Chapter 4. Measurable labels and finite moving blocks

This chapter follows Lemma 3.6 (pp. 13–14), Proposition 4.8 (pp. 22–23), and Section 5 (pp. 23–33) of R. Chapter 3 treated operators with the parameters fixed. We now establish the conditional i.i.d. structure of parameter pairs, recover individual coordinates from the grouped conclusions, construct measurable labels and finite blocks, and turn the original witness into a positive integral of a finite sum of absolute coefficients.

Fixed source, from page 13

Given the local parameter $u$, the conditional probability space $(Y,\eta _u)$ will be called a fibre. Its Hilbert space is $H_u=L^{2}(\eta _u)$. A parameter tuple is a finite list $p=(p_{1},\ldots ,p_k)$ ordered by position.

1. Three distinct steps

Chapter 3 proved that, for a fixed typical parameter square, the intersection of a cell's row-active and column-active spaces is spanned by common exact lines. Three further steps are needed.

First, the entire active space of a single tuple law must be spanned by exact lines. This requires more than the intersection statement: the conditionally independent, identically distributed parameter pairs in Lemma 3.6 and the kernel of an averaged projection exclude any missing vectors.

Second, the countable family of exact lines at each tuple must be listed measurably, with the resulting family depending only on the cell's local parameter $u$. The source first uses compact-operator spectral enumeration over the full parameter tuple $p$, then conditional independence to descend to $u$. Choosing a basis separately in each fibre would not establish measurability.

Third, time must act on the resulting families. The conclusion only requires a permutation of whole finite blocks of labels. It does not assign individual labels a time rule independent of the parameter. Because the final observable sums absolute coefficients over a whole block, the possibly nonmeasurable abstract group identifications disappear from the function being integrated.

All spans are closed linear spans, and Hilbert inner products are linear in the second variable. The local state system $Y$ is assumed neither mixing nor zero entropy, and the individual conditional measures $\eta _u$ need not be invariant under $T_Y$.

2. Lemma 3.6: conditional i.i.d. parameters at the diagonal

Write the square states as $y_{a,j}$ and their local parameters as $u_{a,j}$. Let the row parameters be $p_a=(u_{a,j})_j$ and the column parameters $q_j=(u_{a,j})_a$. Every parameter tuple has marginal law $\pi$, with one-coordinate marginal $\nu$. Put

Write $I=\{1,\ldots,k\}$. The assertion is that, for every $a\in I$ and proper subset $J\subsetneq I$,

First take $J=I\setminus\{j_0\}$. Let $Z=Y_{a,I}$ be the whole row, $O=(Y_{I,j})_{j\in J}$ the indicated whole columns, and $S=(y_{a,j})_{j\in J}$ their shared entries. Use the extension that incorporates the first axis with no parent slots, projecting at position $a$. Its admissibility is the base case of Lemma 3.4. The old $\mathcal O_{1,j_0}$ pulls back to $\sigma (S)$, while the new $\mathcal O_{1,j_0}'$ is $\sigma (O)$. Applying saturation to a whole-row test $F$ gives

Since $S$ is a function of both $Z$ and $O$, for bounded $F,G$,

Thus $Z$ and $O$ are independent given $S$; in particular, $\operatorname{Law}(O|Z)=\operatorname{Law}(O|S)$. Unconditionally, the whole columns indexed by $J$ are independent, each with law $m_{1}$. Conditioning each column on its own shared entry gives

Now take the parameter images of these columns. In one column, the parameter $q$ and the state at position $a$, denoted by $y_a$, have joint law $\pi(\mathrm dq)\eta_{q[a]}(\mathrm dy_a)$. Disintegrating $\pi$ over $q[a]=u$ gives

The measure $\eta _u$ is supported on $u(y_a)=u$, so the conditional kernel of $q$ given $y_a$ is exactly $K_a(u(y_a),\mathrm dq)$. The kernel of the column parameters given $Z$ is therefore the right-hand side of (18). It depends only on $p_a$, and $p_a$ is a function of $Z$. Conditioning further onto $p_a$ proves (18). Smaller $J$ follow by taking marginals.

At a diagonal cell, take $a=i$ and $j=i$. Given $p_i$, the kernel of $q_i$ is $K_i(p_i[i],\mathrm dq_i)$. The marginal of $p_i$ is $\pi$, so, given the shared value $u=u_{i,i}$, the kernel of $p_i$ is also $K_i(u,\cdot )$. Hence

This establishes the conditional i.i.d. pair of parameter tuples needed below. It does not give a product law for the entire parameter square.

3. Proposition 4.8: recovering exact relations at all individual coordinates

Fix a cell $(a,j)$. Make row $a$ and column $j$ singleton groups, and split the remaining rows and columns into two nonempty groups each. The cell's active spaces $V$ and $W$ are independent of this grouping, since the other two groups still contain all remaining coordinates in its row or column. Chapter 3 gives commuting projections onto $V$ and $W$, with $D=V\cap W$ spanned by lines exact for both three-cell groupings.

The same lines must work for every partition. First keep the row grouping fixed and vary the column grouping. The resulting families of common exact lines are subsets of the orthogonal exact-line family of one fixed column law, and they all span the same $D$. Two subsets of an orthogonal family with the same closed span must coincide: a line appearing in only one subset would be orthogonal to the entire span of the other. Thus the lines work for every column partition. Fixing the original column grouping and varying the row grouping gives the corresponding assertion for every row partition.

For $k=3$, no grouped coordinates remain to be split. Suppose $k>3$ and choose a modulus-one representative $\chi _j$ of one line at the cell. For every nontrivial bipartition of $I\setminus\{j\}$, written $S\cup S^c=I\setminus\{j\}$ with complement taken within $I\setminus\{j\}$, there are modulus-one functions $a_S$ and $b_S$ such that

After absorbing constant phases, all products $a_Sb_S$ equal the same function of the other coordinates, $\Phi =\overline{\chi _j(Y_j)}$. Those $k-1$ coordinates have a product marginal, so these function identities hold in their product probability space.

For every singleton partition $\{i\}$, the function $\Phi$ belongs to the tensor product of the one-dimensional line $\operatorname{span}(a_i)$ with the $L^{2}$ space of the other coordinates. The projections onto the individual $\operatorname{span}(a_i)$ act on different tensor factors, so they commute, and they all fix $\Phi$. Their product has a one-dimensional tensor range. Thus $\Phi$ is the pure product of the $a_i$ up to a constant phase, and $\chi _j$ completes to an exact tuple on all $k$ individual coordinates. The column argument is identical. Conversely, lines exact for the full row and column belong to $D$.

The second assertion, intertwining, also follows directly from Chapter 3. Make the given row $a$ a singleton group, make the two target columns $s,t$ singleton groups, and put the other columns in the third group. Split the remaining rows into two groups. The row-edge intertwining in Lemma 4.4 is then exactly

Here $v$ records the other $k-2$ states in the row. The array axes have not been interchanged.

If an exact tuple on the full row satisfies $\prod_{j\in I}\chi_j(Y_j)=c$, then

The scalar has modulus one. Intertwining and conjugation invariance of $P^W$ show that one partner line belongs to its column-active space if and only if the other partner lines belong to their respective column-active spaces. Reverse edges give the converse implication.

There are only finitely many groupings and positions. Intermediate groups or bases may be chosen nonmeasurably with the parameters fixed. The proposition concerns only the active spaces and exact lines of the original conditional kernels; measurable enumeration has yet to be established.

4. Lemma 5.1: from intersections to the full active space

Let $H_u=L^{2}(\eta _u)$. A countable family of bounded simple tests on the state space, followed by Gram–Schmidt, realizes this Hilbert field in a common measurable $\ell ^{2}$ coordinate system, padding zero directions with zeros. The construction depends only on $u$. Thus two tuples sharing the same $u$ use the same $H_u$.

Let $\Pi_i(p)$ be the orthogonal projection onto the active space of $\lambda _p$ at position $i$, and let $C_i(p)$ be the closed span of its exact lines. Countable product tests for conditional expectation give measurable generators of the active space. Their finite-dimensional projections increase to $\Pi_i(p)$, making $\Pi_i$ a measurable operator field. Measurability of $C_i$ is not yet assumed.

By Section 2, at the cell $(i,i)$, the tuples $p,q$ are conditionally i.i.d. given $u$, with common kernel $\pi_i^u=\pi(\mathrm dp\mid p_i=u)$. Section 3 gives, for almost every such parameter pair,

The almost-everywhere assertion does not require measurability of $C_i$. Start with a measurable conull set of parameter squares specified by countably many kernel identities, and disintegrate over their $(p,q)$ images. On a measurable conull set of parameter pairs, the conditional law is concentrated on admissible squares with the specified $(p,q)$. Every such pair therefore has an admissible completion. Conclusion (38) depends only on the two conditional laws specified by $p,q$, and not on the chosen completion. No measurable completion is selected, and no integral involving $C_i$ is taken.

For typical $u$, define the positive contraction

Here $Q_{i,u}$ is the orthogonal projection onto the kernel taken inside $H_u$.

The integral is defined through matrix coefficients. The operator $Q$ is, on $H_u$, the strong limit of $(\operatorname{id}_{H_u}-B_{i,u})^n$, and is therefore measurable. In an ambient $\ell ^{2}$ realization, first project the standard vectors $e_j$ onto $H_u$ and extend the operators by zero outside $H_u$. The ambient orthogonal complement must not be mistaken for part of the fibrewise kernel.

For these $e_j$,

Taking a countable intersection and then adjoints gives, for almost every $p$,

Among the parameters with a conull $q$-section for (38), fix $p$ also satisfying (40). If $v\in \operatorname{Ran} \Pi_i(p)$ and $v\perp C_i(p)$, commutation puts $\Pi_i(q)v$ in the intersection in (38), so

Integration gives $\langle v,B_{i,u}v\rangle =0$, hence $v\in \operatorname{Ker} B_{i,u}$. By (40), $v=0$. Every exact line already lies in the active space, so

For this fixed $p$, (38) is a statement about the entire subspace on a conull $q$-section. The argument therefore applies to every $v$ at once. No measurable dependence on $p$ is required for a counterexample vector $v$.

5. First part of Proposition 5.2: measurable enumeration over tuples

5.1 Measurable compact operators before measurable bases

Fix a position $i$ and another position $j\ne i$. Let

be conditional expectation from cell $i$ onto all the other cells. The other $k-1$ coordinates have a product marginal, so the target is exactly their $L^{2}$ space. The operator $\Delta$ is measurable and contractive. It is the adjoint of reverse conditional expectation and hence vanishes on the orthogonal complement of the active space.

If $\chi _\ell$ is a unit representative of an exact line, then $\Delta\chi _\ell$ is a unit pure tensor of the conjugate partner representatives at the other cells, up to a constant phase. In the unit ball of every $H_u$, choose countably many dense tests $g_n(u)$. Put $\theta _g h=g\langle g,h\rangle$ and define

The matrix coefficients of these operators are already measurable, without enumerating the lines. Now fix $p$ and analyze $A_n$ using the orthogonal basis of exact lines established in Section 4. For distinct labels, the corresponding partner representatives in the two pure tensors are orthogonal at every other position. Since $k\ge 3$, at least one factor outside $i,j$ has no inserted $\theta$ and makes the off-diagonal coefficient vanish. Thus $A_n$ is diagonal on these lines. Let $w_{j,\ell}$ denote the unit tensor factor of $\Delta_i(p)\chi_\ell$ at position $j$. It is the conjugate of the partner representative at that position in the exact tuple, up to a constant phase. The eigenvalues are

The trace bound is Bessel's inequality. The dense tests detect every line and distinguish the coefficient sequences of any two distinct lines.

5.2 Positive random weights give simple spectrum

Choose independent weights with continuous distributions satisfying $0<w_n<2^{-n}$, and put $A(p,w)=\sum_{n\ge1}w_n A_n(p)$. The series converges in operator norm, is jointly measurable and positive, and has finite trace, so it is compact. It may be set to zero on any remaining null parameter set where these properties fail. Finiteness of the trace is a measurable condition, checked by a countable sum of diagonal coefficients.

Repetition of a positive eigenvalue must be detected measurably. For a positive compact operator $A$, its norm $\alpha$ can be computed from countably many dense vectors. If $\alpha >0$, the strong limit $(A/\alpha )^m$ is the top-eigenvalue projection $Q$. Both the matrix coefficients and rank of $Q$ are measurable. Subtract $\alpha Q$ and repeat to enumerate all distinct positive eigenvalues and their finite-dimensional eigenspaces. Thus the event that some positive spectral projection has rank at least two is measurable, without any prior measurable enumeration of exact lines.

For each fixed $p$, distinct exact lines have different sequences $(a_n(\ell ))_n$. Choose a coordinate $n$ at which they differ. Conditional on the other weights, equality of the weighted eigenvalues allows at most one value of $w_n$, an event of probability zero. There are only countably many orthogonal lines at each $p$, so the probability of repeated positive spectrum is zero. Fubini applied to this measurable event gives one deterministic sequence of weights for which all positive eigenvalues are simple for $\pi$-almost every $p$.

Every exact line in the active space has some $a_n>0$. Since all $w_n$ are positive, its weighted eigenvalue is strictly positive. The kernel is exactly the orthogonal complement of the active space. Positive eigenvalues tending to zero do not create additional kernel vectors. The rank-one positive spectral projections therefore enumerate precisely all exact lines.

5.3 Representatives, matching, and measurability

For each rank-one projection, choose the first vector in the countable coordinate list with nonzero projection and normalize its image. This gives a measurable unit representative. Since it belongs to an exact line, it is a modulus-one function almost everywhere.

At this stage, a measurable section is still parametrized by the full tuple $p$. At a fixed position $i$, it takes values in $H_{p_i}$. The approximation below gives function representatives jointly measurable in $(p,y)$. The conclusion that the family of exact lines depends only on the local parameter $u$ is established in Section 6 of this chapter. Producing jointly measurable function representatives from a measurable vector section takes a further approximation. In each fibre, use a countable family of rational simple sections and choose the first approximation with error at most $2^{-m}$; the selection criterion is measurable. Successive differences are summable in integrated $L^{2}$, hence also in integrated $L^{1}$. They yield a jointly measurable almost-everywhere limit representing the original section. Mere existence of representatives separately in every fibre would not suffice.

A listed family of modulus-one representatives $\chi _i$ is an exact tuple if and only if

This is a measurable scalar condition. Uniqueness of partners identifies the unique matching tuple of lines in the countable lists. Both enumeration and exact matching over tuples are therefore measurable.

6. Descending from tuple lists to the local parameter $u$

The preceding spectral enumeration still depends on the whole tuple $p$. A further argument is needed to remove that dependence.

6.1 Membership in column-active spaces is one common $0/1$ variable

Fix a typical square and exact lines matched along row $a$, with representatives $v_j$. Let $Q_j=\Pi_a(q_j)$ be the projection onto the corresponding column-active space. It commutes with the row-active projection, and their intersection is spanned by common exact lines. Orthogonality of the exact lines gives

The equality $\varepsilon _j=1$ means precisely that this row label is also an exact line in its column. Equation (36), projection intertwining, and conjugation invariance imply $\varepsilon _j=\varepsilon _t$ for every $j,t$.

Choose the row labels from the measurable lists of the previous section. Given the row parameter $p_a$, the variable $\varepsilon _j$ depends only on $q_j$ and the fixed representatives. By Lemma 3.6, distinct $q_j$ are pairwise conditionally independent, so the $\varepsilon _j$ are as well. They are also almost everywhere equal. Their common conditional mean $r(p_a)$ therefore satisfies $r=r^{2}$ and takes only the values zero and one.

At the diagonal position $j=a$, the averaged projection from Section 4 gives

Strict positivity follows from (40) and $v_a\ne 0$: vanishing of the positive operator's quadratic form puts the vector in its kernel, whereas the active space is orthogonal to that kernel. Thus $r=1$. Intersecting over countably many row labels and finitely many cells shows that every row exact line is also an exact line in its cell's column.

6.2 Conditional i.i.d. tuples make the whole family depend only on $u$

At $(i,i)$, the exact-line families at position $i$ of the two conditionally i.i.d. tuples satisfy an almost-everywhere inclusion. Interchanging the two i.i.d. copies gives the reverse inclusion, hence equality.

To make this a measurable descent, avoid choosing a tuple as a function of $u$. Use the local dense tests $g_n(u)$ to encode an orthogonal family of unit lines $\mathcal S$ by

This expression is independent of phases and enumeration, and has trace at most one. Its matrix coefficients are measurable by the tuple enumeration. Given $u$, the two independent, identically distributed encodings agree. Every bounded matrix coefficient therefore has zero conditional variance and equals its conditional expectation given $u$. The encodings descend to measurable operator fields depending only on $u$.

The encoding determines $\mathcal S$. In each fixed fibre, all $K_n$ are diagonal on $\mathcal S$, and the dense tests detect and distinguish its lines. The positive-weight simple-spectrum construction recovers the same family from these operators. The measurable spectral recursion therefore gives a local list $\mathcal L_i(u)$ agreeing with the original list for almost every $p$ with $p_i=u$. No measurable representative tuple has been chosen.

At an off-diagonal cell $(a,j)$, the inclusion from Section 6.1 now reads $\mathcal L_j(u_{a,j})\subseteq \mathcal L_a(u_{a,j})$. The single parameter has marginal $\nu$. Running over the finitely many ordered pairs $a,j$ shows that all $\mathcal L_i(u)$ coincide almost everywhere, giving one common local list $\mathcal L(u)$. No exchangeability of the parameter tuple law across positions is assumed.

6.3 Group operations, time, and square matching

Multiply lines modulo constant phase. Lemma 4.2 makes $\mathcal L(u)$ a countable abelian group, whose zero is the constant line. Distinct labels are orthogonal, and nonzero labels have mean zero. Choose measurable representatives $\chi _{u,\ell }$ satisfying $\chi _{u,0}=1$ and

Two unit vectors span the same line exactly when their inner product has absolute value one. Searching the lists with this measurable test for products, inverses, and partners gives measurable group operations and group isomorphisms $\Phi_{st}(p):\mathcal L(p_s)\to\mathcal L(p_t)$.

Covariance of the time kernels, $\eta _{T_Uu}=(T_Y)_*\eta _u$ and $\lambda _{T_U^I p}=(T_Y^I)_*\lambda _p$, makes $[\chi ]\mapsto [\chi \circ T_Y^{-1}]$ define $\tau_u:\mathcal L(u)\to\mathcal L(T_Uu)$, preserving exact matching. The measurable lists and equality test make $\tau$ and its inverse measurable. Intersecting over integer time shifts gives invariant conull sets separately for local parameters, tuples, and squares. It does not imply validity for every tuple assembled from good local parameters.

A label at one cell uniquely completes its row, and then uniquely completes every column. The representative product in every column is constant, as is the product in the initial row. Thus the product of the remaining $k-1$ row products is constant. These remaining whole rows form a conditionally independent proper subfamily, and each row product has modulus one. The product of the absolute values of their means is therefore one, forcing each mean to have absolute value one and each row product to be constant. Every row is consequently matched. Uniqueness follows from uniqueness in the initial row and then each column. Countability of the label lists makes the assertion simultaneous for every initial label on a conull set of square parameters.

7. Lemma 5.3: the role of counting measure

Set $Z=\{(u,\ell):\ell\in\mathcal L(u)\}$ and define $\xi(\mathrm du,\mathrm d\ell)=\nu(\mathrm du)\,\#_{\mathcal L(u)}(\mathrm d\ell)$, where $\#_{\mathcal L(u)}$ denotes counting measure on the label group. The set consisting of the first $N$ labels in each fibre has mass at most $N$, so $\xi$ is $\sigma$-finite. The map $\tau (u,\ell )=(T_Uu,\tau _u\ell )$ preserves $\xi$.

Over a parameter tuple $p$, one position's label determines a unique matching tuple. Counting these matching tuples over $\pi(\mathrm dp)$ gives, on $Z^I$, the measure $\widetilde\pi$. Counting matching label squares similarly gives $\widetilde\vartheta$. Uniqueness ensures that counting from any coordinate produces the same measure. Every coordinate marginal is therefore $\xi$, and every row or column marginal is $\widetilde\pi$. Both lifted spaces are exhausted by the inverse images of the first $N$ labels in one coordinate, so both measures are $\sigma$-finite.

For $s\ne t$, the transition kernel is

This is a probability kernel even when the total measure is infinite. Once the source label is fixed, each compatible parameter tuple has exactly one completion, with no extra counting weight. Likewise, given a labelled row, the conditional parameter-square law remains the original law given that row's parameters: every compatible square has exactly one label completion.

By Lemma 3.6, given a labelled row, every proper subfamily of labelled columns is independent. The parameter columns are conditionally independent, and each column's labels are determined by its own parameters and the label at its shared cell.

Put $(M_{st}F)(z_s)=\int F(z_t)K_{st}(z_s,\mathrm dz_t)$. Both coordinate marginals are $\xi$, so Jensen's inequality gives a contraction on $L^{2}(\xi )$. Interchanging the coordinates gives $M_{st}^*=M_{ts}$. Time equivariance makes $M_{st}$ commute with $V F=F\circ \tau$.

For any $a\ne b$ and $s\ne t$, take, in row $b$ of the labelled square, two $L^{2}$ tests. Conditional on row $a$, the transitions in columns $s,t$ are both $M_{ab}$ and are independent. Hence

Therefore

Cauchy–Schwarz on the two coordinates with marginal $\xi$ controls all integrals. Alternatively, start with bounded tests supported on sets of finite measure and approximate in $L^{2}$. The infinite counting measure is never normalized to a probability measure.

8. Lemma 5.4: from deterministic transport to finite blocks

8.1 The unitary part of a contraction

Fix $s\ne t$ and write $M=M_{st}$. In (47), take $(a,b)=(s,t)$ and $(t,s)$ to obtain

Put $H_1=\overline{\operatorname{Ran}M}$ and $H_2=\overline{\operatorname{Ran}M^*}$. The first identity places $H_{1}$ in the fixed space of $M^*M$, which is contained in $H_{2}$. The adjoint of the second places $H_{2}$ in the fixed space of $MM^*$, which is contained in $H_{1}$. Hence $H_{1}=H_{2}=H$.

On $H$, we have $M^*M=MM^*=\operatorname{id}_H$; on $H^\perp =\operatorname{Ker} M$, we have $M=0$. Thus $M|_H$ is unitary with inverse $M^*$. In particular, for $F\in H$, the equal-norm identity for conditional expectation gives

Equivalently,

The original transition kernel need not be deterministic everywhere. Deterministic transport has been obtained for the observation functions in $H$.

8.2 A strictly positive $L^{2}$ test

Using the local enumeration, set $h(u,\ell _n(u))=2^{-n}$. Then $h>0$ and $\|h\|_2^2\le\sum_{n\ge1}4^{-n}$. The probability kernel $K_{st}$ has target marginal $\xi$, so it assigns zero probability to $\xi$-null sets at almost every source point. Hence $v=Mh>0$ almost everywhere, and $v\in H\cap L^{2}$.

The unitary $V$ commutes with $M$ and $M^*$ and preserves $H$. Define the measurable countable vector

Negative powers of $M$ are taken only on $H$. All coordinates have finite versions on a common conull set, with strictly positive $(0,0)$ coordinate. Define shifts $D_{0}$ and $S_{0}$ on these sequence coordinates by increasing $n$ and $r$, respectively, by one. Equation (51) and commutation with time give

The maps $D_{0}$ and $S_{0}$ are commuting invertible coordinate permutations.

8.3 Countability from an unchanged target-parameter law

Given a source point $z=(u,\ell )$, the target parameter of $K_{st}$ still has law $\nu$. This uses the law $\pi$ and independence of its coordinates $p_s,p_t$ and the fact that fixing a label does not alter the parameter kernel. The independence holds because, for $k\ge 3$, two positions form a proper subfamily; the unchanged kernel follows from the counting construction with unique completions.

Define the measurable relation $R(z,u')$ by existence, in the fibre over $u'$, of a label $\ell _n(u')$ satisfying $\psi (u',\ell _n(u'))=D_{0}\psi (z)$. Existence is a countable union, and equality of sequences is a countable conjunction of coordinate equalities. By (53), for $\xi$-almost every $z$, the relation $R(z,u')$ holds for a set of $u'$ of full $\nu$-measure.

Apply Tonelli to the $\sigma$-finite measure $\xi\otimes\nu$. Choose one fixed $u'$ such that $R(z,u')$ holds for $\xi$-almost every $z$. This target fibre has only countably many values of $\psi$. Since $D_{0}$ is invertible, $\psi$ has countable essential range. Only one fixed target fibre is selected; no measurable selection of a target label for each $z$ is required.

Let $\mathcal B$ be the set of values of positive mass in this countable range. Define

For $b$, its $(0,0)$ coordinate $b_{00}$ is strictly positive, and

Thus $B_b(u)$ is finite for almost every $u$. A countable intersection puts every label in each good fibre into one of these finite blocks. This alone does not yet make each block nonempty or give it the same cardinality in every fibre.

Equality of the two marginals and invariance of $\xi$ under time make $D_{0}$ and $S_{0}$ permutations of $\mathcal B$, preserving $\xi (E_b)$ along their orbits. Equation (53) and countability of the labels show that, for $\pi$-almost every $p$, the map $\Phi_{st}(p)$ carries each $B_b(p_s)$ bijectively onto $B_{D_{0}b}(p_t)$. The reverse inclusion uses bijectivity of $\Phi_{st}$ and invertibility of $D_{0}$; a single inclusion would not establish equality.

8.4 Cancelling the common block shift gives fixed finite cardinalities

Disintegrate the measurable conull set of admissible parameter tuples over $(p_s,p_t)$. Its marginal is $\nu\otimes\nu$, so almost every pair $(u,w)$ has a tuple completion satisfying the required conditions. Fubini gives a common source $u_*$ with such completions for almost every target $u$.

For each target $u$, choose one resulting group isomorphism $I_u:\mathcal L(u_*)\to\mathcal L(u)$ pointwise. Every $I_u$ sends block $b$ to block $D_{0}b$. No measurability of $I_u$ in $u$ is required.

Fix one good target $u_{0}$, take the abstract group $\mathcal L=\mathcal L(u_0)$, and put $J_u=I_u I_{u_{0}}^{-1}$. The inverse of $I_{u_{0}}$ pulls target block $b$ back to source block $D_{0}^{-1}b$, which $I_u$ sends to target block $b$. Thus $J_u$ preserves every block index. Each cardinality $|B_b(u)|$ is consequently almost everywhere the same integer $m_b$. By (54), that integer is finite and strictly positive.

Write the abstract blocks as $B_b=B_b(u_{0})$. Choose one $u$ in the common invariant conull set for local time equivariance, and define

This is an automorphism of $\mathcal L$ sending each $B_b$ to $B_{S_{0}b}$. Taking $G=A^{-1}$ gives

Once fixed, $G$ implements the permutation of whole blocks. The proof does not assert $\tau _uJ_u=J_{T_Uu}G^{-1}$ for every $u$, or identify the actual time orbit of each individual fibre label with an abstract $G$ orbit. The block-level conclusion is exactly what the next step needs.

9. Proposition 5.5: locating the witness in a finite block

Suppose the original $X$ is the zero-entropy finite-alphabet mixing process from Proposition 2.1 and the first level retains the nonzero witness correlation. Fix a position $i$. Given a parameter tuple $p$, condition the product of the other witness functions onto cell $i$. The result lies in the active space of $\Pi_i(p)$. If $f_i\circ \kappa$ were orthogonal to that space for almost every $p$, the original witness integral would vanish. The product of the other functions may be conjugated to respect the second-variable-linear inner product; the active space is invariant under conjugation.

Lemma 5.1 and the local label lists therefore give a witness $f=f_i$ with a nonzero coefficient at some local label on a set of parameters of positive $\nu$-measure. Since $\kappa _*\eta _u=\mu _X$,

The coefficient of the identity label vanishes, so nonzero labels carry the correlation.

We continue to write $\kappa(y)$ for what the source abbreviates here as $x(y)$. For each block $b$, define the intrinsic measurable function

There are countably many blocks, and they cover all labels. If every $c_b$ had zero integral, each would vanish almost everywhere; a countable intersection would make all coefficients vanish, contradicting the preceding conclusion. Thus some fixed $b$ satisfies $c=\int c_b\,\mathrm d\nu>0$. The block has fixed finite size, and $c_b\le m_b\|f\|_{\infty }$, so $c$ is finite. Set $B=B_b\subseteq\mathcal L$.

For each good $u$, use the existing block-preserving identification $J_u$ and define $\chi _\ell ^u=\chi _{u,J_u\ell }$. Although measurability of $J_u$ has not been established, within each fixed fibre these representatives satisfy

For $r\ge 0$, equation (56) of the source defines

This expression involves the possibly nonmeasurable $J_u$ and does not by itself establish measurability of $s_r$. By (49) and the block-preserving property of $J_u$, it equals

The right-hand side sums over a measurable finite block in the measurable local list, so it is measurable. Changing a representative by a constant phase, or permuting labels within the block, leaves the absolute coefficients and their sum unchanged. This removes the need for measurable $J_u$.

Finally, use covariance of the conditional kernels:

Put $y'=T_Y^{r}y$. Since $\kappa$ intertwines time, $f(T_X^{r}\kappa (y))=f(\kappa (y'))$. The representative $\chi _{u,\ell '}\circ T_Y^{-r}$ differs from the representative of the time-transported label only by a constant phase. Time maps $B_{S_{0}^{-r}b}(u)$ bijectively onto $B_b(T_U^{r}u)$, giving

The measure $\nu$ is time invariant. Thus, for every $r\ge 0$,

The integer times $r$ are countable, so these identities hold on one invariant local conull set. Labels and label pairs in every fibre are also countable. Modulus one, orthogonality, and all required product relations can therefore hold simultaneously on a conull set of local parameters, and then on a common $\eta _u$-conull set in each typical fibre. No assertion is made for every parameter value or every parameter tuple.

The original nonzero correlation has now become a finite sum of absolute coefficients whose integral is the same positive number at every time.

This step uses neither mixing nor zero entropy of the extension $Y$, and does not make $\eta _u$ an invariant system. Section 6 of R will use only the fact that each fibre's $X$ marginal is the original $\mu _X$, together with the label-function relations above.