{H_\lambda}^{V}=\varnothing.
\]
In particular, over $L$ the conclusion of
\cref{thm:generalized-small-family} fails at the next cardinal
after $\lambda$.
\end{theorem}
\begin{proof}
The family is definable from the ordinal $\kappa$. The absence
of its members from $\OD_{H_\lambda}^V$ is exactly
\cref{thm:generalized-no-uniform-ultrafilter}.

For every infinite $\kappa$ in any ZFC universe, the following
argument gives $|\mathscr U_\kappa|=2^{2^\kappa}$.
There is a family $(A_x)_{x\subseteq\kappa}$ of subsets of a
set of size $\kappa$ such that every finite Boolean combination
of distinct members has size $\kappa$. Take the underlying set
\[
 I=\{(s,H,\gamma):s\in[\kappa]^{<\omega},
       H\subseteq\mathcal P(s),\ \gamma<\kappa\}
\]
and put
\[
 A_x=\{(s,H,\gamma)\in I:x\cap s\in H\}.
\]
For finitely many distinct $x$'s, choose finite $s$ distinguishing
their restrictions. Choosing $H$ gives any desired membership
pattern, and varying $\gamma$ gives $\kappa$ points with that
pattern. Since $|I|=\kappa$, transport the family to $\kappa$.

Every binary assignment to this $2^\kappa$-sized family, together
with the co-$<\kappa$ filter, generates a proper filter. Extend
it to an ultrafilter using choice in $V$. It is uniform because
it contains the co-$<\kappa$ filter, and distinct assignments
give distinct ultrafilters. The reverse bound counts all subsets
of $\mathcal P(\kappa)$, so
\[
 |\mathscr U_\kappa|=2^{2^\kappa}.
\]

The forcing preserves cardinals. Its $<\kappa$-closure preserves those
at most $\kappa$, and its $\kappa^+$-chain condition preserves those at
least $\kappa^+$. Its $\lambda$ distinct columns and the nice-name
bound $(\lambda^\kappa)^{W_0}=\lambda$ give
$(2^\kappa)^V=\lambda$. Finally a nice name for a subset of
$\lambda$ is a subset of the fixed ground set of pairs consisting of
an ordinal below $\lambda$ and a condition. This set has size $\lambda$.
Ground GCH bounds the number of these names by
$(2^\lambda)^{W_0}=(\lambda^+)^{W_0}$. Cardinal preservation and
Cantor's theorem give the matching lower bound in $V$. Hence
\[
 |\mathscr U_\kappa|^V=2^{2^\kappa}=2^\lambda
 =\lambda^+,
\]
which proves the claim.
\end{proof}


\section{Saturation in generalized Cohen extensions}
\label{sec:generalized-category-obstruction}\label{sec:generalized-arithmetic}

\subsection{Copies of small definable structures}

For the following corollary, specialize
\eqref{eq:generalized-direct-setting} to width $\Lambda=\lambda=\kappa^+$.

\begin{corollary}\label{cor:add-structure-copy}
Every $\mathrm{OD}_{H_\lambda}$ structure of cardinality at most
$\lambda$ in a fixed countable language has an ambiently isomorphic
copy in $\Sstar$.
\end{corollary}
\begin{proof}
Its domain belongs to $\mathcal D$ and is pointwise in $\mathcal D$ by
\cref{thm:generalized-small-family}. Apply \cref{lem:add-copy}.
\end{proof}

The isomorphism is asserted in the ambient universe. In particular,
ambient cardinality and saturation pass to the copy; a well-order of
its domain need not belong to $\Sstar$.

\subsection{The generalized Cohen category environment}
\label{subsec:gce-setting}

Apply \cref{subsec:common-cohen-category} with
$\kappa>\omega$, width $\Lambda$, and $\mathcal S=\Sstar$.
The good bases form the set $D$ of
\cref{def:generalized-good-base}. For every ground Cohen tree
$\mathcal T\cong\kappa^{<\kappa}$, the associated ideals are closed under
$\kappa$ unions and satisfy the finite-product category theorem,
local decision, and conull uniformization into arbitrary codomains.
The next argument uses these properties on a tree of arithmetic residues.

\subsection{The generalized arithmetic obstruction}
\label{subsec:gca-main}

Return to the setting of \eqref{eq:generalized-direct-setting},
with arbitrary $\Lambda>\kappa$.  Saturation is computed in $V$;
$\kappa^+$-saturation realizes types over at most $\kappa$ parameters.

\begin{theorem}\label{thm:generalized-pa-obstruction}
Let $\kappa$ be regular uncountable in $L$, let $\Lambda>\kappa$
be any cardinal of $L$, and let
\[
 V=L[G],\qquad G\subseteq\Add(\kappa,\Lambda)^L
 \text{ be generic},\qquad
 \Sstar=\operatorname{HOD}_{H_{\kappa^+}^V}^{V}.
\]
No model of $I\Delta_0$ belonging to $\Sstar$ is $\kappa^+$-saturated in $V$.
There is no restriction on the cardinality of the model.
The same nonexistence statement holds for models belonging to
$L(\mathcal P(\kappa)^V)$.
\end{theorem}

We prove the theorem over the fixed stable ground $W_0$ from the
convention following \eqref{eq:generalized-direct-setting}.
We embed a ground arithmetic model into a hypothetical saturated model.
We work in their canonical exponential cuts, while realizing all types
in the original models. The category argument gives a small coinitial
family above a cofinal hull, which contradicts saturation.

\subsection{A ground arithmetic model and its cofinal hull}
\label{subsec:gca-hull}

Suppose that $N\in\Sstar$ is a $\kappa^+$-saturated model of
$I\Delta_0$.  Put $C=C(N)$, using
\cref{lem:canonical-exponential-cut}.  Arithmetic calculations below
take place in the EFA model $C$, in its conservative exponential
language with bounded induction.  No saturation of $C$ is assumed.
Write $\operatorname{Bit}(b,i)$ for
\[
 \exists u\leq b\ \exists v<2^i\,
       \bigl(b=(2u+1)2^i+v\bigr).
\]
Bounded induction proves division with remainder and the exponent
laws. It also constructs the XOR of two strings below $2^m$.
Induct on $m$ with the bounded assertion that some $z<2^m$ has
the prescribed XOR digits at all $i<m$. At a successor step
append the next digit by retaining $z$ or adding $2^m$.
The same induction proves uniqueness from the digits, hence
associativity, self-inverseness, and compatibility with reduction
modulo a smaller power of two. These identities hold at every
internal position, including nonstandard ones. We will also use
\[
 b<2^d,\quad i\geq d\quad\Longrightarrow\quad
 \neg\operatorname{Bit}(b,i).
\]
All arithmetic searches below are explicitly bounded.

The complete theory $T_0=\operatorname{Th}(N)$ is a real.  The
forcing adds no reals, so $T_0\in W_0$. In $W_0$ choose a
$\kappa$-saturated $M_0\models T_0$ of cardinality $\kappa$.
To construct it, start with a model of size $\kappa$ and build an
elementary chain of length $\kappa$. At a successor stage realize every
type over every
parameter subset of size less than $\kappa$ from the preceding
model.  There are at most $\kappa$ such types, since
$\kappa^{<\kappa}=\kappa$ and the language is countable.
Compactness and downward L\"owenheim--Skolem give the next model
of size $\kappa$.  Regularity ensures that every small parameter
set in the union is contained in a stage.  The union is therefore
$\kappa$-saturated.  This construction and its choices take place in $W_0$.

Let $C_0=C(M_0)$, computed in $W_0$.  It is unchanged in $V$,
by absoluteness of satisfaction for the ground set-sized structure.
In $W_0$, for any $D\subseteq C_0$ of size less than $\kappa$, the type
\[
 \{T_n(x):n<\omega\}\ \cup\ \{a<x:a\in D\}
\]
is finitely satisfiable in $C_0$.  Saturation of $M_0$ realizes it
in $C_0$.  The analogous type above all standard numerals supplies
a nonstandard point.  Therefore $|C_0|=\operatorname{cf}(C_0)=\kappa$.

Enumerate $M_0$ in $V$ and use saturation of $N$ to extend partial
elementary maps, obtaining an elementary embedding
\[
 e:M_0\longrightarrow N.
\]
At each stage fewer than $\kappa$ previously chosen images occur
as parameters.  The graph of $e$ has $\kappa$ hereditary entries
in $\Sstar$, so it belongs to $\Sstar$ by
\cref{lem:gce-small-codes}.  Put $M=e[C_0]\subseteq C$.
This inclusion follows by transferring each standard formula $T_n$.
The image $M$ is closed under arithmetic and exponentiation, since
$e$ preserves the arithmetic operations and the graph $E$.
No elementarity between the two exponential cuts is needed.  Define
\begin{equation}\label{eq:gca-hull}
 K=\{b\in C:\exists m\in M\ (b<m)\},
 \qquad A_N=C\setminus K.
\end{equation}
This is a cut in $C$, and all its displayed objects belong to
$\Sstar$. The initial segment $K$ contains $M$ and is closed
under successor, addition, and $b\mapsto2^b$. Apply each of these
operations to bounds in $M$ to obtain the required new bounds.
Saturation of $N$ realizes
\[
 \{T_n(d):n<\omega\}\ \cup\ \{m<d:m\in M\}.
\]
Every finite fragment has a witness in $M$.  Fix a realization
$d\in C$.  Thus $d\in A_N$, and $2^d$ exists in $C$.
Every element of $A_N$ is above every element of $M$, and
\begin{equation}\label{eq:gca-half}
 h\in A_N\quad\Longrightarrow\quad
 \lfloor h/2\rfloor\in A_N.
\end{equation}
Indeed, if the half belonged to $K$, closure under addition and
successor would put $h\leq2\lfloor h/2\rfloor+1$ in $K$.

\subsection{The Cohen tree of coherent residues}
\label{subsec:gca-residue-tree}

In $W_0$, choose an increasing cofinal sequence
$(c_\alpha:\alpha<\kappa)$ in $C_0$, with $c_0$ nonstandard and
\begin{equation}\label{eq:gca-gaps}
 c_\beta-c_\alpha>n
 \qquad(\alpha<\beta<\kappa,\ n<\omega).
\end{equation}
To construct it, enumerate $C_0$ in order type $\kappa$.  At stage
$\beta$ require an element above the next enumerated element and
above $c_\alpha+n$ for every $\alpha<\beta$ and $n<\omega$.
Include every $T_n(x)$.  The resulting type has size below $\kappa$
and every finite part has a witness in $C_0$.  Saturation of $M_0$
realizes it in $C_0$.  At stage zero include all inequalities asserting
nonstandardness.

Let $\mathcal T_{\rm res}$ consist of the sequences
$r=\langle r_\alpha:\alpha<\delta\rangle$, $\delta<\kappa$,
with entries in $M_0$ satisfying
\begin{equation}\label{eq:gca-residues}
 0\leq r_\alpha<2^{c_\alpha},\qquad
 r_\beta\bmod2^{c_\alpha}=r_\alpha
 \quad(\alpha<\beta<\delta).
\end{equation}
Every displayed residue lies in $C_0$ by its upper bound.  The
calculations take place in that EFA cut, and the tree order is extension.

\begin{lemma}\label{lem:gca-residue-tree}
In $W_0$, the tree $\mathcal T_{\rm res}$ is isomorphic to
$\kappa^{<\kappa}$.
\end{lemma}
\begin{proof}
Unions of increasing chains of length below $\kappa$ are nodes.
The tree has cardinality at most $\kappa$, since
$|M_0|=\kappa^{<\kappa}=\kappa$.
For a node $r$ of length $\delta$, its immediate successors are
the solutions in $M_0$ of
\[
 b<2^{c_\delta},\qquad
 b\bmod2^{c_\alpha}=r_\alpha\quad(\alpha<\delta).
\]
There are $\kappa$ solutions.  To prove the lower bound, append
inequalities excluding any prescribed set of fewer than $\kappa$
elements.  The type still has size below $\kappa$.  Its exponent values
are named parameters, so its congruences are formulas in the ordinary
arithmetic language of $M_0$.
A finite subsystem of the congruences reduces to the one at its
largest index $\alpha$.  If there are $n$ exclusions, the $n+1$ distinct elements
$r_\alpha+j2^{c_\alpha}$, $j=0,\ldots,n$, satisfy that congruence
and lie below $2^{c_\delta}$ by \eqref{eq:gca-gaps}.
At least one survives the exclusions.  If no congruence occurs,
nonstandardness of $c_\delta$ gives the same conclusion.
Saturation of $M_0$ realizes the full type; the upper bound puts the
realization in $C_0$.  Thus the solution set cannot
have size below $\kappa$, and its size is at most $|M_0|=\kappa$.

Choose a bijection from $\kappa$ onto each successor set in $W_0$.
Use them recursively at successor levels and take unions at
limit levels.  This gives the required tree isomorphism.
Surjectivity at a limit level follows from the coherent inverse
images of all proper initial segments.
\end{proof}

Put $X=[\mathcal T_{\rm res}]^V$.  The ground tree and its short
nodes are unchanged in $V$, and its branches are coded by subsets
of $\kappa$.  Apply the environment of
\cref{subsec:gce-tree-points,subsec:gce-traces,subsec:gce-uniformization}
to this tree.  Write $\mathcal I$ for its ideal on $X$ and
$\mathcal I_2$ for the ideal on $X^2$.
Nonempty cylinders and rectangles are positive, binary functions
admit local decisions, and \cref{thm:gce-ku} applies to every
relation in $\Sstar$.

\begin{lemma}\label{lem:gca-residue-lifts}
There is a function $F:X\to C$ in $\Sstar$ with $F(r)<2^d$
for every $r\in X$, and a set $X_0\subseteq X$ with
$X\setminus X_0\in\mathcal I$ such that for $r\in X_0$,
\begin{equation}\label{eq:gca-lift}
 F(r)<2^d,\qquad
 F(r)\bmod2^{e(c_\alpha)}=e(r_\alpha)
 \quad(\alpha<\kappa).
\end{equation}
\end{lemma}
\begin{proof}
For any branch $r$, the displayed conditions form a finitely
satisfiable ordinary-language type of size $\kappa$ in $N$, using
the displayed exponent values as named parameters.
For a finite subsystem, the residue at the greatest mentioned
index is itself a witness below $2^d$.
Saturation therefore gives a witness for each branch.  The bound
below $2^d\in C$ puts every witness in $C$ by initiality.
This defines a relation in $\Sstar$ with all sections nonempty,
since its ambient witnesses are elements of $C$ and therefore
belong to $\Sstar$.
Apply \cref{thm:gce-uniformization}, and extend its partial
uniformizer by $0$ off its domain. The bound $F(r)<2^d$ therefore
holds on all of $X$.
\end{proof}

Write $\operatorname{Ext}(r)(i)\in\{0,1\}$ for the truth value of
$C\models\operatorname{Bit}(F(r),i)$.
Coordinatewise bitwise XOR of residues makes $X$ a Boolean group.
Reduction modulo a smaller power of two commutes with XOR, so
the operation preserves coherence and translations act on the tree.
If $r,r+t\in X_0$, their residues determine the relation between
these two bit strings at \emph{every} position $i\in K$.
Choose $\alpha$ with $i<e(c_\alpha)$ and use
\eqref{eq:gca-lift} at that modulus.  The EFA identity for XOR
computes the bits throughout that interval in $C$, including
positions $i\notin M$.

\subsection{Translations and a bound above the cut}
\label{subsec:gca-local-bound}

For $\alpha<\kappa$, define ground branches $a_\alpha,a_\infty\in X$ by
\begin{equation}\label{eq:gca-translations}
 (a_\alpha)_\beta
   =2^{\min(c_\alpha,c_\beta)}-2^{c_0},\qquad
 (a_\infty)_\beta=2^{c_\beta}-2^{c_0},\qquad
 t_\alpha=a_\alpha+a_\infty.
\end{equation}
All these arithmetic calculations and the entire branches belong
to $W_0$.  The pattern $a_\alpha$ has ones on $[c_0,c_\alpha)$,
and $a_\infty$ has ones at positions at least $c_0$.
For every cylinder $U$, translation by $t_\alpha$ fixes $U$
for all sufficiently large $\alpha$. If $U$ specifies a node
of length $\delta$, then the $\beta$th residue of $t_\alpha$
is zero for $\beta<\delta\leq\alpha$.
These translations are ground tree automorphisms, so preserve
$\mathcal I$ by \cref{prop:gce-ideals}.

For $t=a_\alpha$ or $t=a_\infty$, let $\theta_t(i)$ denote in $C$
the characteristic function of $[e(c_0),e(c_\alpha))$ or of
$[e(c_0),\infty)$, respectively.  The residue identities give
\begin{equation}\label{eq:gca-bit-identity}
 \operatorname{Ext}(r+t)(i)
  =\operatorname{Ext}(r)(i)\mathbin\oplus\theta_t(i)
 \qquad(i\in K)
\end{equation}
whenever $r,r+t\in X_0$.
The union of the two exceptional sets where coding fails belongs to
$\mathcal I$.

\begin{lemma}[Local bound above the cut]\label{lem:gca-local-bound}
For such a $t$ and every nonempty basic cylinder $U$, there are
$H\in A_N$ and a nonempty basic $U'\subseteq U$ such that
\eqref{eq:gca-bit-identity} holds simultaneously at every $i<H$,
for $r\in U'$ outside one member of $\mathcal I$.
\end{lemma}
\begin{proof}
For each $r$, search for the least failure of
\eqref{eq:gca-bit-identity} at a position $i<d$, and take $d$
if there is none. The failure predicate is bounded in the
exponential language, with the two integer codes and interval
endpoints as parameters, so EFA supplies this bounded minimum.
When both codes are valid, the identity throughout $K$ puts
the resulting value in $A_N$. Replace it by $d$ if it belongs
to $K$. This defines a function $h:X\to A_N$ in $\Sstar$;
outside the coding exceptions the identity holds below $h(r)$.
Neither the external function $F$ nor the cut $K$ occurs in
the formula to which bounded minimization is applied.

Apply the general form of \cref{lem:cso-local-overspill} to this
function $h$. Its hypotheses are supplied by
\cref{prop:gce-ideals,cor:gce-local-decision,thm:gce-ku} and
\eqref{eq:gca-half}. Removing the coding exceptions gives the claim.
\end{proof}

\subsection{A small coinitial family and the omitted type}
\label{subsec:gca-coinitiality}

For each $\alpha<\kappa$, let $u_\alpha(r)$ be the least position
$e(c_\alpha)<i<d$ at which
\[
 \operatorname{Ext}(r)(i)\ne
 \operatorname{Ext}(r+a_\alpha)(i).
\]
This is a bounded search. Since both codes are below $2^d$,
their bits vanish at every position $i\geq d$, so the search
finds every possible disagreement above $e(c_\alpha)$.
Use a formal value $\infty$, greater than every element of $C$,
if there is no such position. For a basic cylinder $U$, put
\begin{equation}\label{eq:gca-final-segments}
 I_{\alpha,U}=
 \{b\in A_N:\{r\in U:u_\alpha(r)\geq b\}\in\mathcal I\}.
\end{equation}
Each $I_{\alpha,U}$ is a final segment of $A_N$.

\begin{lemma}\label{lem:gca-final-cover}
Every $I_{\alpha,U}$ is proper, and
\[
 A_N=\bigcup_{\alpha<\kappa,\ U\text{ basic}}I_{\alpha,U}.
\]
\end{lemma}
\begin{proof}
This is the final-segment argument of \cref{lem:cso-coinitiality},
with $(C,\omega,M\setminus\omega,n+1)$ replaced by
$(X,K,A_N,e(c_\alpha))$. We verify the steps in this setting.
\Cref{lem:gca-local-bound} for $a_\alpha$ gives $u_\alpha\geq H$
on a positive subcylinder, proving properness.
If $b$ misses the union, apply that lemma to $a_\infty$ and choose
$i\in A_N$ with $i+1<\min(b,H)$, using \eqref{eq:gca-half}.
Locally decide the bit $\operatorname{Ext}(r+a_\infty)(i)$ and
choose $\alpha$ so that $t_\alpha$ fixes the resulting cylinder.
Translation invariance makes the bits at $r+a_\alpha$ and
$r+a_\infty$ equal there modulo $\mathcal I$, while the latter
differs from the bit at $r$. Thus $u_\alpha\leq i$ locally,
$i+1\in I_{\alpha,U'}$, and finality puts $b$ in that segment,
a contradiction. All thresholds $e(c_\alpha)$ lie in
$K$, so $i$ is above each of them, as required in the cited proof.
\end{proof}

\begin{proof}[Proof of \cref{thm:generalized-pa-obstruction}]
Use the constructions above for a hypothetical saturated $N\in\Sstar$.
There are $\kappa$ pairs $(\alpha,U)$, since the cylinder basis has
size $\kappa$.  In $V$, choose
$b_{\alpha,U}\in A_N\setminus I_{\alpha,U}$ for each pair.
The sequence belongs to $\Sstar$ by \cref{lem:gce-small-codes}.
It is coinitial in $A_N$, since $b\in I_{\alpha,U}$ implies
$b_{\alpha,U}<b$ by finality.

Consider the type
\begin{equation}\label{eq:gca-omitted-type}
 \{T_n(x):n<\omega\}\ \cup\ \{m<x:m\in M\}\ \cup\
 \{x<b_{\alpha,U}:\alpha<\kappa,\ U\text{ basic}\}.
\end{equation}
It has at most $\kappa$ parameters and is finitely satisfiable.
Indeed, choose an element of $M$ above the finitely many lower
bounds; it satisfies every $T_n$, and every upper bound lies in
$A_N$, hence is above all of $M$.  A realization by
$\kappa^+$-saturation of $N$ lies in $C$, is above every member
of $M$, and hence lies in $A_N$, but is below its coinitial family.
This is impossible.

Finally, \cref{lem:general-hereditary-model} gives
$L(\mathcal P(\kappa)^V)\subseteq\Sstar$, so the same
nonexistence conclusion holds in that inner model.
\end{proof}

\subsection{The exact saturation threshold and definable presentations}
\label{subsec:gca-consequences}

\begin{corollary}\label{cor:generalized-saturation-sharp}
Let $W_0\models\mathsf{ZFC}+\mathsf{GCH}$, let $\kappa$ be regular
uncountable in $W_0$, and let $V=W_0[G]$ for
$\Add(\kappa,\Lambda)^{W_0}$, where $\Lambda>\kappa$ is any ground
cardinal. Every completion of $I\Delta_0$ has a model in
$L(\mathcal P(\kappa)^V)$ of cardinality $\kappa$ which is
$\kappa$-saturated in $V$.
In the setting of \cref{thm:generalized-pa-obstruction}, this is
also a model in $\Sstar$. Thus $\kappa$-saturation is possible in
both inner models, whereas $\kappa^+$-saturation is impossible.
\end{corollary}
\begin{proof}
The completion belongs to $W_0$ because there are no new reals.
The elementary-chain construction above gives a ground
$\kappa$-saturated model $M_0$ with domain the ordinal $\kappa$.
Every parameter set of size below $\kappa$ from $M_0$ has a
ground enumeration, by the absence of new short ordinal sequences.
Every type over such a set is also ground, since the countable
language supplies fewer than $\kappa$ formulas with those
parameters. The ground witnesses therefore show that $M_0$
remains $\kappa$-saturated in $V$.
Using the canonical ordinal pairing functions, its countably many
relations and operation graphs have one subset-of-$\kappa$ code.
That code and its decoded structure belong to
$L(\mathcal P(\kappa)^V)$; \cref{lem:general-hereditary-model}
puts this model in $\Sstar$ as well.
\end{proof}

\begin{corollary}\label{cor:generalized-no-definable-saturated}
In the setting of \cref{thm:generalized-pa-obstruction}, if
$\Lambda=\kappa^+$, there is no
$\mathrm{OD}_{H_{\kappa^+}}^V$ presentation of an $I\Delta_0$ model of
ambient cardinality at most $\kappa^+$ that is
$\kappa^+$-saturated in $V$.  In particular, there is no such
presentation of a model of true arithmetic.
More generally, at any $\Lambda>\kappa$, an arbitrary-size
$\mathrm{OD}_{H_{\kappa^+}}^V$ presentation whose domain is
pointwise $\mathrm{OD}_{H_{\kappa^+}}^V$ cannot be
$\kappa^+$-saturated.
\end{corollary}
\begin{proof}
For $\Lambda=\kappa^+$, apply
\cref{thm:generalized-small-family,cor:add-structure-copy}
to obtain an ambiently isomorphic copy in $\Sstar$.
\Cref{thm:generalized-pa-obstruction} rules it out.
For the final assertion, the definition-code construction of
\cref{lem:add-copy} applies directly to the pointwise definable
domain.  That construction uses only closure of the definability
class under finite parameters and ZF in its hereditary part,
both available by \cref{lem:gce-small-codes}.
Its $\Sstar$-copy contradicts the same theorem.
\end{proof}

\begin{corollary}\label{cor:uniform-definable-saturation-obstruction}
Let $\kappa$ be any infinite regular cardinal of $L$. The generic
extension $V=L[G]$ by $\Add(\kappa,\kappa^+)^L$ preserves all
cardinals and cofinalities and satisfies GCH. In $V$, every complete
consistent extension of $I\Delta_0$ has a $\kappa^+$-saturated model of
cardinality $\kappa^+$, but none of these models has an
$\mathrm{OD}_{\mathcal P(\kappa)}$ copy. In particular there is no
$\mathrm{OD}_{\mathcal P(\kappa)}$ $\kappa^+$-saturated elementary
extension of $\mathbb N$ of cardinality $\kappa^+$.
\end{corollary}
\begin{proof}
The forcing has size $\kappa^+$, is $<\kappa$-closed, and is
$\kappa^+$-cc. These properties preserve all cardinals and
cofinalities. No subsets of infinite cardinals below $\kappa$
are added. For each cardinal $\tau\geq\kappa$, a nice name for a
subset of $\tau$ uses $\tau$ antichains, each of size at most
$\kappa$. The number of such names in $L$ is at most
\[
 ((\kappa^+)^\kappa)^\tau=(\kappa^+)^\tau=\tau^+.
\]
The ground subsets give the reverse bound, so every instance of
GCH is preserved. Consequently
$(\kappa^+)^{<\kappa^+}=\kappa^+$, and the standard saturated-model
construction supplies the asserted models of every complete
countable theory extending $I\Delta_0$.

For $\kappa=\omega$, \cref{cor:saturated-no-od} excludes the
definable copies. For uncountable $\kappa$, apply
\cref{cor:generalized-no-definable-saturated}, since every subset
of $\kappa$ belongs to $H_{\kappa^+}$. As in
\cref{rem:ground-convention}, the proof applies to every original
ZFC ground satisfying GCH, GA, and $V=\operatorname{HOD}$.
\end{proof}

\begin{remark}
In $L$ itself, its canonical well-order makes the usual
saturated-model construction definable, giving an OD
$\kappa^+$-saturated elementary extension of $\mathbb N$ of size
$\kappa^+$ for every infinite $\kappa$. The regularity restriction
in the corollary is exact. If $\kappa$ is any singular cardinal of $L$, a
cardinal- and cofinality-preserving extension satisfying GCH
still makes $\kappa$ a singular strong limit. Then
\cref{cor:gch-saturation} supplies such an OD model and rules
out the analogous negative conclusion.
\end{remark}


\subsection{Trace coding and finite-support models}\label{subsec:ga-quotients}

The following observations hold in ZFC, independently of forcing.
In PA, decode $z$ as $(b,c-2)$, $c\geq2$, and define
$\operatorname{ap}(z,m)=\lfloor b/c^m\rfloor\bmod c$.
This total operation interpolates at finitely many arbitrary model
positions. For distinct $m_1,\ldots,m_r$ and prescribed $v_i$,
choose $c>\max_i v_i$ and $b=\sum_{i=1}^r v_ic^{m_i}$.
PA verifies the required digits for each standard finite $r$.

\begin{proposition}\label{prop:generalized-arithmetic-quotients}
If $\kappa$ is infinite, $N\models\mathrm{PA}$ is
$\kappa^+$-saturated, and $M\subseteq N$ has size at most $\kappa$,
then
\[
 \operatorname{Ev}(z)(m)=\operatorname{ap}(z,m),\qquad
 \pi(z)=\{m\in M:\operatorname{ap}(z,m)=1\}
\]
map $N$ onto $\,{}^M N$ and $\mathcal P(M)$, respectively.
Their equality kernels are
\[
 \begin{aligned}
 zE_M^{\rm fun}w&\ \Longleftrightarrow\
   \forall m\in M\ [\operatorname{ap}(z,m)=\operatorname{ap}(w,m)],\\
 zE_M^{\rm set}w&\ \Longleftrightarrow\
   \forall m\in M\ [\operatorname{ap}(z,m)=1\leftrightarrow
                                      \operatorname{ap}(w,m)=1].
 \end{aligned}
\]
Consequently $N/E_M^{\rm fun}\cong{}^M N$ and
$N/E_M^{\rm set}\cong\mathcal P(M)$.
\end{proposition}
\begin{proof}
For $f:M\to N$, the type
$\{\operatorname{ap}(z,m)=f(m):m\in M\}$ has at most $\kappa$
parameters and is finitely satisfiable by interpolation.
Saturation realizes it. Characteristic functions give the second
surjection; the displayed equivalence relations are their kernels.
\end{proof}

\begin{corollary}\label{cor:generalized-quotients-in-inner-model}
If $S$ is a transitive ZF inner model closed under ambient sequences
of length at most $\kappa$, and $M,N\in S$ with $|M|\leq\kappa$,
then $({}^M N)^S=({}^M N)^V$ and
$\mathcal P(M)^S=\mathcal P(M)^V$.
When $N$ satisfies the preceding proposition, its trace maps,
kernels, quotients and induced bijections belong to $S$.
\end{corollary}
\begin{proof}
An ambient enumeration of $M$ belongs to $S$. Composing any
$f:M\to N$ with it proves the first equality, and characteristic
functions prove the second. The absolute trace definitions and
ZF in $S$ supply the indicated maps and quotients.
\end{proof}

\begin{proposition}\label{prop:ga-em-omission}
No $\aleph_1$-saturated PA model has an expansion generated
definably or algebraically by an infinite sequence of distinct
order-indiscernibles. In particular an ordinary Skolemized
Ehrenfeucht--Mostowski model on such a sequence is not
$\aleph_1$-saturated.
\end{proposition}
\begin{proof}
Choose a countably infinite monotone subsequence $(a_{i_n})$ of
the indiscernibles. If $b$ is uniquely defined by
$\theta(y,a_{j_1},\ldots,a_{j_r})$, the relative order of $i_n$
and the finitely many $j_s$ is eventually constant. Apply
indiscernibility to
$\exists y\,[\theta(y,y_1,\ldots,y_r)\wedge\operatorname{Bit}(y,z)]$.
It makes $\operatorname{Bit}(b,a_{i_n})$ eventually constant.
Thus the countable type requiring alternating bits at the
$a_{i_n}$ is omitted, although every finite part is realized
by binary interpolation. Algebraic generation reduces to
unique definition, since each element of a finite definable set
is its $r$th member in the PA order for some standard finite $r$.
\end{proof}

Indiscernibility is in the generating language. For a quantifier-free
EM blueprint, use its Skolemized language with the definable bit
function, so the formula after term substitution is atomic.
The argument also covers distinct fixed-length tuples. One fixed
coordinate differs on every ordered pair and supplies the bit
positions. The omitted type uses countably many parameters;
the conclusion concerns $\aleph_1$-saturation, not recursive
saturation over a finite tuple.


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\end{thebibliography}

\end{document}