Unit 4, continued: Maximizing auxiliary positions and stabilizing conditional expectations

This part studies B §§6.1–6.3, pp.18–21. Pointwise failure has already supplied fixed selectors, a common ultrafilter, and a nonzero correlation. Preservation of the three auxiliary groups remains conditional on the higher-order mixing and relative structure results stated earlier in Unit 4.

We choose a nonzero correlation with as many auxiliary positions as possible, then extend the system so that specified conditional expectations no longer change under further extensions. The final construction regroups the auxiliary variables; the next part computes row and column conditional laws in that system.

1. Which positions use auxiliary variables?

Source: B p.18, equation (6.1).

For a system $Y$ with three auxiliary groups, write its original spatial coordinate as $x_Y(y)$ and the finite list of all auxiliary spatial coordinates as $d_Y(y)$ . If the auxiliary list has length $m$ , its spatial law is $\mu^{\otimes m}$ . When $m=0$ , use a one-point space.

Let $I=\{1,2,3,4\}$ and $A\subseteq I$ . Consider the correlation

\int_{Y^4}\Phi(P) \prod_{i\in A}h_i(d_Y(y_i)) \prod_{i\notin A}h_i(x_Y(y_i)) \,\mathrm{d}\lambda_Y, \quad P=(\omega_Y(y_i))_{i\in I}. \tag{Corr}

All functions are bounded and real-valued. The weight $\Phi$ may be any bounded function of the full old parameter array. Each $h_i$ is centered for the spatial law it uses: if $i\in A$ , require $\int h_i\,\mathrm{d}\mu^{\otimes m}=0$ ; otherwise require $\int h_i\,\mathrm{d}\mu=0$ . A function at an auxiliary position may depend on the entire auxiliary list.

The paper calls $i\in A$ an output role. We call it an auxiliary position: its function uses $d_Y(y_i)$ rather than $x_Y(y_i)$ .

By B Lemma 5.2, each position's function is also centered conditional on its own parameter $P_i$ . The initial suspension has no auxiliary variables, and the positive correlation from Part I supplies a nonzero example with $A=\varnothing$ . Its phase weight is a function of the distal parameters.

Among all such systems and bounded tests, let $r$ be the largest number of auxiliary positions admitting a nonzero correlation. It exists because the possible values are $0,1,2,3,4$ and $0$ already occurs. No maximum over a family of real-valued correlations is required.

2. Four auxiliary positions are impossible

Source: B Proposition 6.1, p.18, the first paragraph of the proof.

If $A=I$ , preservation of the three groups and Corollary 5.4 give a product law for all auxiliary spatial variables in the four coordinates, conditional on the full old parameter array $P$ . Thus

\mathbb E_{\lambda_Y}\left[ \prod_{i=1}^{4}h_i(d_Y(y_i))\mid P\right] =\prod_{i=1}^{4}\int h_i\,\mathrm{d}\mu^{\otimes m}=0.

Multiplication by any $\Phi(P)$ and integration still gives zero, contradicting the nonzero correlation in (Corr).

3. Three auxiliary positions would yield too many orthogonal functions

Source: the remainder of B Proposition 6.1, pp.18–19.

3.1 Repeatedly retaining one coordinate at speed one

Suppose $|A|=3$ , and let $b$ be the unique nonauxiliary position. Set

Y_0=Y, \quad Y_{n+1}=M_b(Y_n).

At every step, use all auxiliary copies supplied by the group-preservation proposition. Projection from the new system onto coordinate $b$ has speed $b/b=1$ , so it is a flow factor onto the preceding system.

A state of $Y_n$ is an array indexed by $I^n$ whose entries are original $Y$ -states. Through the corner $(b,\ldots,b)$ , consider the $n$ coordinate lines.

3.2 Every line has the same selected law

The ordered four states on every coordinate line have law $\lambda_Y$ . This follows by induction from the construction, without assuming symmetry between rows and columns.

If the varying coordinate is the newly added one and the others are fixed at $b$ , successive old-coordinate projections define a speed-one factor. Compatibility of the selected law with projections gives $\lambda_Y$ as the law of this new line.

If an old coordinate varies while the new coordinate is fixed at $b$ , the new law's marginal at coordinate $b$ is the preceding array measure. The induction hypothesis then identifies the line law.

Thus every line has the required distribution, without interchanging coordinate axes.

3.3 Different lines use disjoint lists away from the corner

Write the states on line $a$ as $(y_i^{(a)})_{i\in I}$ , with parameter array $P^{(a)}$ . Define

H_a=\Phi(P^{(a)})\prod_{i\ne b}h_i(d_Y(y_i^{(a)})), \quad H=h_b(x_Y(\text{corner})).

Two different lines intersect only at the corner, and the product defining $H_a$ does not use the corner's auxiliary list. The finitely iterated form of Corollary 5.4 gives a product law for the auxiliary lists throughout the array, conditional on all old parameters.

For $a\ne a'$ , the auxiliary variables used by $H_a$ and $H_{a'}$ are disjoint. Their parameter weights are fixed under the conditioning, and the spatial functions are centered. Hence

\mathbb E[H_aH_{a'}]=0.

Moreover, $\|H_a\|_2\le K$ , where $K$ depends only on the fixed $\Phi,h_i$ and not on the array dimension $n$ .

3.4 The Bessel contradiction

Denote the original nonzero correlation by $c\ne0$ . Since every line has the same law,

\langle H,H_a\rangle=c \quad(1\le a\le n).

If some $H_a$ has zero norm, this already contradicts $c\ne0$ . Otherwise normalize these pairwise orthogonal vectors and apply Bessel's inequality:

\sum_{a=1}^{n}\frac{|c|^2}{\|H_a\|_2^2}\le\|H\|_2^2, \quad n|c|^2\le K^2\|H\|_2^2.

The corner's one-coordinate marginal is always the original measure, so $\|H\|_2$ is also independent of $n$ . Taking arbitrarily large $n$ gives a contradiction.

The argument is applied separately for each finite $n$ . No common infinite-array space is needed. A nonzero correlation can therefore have at most two auxiliary positions: $r\le2$ .

4. Which conditional information must survive extensions?

Source: B §6.2, pp.19–20, equations (6.2)–(6.3).

On $(Y^4,\lambda_Y)$ , define five $\sigma$ -algebras:

\mathcal D_0(Y)=\sigma(P), \quad \mathcal D_l(Y)=\sigma(P,(y_i)_{i\ne l}), \quad l\in I.

The first, $\mathcal D_0$ , contains only the parameters. The algebra $\mathcal D_l$ contains all parameters and the three complete states other than state $l$ .

If $\rho:Y'\to Y$ is a flow extension over $C$ , the coordinatewise map $\rho^4$ pushes $\lambda_{Y'}$ to $\lambda_Y$ . The new distal factor contains the pullback of the old one, and the corresponding new states determine the old states. Therefore

(\rho^4)^{-1}\mathcal D_l(Y)\subseteq\mathcal D_l(Y') \quad(l=0,1,2,3,4)

with inclusion understood modulo null sets.

4.1 The full statement of Proposition 6.2

Every standard flow extension $Y_0\to C$ has a further standard flow extension $Y\to Y_0$ such that, for every further extension $\rho:Y'\to Y$ , every $f\in L^2(\lambda_Y)$ , and every $l\in\{0\}\cup I$ ,

\mathbb E_{\lambda_{Y'}}[f\circ\rho^4\mid\mathcal D_l(Y')] = \mathbb E_{\lambda_Y}[f\mid\mathcal D_l(Y)]\circ\rho^4. \tag{Stable}

Equation (Stable) is an equality of function classes after pulling back to $L^2(\lambda_{Y'})$ . It is enough that it holds for each fixed extension and test. No pointwise full-measure set common to all extensions and all $L^2$ functions is asserted.

If the initial system has three auxiliary groups, their pullbacks can be retained in the extension with the same three-group conditional laws.

The paper calls a system satisfying (Stable) saturated. This means only that the five specified conditional expectations are stable under the allowed extensions; it does not fix every conditional distribution.

5. The increasing-energy construction

5.1 Monotonicity of the energy

Fix a function $f$ on a selected-law space and a type $l$ . Define

\mathcal E_Y(f,l) =\|\mathbb E_{\lambda_Y}(f\mid\mathcal D_l(Y))\|_2^2.

Factor pullback is an isometric embedding in $L^2$ , and extensions increase the conditioning information. The orthogonal projection norm is therefore nondecreasing, while

0\le\mathcal E_Y(f,l)\le\|f\|_2^2.

5.2 Countably many tests suffice

Starting from $Y_0$ , construct an extension chain $Y_0\leftarrow Y_1\leftarrow\cdots$ . Each space $L^2(\lambda_{Y_n})$ is separable. At every new stage, choose a countable dense set of functions and retain all previously chosen functions by pullback.

Use a countable interleaving so that every introduced function and every type $l$ is revisited infinitely often with errors tending to zero. For example, enumerate quadruples consisting of introduction stage, function index, type, and accuracy, processing each only after its stage exists.

At a step treating $(f,l)$ , choose a standard extension whose energy is within the prescribed error $\varepsilon_n$ of the supremum over all standard extensions. The supremum is finite, and the identity extension makes the collection nonempty. The definition of supremum therefore permits this choice.

This is an existence argument. It does not assume that any extension attains the supremum exactly.

5.3 The inverse limit retains the selected tests

Take the countable inverse limit $Y$ using the full-countable-product construction from Part I. It accommodates fixed-time factor relations modulo null sets without requiring a compatibility set invariant pointwise at all real times.

The selected laws are compatible with the stage projections, and the union of the pulled-back stage $L^2$ spaces is dense in $L^2(\lambda_Y)$ . The retained dense tests are consequently dense in the limit space.

No claim is needed that $W_Y$ equals the inverse limit of the stage maximal distal factors. Only containment of old factors in new ones is used.

5.4 Comparison with any further extension

Fix a retained function $f$ , a type $l$ , and any extension $Y'\to Y$ . If stage $n+1$ optimized these data to error $\varepsilon_n$ , then $Y'$ is also an extension of $Y_n$ . Hence

\begin{aligned} \mathcal E_{Y'}(f,l) &\le\sup_{R\to Y_n}\mathcal E_R(f,l)\\ &\le\mathcal E_{Y_{n+1}}(f,l)+\varepsilon_n \le\mathcal E_Y(f,l)+\varepsilon_n. \end{aligned}

Along the subsequence revisiting these data, $\varepsilon_n\to0$ . Monotonicity gives the reverse inequality, so the energies agree.

Let $P_0f$ and $P_1f$ be the old and new conditional projections. Since the old subspace is contained in the new one,

\|P_1f\|_2^2 =\|P_0f\|_2^2+\|P_1f-P_0f\|_2^2.

Equality of energies forces equality of the projections, proving (Stable) for retained functions. Conditional expectation is an $L^2$ contraction and pullback is isometric, so density extends the conclusion to every $f\in L^2(\lambda_Y)$ .

Pullback retains the auxiliary groups through extensions and the limit without changing their joint law. B Lemma 5.2 supplies the required laws with the enlarged parameter conditioning. The stable extension therefore remains in the same class of three-group systems.

6. A stable extension retains the maximal nonzero correlation

Choose a nonzero correlation attaining the maximal auxiliary-position count $r$ , then take a stable extension $Y$ by Proposition 6.2. Pull back all old functions. The old parameter weight remains measurable in the new parameters, so the correlation and its nonvanishing are unchanged.

Since $r\le2$ , choose $b\in I\setminus A$ . Set

Z=M_b(Y), \quad \theta:Z\to Y, \quad \theta((y_j)_j)=y_b.

The one-state measure of $Z$ is $m_Z=\lambda_Y$ . We now give it a three-group auxiliary structure adapted to the chosen correlation, instead of automatically retaining the preceding grouping of all copies.

7. Regrouping the three lists away from the corner

Source: B §6.3, Lemma 6.3, pp.20–21, equation (6.4).

In coordinate $b$ , retain all old auxiliary variables $d_Y(y_b)$ . For each $j\ne b$ , add the list

Q_j= \begin{cases} d_Y(y_j),&j\in A,\\ x_Y(y_j),&j\notin A. \end{cases}

In the second case, the added object is the suspension factor at that coordinate; its spatial variable is the first coordinate of the factor. Label the new groups by $j\in I\setminus\{b\}$ and put $Q_j$ into new group $j$ . Assign the three old groups at coordinate $b$ bijectively to the new groups. Every factor speed remains positive and rational.

7.1 Independence of all new auxiliary variables

First condition on the full old parameter array $P$ . View the new auxiliary variables as lists at four positions: at position $b$ the list is $d_Y(y_b)$ , and elsewhere the lists are $Q_j$ .

Any proper subset of these lists has its fixed spatial product law. Equation (Full) from Part II separates the states conditional on their parameters, and B Lemma 5.2 supplies the fixed product law within each list conditional on its own parameter.

Take bounded real tests of the four lists and split each into its mean and centered part. Every term except the fully centered one is determined by the proper-subset laws.

If the conditional expectation of the fully centered product were nonzero, its sign could be used as a bounded parameter weight to give a nonzero integral. This would be a correlation of the form (Corr) on the original system $Y$ , with auxiliary positions $A\cup\{b\}$ , one more than $r$ . That contradicts maximality.

The four lists therefore also have their full spatial product law conditional on $P$ .

7.2 Independence of the original variable and any two new groups

Retain the new original variable $x_Y(y_b)$ and any proper subset of new groups. At least one position $j\ne b$ is omitted. At position $b$ , only a proper subset of the old three groups remains, since the old and new group labels correspond bijectively.

The complete state positions involved form a proper subset of $I$ . First apply (Full), conditional on $P$ , to separate the positions. At position $b$ , use the old three-group rule already upgraded by Lemma 5.2 to conditioning on $W_Y$ , separating the original variable from the retained old groups. Precisely, for every proper subset $J\subsetneq\{1,2,3\}$ of old groups,

\operatorname{Law}_{m_{Y,w}} \left(x_0,(x_\alpha)_{\alpha\in\cup_{j\in J}\mathcal D_j}\right) =\mu\otimes \bigotimes_{\alpha\in\cup_{j\in J}\mathcal D_j}\mu

for almost every $w$ . Since $P_b=w$ has already been given, the original phase-conditional definition would not suffice. The stronger conclusion of Lemma 5.2 is what supplies all the required spatial product laws.

All new phases are functions of $P$ . Conditioning the two product laws again from $P$ onto the new phases leaves the constant product measure unchanged. The new system $Z$ thus satisfies both conditions in Definition 5.1.

8. The two conditional expectations needed next

Source: B p.21, equations (6.5)–(6.6).

Define the product of the three factors outside position $b$ :

q((y_j)_j)= \prod_{j\in A}h_j(d_Y(y_j)) \prod_{j\notin A\cup\{b\}}h_j(x_Y(y_j)).

In a single $Z$ -state, this is also a function of the three added lists $Q_j$ . Set $y=y_b$ and $e=\omega_Z$ , and choose Borel versions of the conditional expectations

g(P,y)=\mathbb E_{\lambda_Y}[q\mid P,y_b], \quad u(e,y)=\mathbb E_{m_Z}[q\mid\omega_Z,y_b].

Standard Borel factor realizations and measurable disintegration permit globally bounded Borel representatives with

|g|,|u|\le\|q\|_\infty.

They may be set to zero off the sets carrying the actual distributions. Both expectations use the state measure $m_Z=\lambda_Y$ . The difference is that the new parameter $e$ contains the old parameter array $P$ and may carry more information. Write the factor map as $R$ , so $P=R(e)$ holds modulo null sets. The tower property gives

g(P,y)=\mathbb E[u(e,y)\mid P,y] \tag{Tower}

under the actual distribution of $(P,y)$ . This does not imply $u(e,y)=g(P,y)$ ; extra conditioning can change a conditional expectation. The use of (Stable) must follow the specific conditioning fields in the square array below.

The original nonzero correlation equals

\mathbb E[\Phi(P)h_b(x_Y(y_b))g(P,y_b)],

Consequently, the function $z\mapsto g(P(z),y_b(z))$ is nonzero in $L^2(\lambda_Y)$ . The next part uses stability to compute row and column conditional laws, then uses nonvanishing of $g$ to produce an additional auxiliary position.

Guide to the original text and scope

B pp.18–19, equation (6.1), Proposition 6.1: auxiliary-position counts, iterated arrays, and the Bessel contradiction.
B pp.19–20, equation (6.2), Proposition 6.2, equation (6.3): five conditioning fields and a stable extension.
B pp.20–21, equation (6.4), Lemma 6.3: regrouping three lists using the maximal correlation.
B p.21, equations (6.5)–(6.6): the functions $q,g,u$ and nonvanishing of $g$ .

The conclusions use the previously specified selected laws, three-group preservation, and distal conditional laws, retaining their dependence on higher-order mixing and the other structural inputs. The square-array conditioning and final nonvanishing argument are deferred to the next part.