Unit 5, Part II: Minimal support and the maximal number of auxiliary positions

This part studies A §§4–5, pp.10–17. At arbitrary finite order, no pointwise theorem for every proper-subset average on general auxiliary systems is available. A first minimizes support among all nonzero correlations, then maximizes the number of auxiliary positions at that fixed support size. Smaller-support correlations must vanish. This determines joint laws of admissible spatial lists, not joint laws of complete states.

Throughout, fix the same $n$, suspension system $C$, selector sequence $(N_a)$, and free ultrafilter $\mathcal U$. Every flow extension uses their pullbacks; they are not chosen anew when comparing different admissible systems.

Preservation of the three auxiliary groups under selection still depends on the companion manuscript R, invoked by A Theorem 2.1. Its precise assertion is that, for every fixed finite $s\ge2$ and bounded functions $g_1,\ldots,g_s$,

The deduction of this assertion from ordinary mixing in R is developed in the companion reading on multiple mixing. The argument below explicitly assumes (HM-A), the conditional laws from Part I, and the previously stated relative structure results.

1. Defining the three auxiliary groups

Source: A Definition 4.1, pp.10–11, equation (4.1).

Fix $I=\{1,\ldots,n\}$. Let $Y$ be a standard probability-preserving flow over the suspension $C$, with original factor $\pi_Y:Y\to C$. Give finitely many probability-preserving maps $D_\gamma:Y\to C$ with positive rational speeds $q_\gamma$, satisfying, for each fixed $t$,

Partition the auxiliary indices into three groups $\mathcal D_1,\mathcal D_2,\mathcal D_3$, allowing empty groups. Write

Let $V$ collect the phases of the original and all auxiliary factors. Require

and, for every proper subset $H\subsetneq\{1,2,3\}$,

An empty list is represented by a one-point probability space. The second condition includes empty $H$, making $x_0$ independent of the full phase array. It does not make $x_0$ independent of all three groups jointly.

The paper calls the flow with all these data a channel. Below we use the direct description: a system with three auxiliary groups.

2. Which parameters change in the results from Unit 4?

2.1 From phase conditioning to distal-parameter conditioning

A Lemma 4.2 (p.11) asserts that (G1-A) and (G2-A) remain valid when conditioning on $V$ is replaced by conditioning on the maximal distal factor $W_Y$. Further flow extensions retain these laws.

The proof is the same as for B Lemma 5.2 in Unit 4. Clearing the rational speed denominators gives an integer time $h$ that fixes every phase. The spatial tuple then evolves under a product of positive integer powers. Ordinary mixing makes this product system mixing. Correlations tend to zero at each fixed phase; a uniform bound and dominated convergence give relative weak mixing over the phase base. The factor $W_Y$ is relatively distal at the same time $h$, so relative disjointness preserves the fixed spatial product law.

This step does not use (HM-A). Higher-order mixing enters the repeated spatial readings in the next step. On extensions, first pull back all existing variables without changing their joint law, then repeat the argument for the new maximal distal factor.

2.2 Copying the variables to all $n$ positions

A Proposition 4.3 (pp.11–14) states that, on $M_b(Y)$, retaining the original factor $\pi_Y(y_b)$ and taking all $D_\gamma(y_i)$, $i\in I$ as auxiliary factors, with the old group label of $\gamma$, preserves both conditional laws.

Unit 4 developed every estimate in this preservation argument. Passing from four positions to a fixed $n$ positions requires the following finiteness checks, rather than a new arbitrary-order pointwise theorem.

Write $t=hl+s$ and $0\le s<h$, and set $e_\gamma=hq_\gamma\in\mathbb N$. The time of each auxiliary reading is

The shifts $a_{\gamma,i}$ have finitely many patterns. The old original-variable reading occurs at $hbl+a_0$. For a selected length $N=c2^k$, removing an incomplete final interval of length $h$ and changing normalization costs at most $2h/N$.

For a proper subset of auxiliary groups, the original variable is independent of those groups conditional on phase. Whenever an auxiliary coordinate contains a centered function, (HM-A) makes its correlations at different positions tend to zero. The finite shift patterns admit a common threshold. The tail supremum of the Cesàro means tends to zero, controlling any sufficiently large point-dependent length. If only the original variable is centered, apply Birkhoff to $T^{hb}$.

If all three groups contain centered functions, obtain unit-ball vectors $u_l,v_l,w_l$ in three real Hilbert spaces. After finitely many initial indices are deleted, the first two sequences have inner products bounded by $\eta$ outside bounded-degree exceptional graphs. Bad pairs can only lie in finitely many constraints

For fixed $l$, each constraint permits boundedly many $l'$ because $e_\gamma i'>0$. The mixing order in (HM-A) and the graph degree may depend on the fixed $n$ and tests. Uniformity in $n$ is not required.

For a length event, let $p$ be its conditional probability, let $m=\lfloor c2^k/h\rfloor$, and set

Fix $0<\varepsilon\le1$, choose $0<\eta\le\varepsilon^2/10^4$, then determine the exceptional degree and initial cutoff. Project the covariance of $u_l\otimes v_l$ onto its first $d_m$ eigendirections. Theorem A.1 of A Appendix A is the same projection-counting estimate as B Theorem 4.1, proved in detail in Unit 4. The leading part contributes at most $p(\varepsilon+o(1))$ on this event; the covariance trace and subprobability density bound the remainder by $\sqrt{p/d_m}$.

The dimension growth and summation depend only on $m$, not on there being exactly four positions. Fix $\varepsilon,\eta$ and the resulting degree and cutoff, then take the length-tail supremum. The leading terms sum to at most $\varepsilon+o(1)$. The remainder satisfies

A countable generating algebra gives a common exceptional null set for the two-group event marginals. Projections need only be preselected for countably many lengths and finitely many shift patterns. Finally take $a\to\infty$ and then $\varepsilon\downarrow0$. These checks justify the previous estimates for every fixed $n$ here.

2.3 The full auxiliary product law given the old parameters

A Corollary 4.4 (p.14) concludes that

First use the new three-group structure on $M_b(Y)$, then apply Lemma 4.2 conditional on its new maximal distal factor. That factor contains the old parameter array $P$. Conditioning again on the smaller $P$ leaves the fixed product measure unchanged.

3. Two admissible spatial lists at each position

Source: A §5.1, p.14.

At each position $i$, specify in advance one of the following two types:

or

The paper calls these an output role and an input role, respectively. The second type permits dependence on the original variable together with a proper subset of the old groups, making it broader than the corresponding choice in B. A function may ignore some listed coordinates, but its position type is determined by the list fixed in advance.

Each admissible list has a fixed spatial product law, denoted $\kappa_i$. For nonempty $J\subseteq I$, choose bounded real functions $h_i$ satisfying

and consider

where $\Phi$ is any bounded real function of the full parameter array. Lemma 4.2 also gives

4. Product laws from minimal support

Source: A Lemma 5.1, p.15.

4.1 The minimum exists and is at least four

The positive correlation from Part I gives a nonzero example of (Corr-A) on the suspension without auxiliary variables. Across all admissible systems, role lists, and tests, let $k$ be the smallest support of a nonzero correlation. Possible sizes lie in the finite set $\{1,\ldots,n\}$. The initial example makes the attainable set nonempty, so a minimum exists.

If $|J|\le3$, the full-parameter law in A Proposition 3.3 makes these complete states independent conditional on $P$. Centering at each position then gives

Multiplying by any $\Phi(P)$ still gives zero, hence $k\ge4$. If $n<4$, this already contradicts pointwise failure. From now on, consider $4\le k\le n$.

4.2 Expanding every smaller collection of list tests

In any admissible system, choose $s<k$ distinct positions $i_1,\ldots,i_s$ and an admissible spatial list at each. For bounded real tests $F_j$, write

The finite expansion is

For every nonempty $K$, the centered product on the right uses only $|K|\le s<k$ positions. Its integral against any bounded real $\Phi(P)$ vanishes by global minimality of $k$. Thus

Take a common full-measure parameter set for countable Borel generating algebras on the standard spaces. Product tests determine the probability measure, and uniqueness extends the identity to all Borel sets. Hence

This holds on every admissible system because the minimum was chosen globally over those systems. It concerns the specified spatial lists only. Independence of $s$ arbitrary complete states conditional on parameters has not been proved.

5. With minimal support fixed, maximize auxiliary positions

Source: A p.15.

Among nonzero correlations with support exactly $k$, let $r$ be the largest number of auxiliary-variable positions. It lies in $\{0,\ldots,k\}$, so a system and correlation attaining that count can be chosen.

If $r=k$, all lists use only auxiliary variables. Equation (All-outputs) and their zero spatial means make the correlation vanish. Therefore

Fix a correlation attaining $(k,r)$. Let $A\subseteq J$ be its auxiliary-position set, and choose $b\in J\setminus A$. Then

Next, extend only this chosen system while retaining its correlation. The global extremal choices $k,r$ still constrain every admissible extension.

6. Stabilize the parameters together with one complete state

Source: A §5.2, Lemma 5.2, pp.15–16, equations (5.2)–(5.3).

Define

A conditions on one state. B §6.2 instead conditions on the other three states after omitting one position. These conditioning fields are different. A's square-array argument uses the single-state form defined here.

6.1 Inclusion under extensions

For a flow extension $\rho:Y'\to Y$ over $C$, let $(\rho^I)_*\lambda_{Y'}=\lambda_Y$. The pullback of the old $W_Y$ is a distal factor of the new flow and is contained in $W_{Y'}$. The old position $y_i$ is the image of the corresponding new position. Thus

with inclusions understood modulo null sets.

Lemma 5.2 asserts the existence of a standard flow extension $Y_*\to Y$ such that, for every further standard flow extension $\rho:Y'\to Y_*$, every $f\in L^2(\lambda_{Y_*})$, and every $l\in\{0\}\cup I$,

This is an equality of function classes in $L^2(\lambda_{Y'})$. No pointwise full-measure set common to all extensions and tests is required.

6.2 Bounded energies and a countable construction

For a fixed stage test $f$ and type $l$, define the energy

The inclusion of conditioning fields makes this energy nondecreasing under extension, and $\|f\|_2^2$ bounds it. Each stage $L^2$ is separable. Choose a countable dense set and retain all earlier tests by pullback. A countable diagonal arrangement revisits each retained test and each $l$, with prescribed errors tending to zero.

At a stage treating $(f,l)$, choose among all standard extensions one whose energy is within $\varepsilon$ of the attainable supremum. Standard Borel spaces, measures, Borel actions, and factor maps can be coded on a fixed standard space, so this is a supremum over a set. The identity extension makes the candidates nonempty, and boundedness makes the supremum finite. An approximately optimal extension suffices; the maximum need not be attained.

6.3 Comparing the inverse limit with any further extension

Take the countable inverse limit $Y_*$. In the full-product realization from Part I, each fixed time preserves measure, factor relations hold modulo null sets, the stage selected laws are compatible, and stage functions are dense in the limiting $L^2$ space.

Fix a retained test, a type, and any extension $Y'\to Y_*$. If the stages immediately before and after one optimization are $Y_a,Y_{a+1}$ and its error is $\varepsilon_a$, then $Y'$ is also an extension of $Y_a$. Hence

Along the subsequence revisiting these data, let $\varepsilon_a\to0$ to obtain one inequality. Energy monotonicity gives the reverse inequality, so the energies are equal.

Identify the old $L^2$ with an isometric subspace of the new $L^2$. The old and new conditional expectations are orthogonal projections $P_0,P_1$ onto nested closed subspaces. We have

Equality of energies forces equality of projections. Stage density and the contraction property of conditional expectation extend the conclusion to every $f$, proving (Stable-A). No claim that maximal distal factors commute with inverse limits is used.

6.4 The extremal correlation survives the extension

Pullbacks of the original factor and three groups retain the required laws. Factor compatibility of the selected measure preserves the correlation integral, and the old parameter weight becomes a function of the new parameters. The same nonzero correlation still has support $k$ and auxiliary-position count $r$. Denote the stable system again by $Y$.

7. Turning the other support positions into new auxiliary lists

Source: A §5.3, p.16.

Take $Z=M_b(Y)$, whose states are $z=(y_j)_{j\in I}$, measure is $m_Z=\lambda_Y$, and original factor is $\pi_Y(y_b)$.

First retain all old auxiliary factors at position $b$ with their original group labels. For every $j\in J\setminus\{b\}$, add the full list

If $j$ is an original-variable position, its original variable in this list comes from the genuine $C$-valued factor $\pi_Y(y_j)$. We do not add a bare spatial coordinate without a flow structure.

Since $k\ge4$, there is a surjection

Put all of $Q_j$ into new group $\tau(j)$, without splitting the list. At position $j$, old factor speeds are multiplied by $j/b$, so the new speeds remain positive and rational. Their probability-preserving marginals are unchanged.

Two rules are distinct: position $b$ retains the old group labels, while the complete lists at other positions are assigned using $\tau$. Each rule has its own use below.

8. A Lemma 5.3: Proving both new conditional laws

Source: A pp.16–17.

All new phases are functions of the full old parameter array $P$. It is enough to prove the fixed spatial product laws conditional on $P$, then condition again on the full new phase array.

8.1 All new auxiliary lists

The new auxiliary variables form lists at $k$ old positions:

Each is an admissible list in the old system. Take a bounded real test of each list, split it into its mean for the fixed spatial product law and its centered part, and expand.

If an expansion term involves fewer than $k$ centered positions, (Below-k) determines its conditional mean: it vanishes whenever a centered term occurs. Only the term with all $k$ tests centered remains.

If its conditional expectation given $P$ were nonzero, use its sign as a parameter weight. The integral would equal the positive integral of its absolute value. This gives a nonzero correlation on the old system, still with support $k$. All other position types are unchanged, but position $b$ now uses only auxiliary variables. There are $r+1$ auxiliary positions, contradicting maximality of $r$.

If $d_Y(y_b)$ is empty, a centered test on its one-point space is zero, so the conclusion still holds. Thus all new auxiliary variables have their spatial product law conditional on $P$, proving the first new rule.

8.2 The new original variable and any proper subset of new groups

Take $H\subsetneq\{1,2,3\}$ and include the new original variable $x_0(y_b)$. At old position $b$, the retained list is exactly

because the old group labels are unchanged there. This is an admissible original-variable list.

A remaining position $j$ is included exactly when $\tau(j)\in H$; if included, its complete $Q_j$ is retained. Since $\tau$ is surjective and $H$ is proper, at least one $j\in J\setminus\{b\}$ is omitted. Fewer than $k$ old positions are therefore involved.

Apply (Below-k) at those positions to obtain the required full spatial product law. Lemma 4.2 fixes each list's internal spatial law. The new original variable and retained new groups therefore satisfy the second rule.

8.3 Reducing the conditioning and distinguishing the two grouping rules

The two product measures just obtained are constant, independent of $P$. The new phases are $P$-measurable, so the tower property reduces both laws to conditioning on those phases. This proves that $Z$ has the required three-group structure.

Surjectivity of $\tau$ makes the second rule omit a support position. Keeping the old group labels at position $b$ makes its original variable and retained groups an admissible list. Mixing up these two assignments would remove the justification for applying minimal support.

9. The resulting nonzero conditional average

Source: the start of A §6.1, p.17.

Set

The original nonzero correlation is

If $g(P,y_b)=0$ in $L^2(\lambda_Y)$, this integral would vanish, a contradiction. Hence $g\ne0$. Regard the same state space as $Z$; the function $q$ then uses only newly added auxiliary lists. The next part uses the single-state conditioning in (Stable-A) to compare directly the row and column versions of $q$ in the square array.

Scope and remaining work

• A pp.10–14, Definition 4.1, Lemma 4.2, Proposition 4.3, and Corollary 4.4: the three groups and their preservation. Unit 4 supplies the detailed tensor estimate; this part checks its use at any fixed number of positions.
• A pp.14–15, equation (5.1), Lemma 5.1: admissible lists, globally minimal support, and conditional products for smaller lists.
• A pp.15–16, equations (5.2)–(5.3), Lemma 5.2: simultaneous stability of parameter and single-state conditional expectations.
• A pp.16–17, Lemma 5.3: the two conditional laws after regrouping by a surjection.

Nonvanishing of the row–column correlation and the final increase in auxiliary positions remain for the next part, on A §6. Higher-order mixing and the stated structural theorems remain explicit external assumptions.