Unit 5, Part III: Single-state conditioning and nonzero correlations in the square array

This part studies A §6, pp.17–21. Retain the minimal support $k$ , maximal auxiliary-position count $r$ , stable system $Y$ , and regrouped system $Z=M_b(Y)$ from Part II. Stability conditional on all parameters and one state suffices to compute the row–column correlation. Conditional independence of parameters and a Hilbert-space lemma then prove its nonvanishing, increasing the auxiliary-position count to $r+1$ .

The assumptions remain those of Part II. Three-group preservation uses the stated higher-order mixing result; the parameter laws use the low-order pointwise and relative structure theorems. We complete A's deduction under these assumptions, without independently certifying R, H, or every structural theorem.

1. Notation for the square array

Source: A §6.1, p.17, equations (6.1)–(6.3).

The old selected-law space is $(Y^I,\lambda_Y)$ , with $I=\{1,\ldots,n\}$ . Fix support $J\subseteq I$ with $|J|=k$ , auxiliary-position set $A\subseteq J$ with $|A|=r$ , and $b\in J\setminus A$ . Set

q((y_i)_{i\in I})= \prod_{j\in J\setminus\{b\}}h_j(L_j(y_j)), \quad g(P,y_b)=\mathbb E_{\lambda_Y}[q\mid P,y_b]. \tag{g-A}

Part II proved that the function $g(P,y_b)$ is nonzero in $L^2(\lambda_Y)$ .

Regard the same $Y^I$ as the one-state space of the new system $Z$ , with measure $m_Z=\lambda_Y$ . Write $e=\omega_Z(z)$ and define

u(e,y_b)=\mathbb E_{m_Z}[q\mid e,y_b]. \tag{u-A}

The new maximal distal factor contains the old parameter array $P$ . Choose a Borel factor map $R$ such that

R(e)=P, \quad\kappa(e)=R(e)_b \quad\text{modulo null sets}.

The map $\kappa$ is induced on old distal parameters by the flow factor at position $b$ , namely $\theta:Z\to Y$ , where $\theta(z)=y_b$ .

Choose globally bounded real Borel representatives $g,u$ satisfying $|g|,|u|\le\|q\|_\infty$ , setting them to zero off the sets carrying the actual distributions if needed. The definitions are conditional-expectation identities between function classes, with no meaning asserted at every exceptional parameter value. Since $P=R(e)$ , the one-state tower property gives only

\mathbb E[u(e,y_b)\mid P,y_b]=g(P,y_b),

The functions $u$ and $g$ cannot yet be identified.

Now, on $(Z^I,\lambda_Z)$ , write

z_i=(y_{ij})_{j\in I}, \quad E_i=\omega_Z(z_i), \quad E=(E_i)_{i\in I}.

The first index $i$ specifies a row, and the second $j$ a position within it. Write column $b$ and its parameters as

y_i=y_{ib}, \quad p=(\kappa(E_i))_{i\in I}, \quad w=\kappa(E_b). \tag{Square-data}

The common corner is $y_b=y_{bb}$ . Row $b$ has old parameter array $R(E_b)$ , while column $b$ has old parameter array $p$ . Their parameter at position $b$ is the same $w$ .

Column $b$ has law $\lambda_Y$ because $\theta:Z\to Y$ is a speed-one flow factor and selected laws respect factors. Row $b$ has law $\lambda_Y$ because the one-coordinate marginal of $\lambda_Z$ is $m_Z=\lambda_Y$ . The two law identities have different sources and do not imply transpose invariance of the full array.

2. Type-zero stability determines the full column law

Source: the first paragraph of A Lemma 6.1, p.18.

Apply (Stable-A) from Part II to the extension $\theta:Z\to Y$ , first using type $0$ . The old conditioning field is $\sigma(P)$ , which pulls back to $\sigma(p)$ in the array. The new conditioning field is $\sigma(E)$ . Thus, for every $f\in L^2(\lambda_Y)$ ,

\mathbb E_{\lambda_Z}[f((y_i)_{i\in I})\mid E] =\mathbb E_{\lambda_Y}[f\mid P]\big|_{P=p}.

Apply this first to indicators from a countable Borel generating algebra of the old state space $Y^I$ . Take a common $E$ -full-measure set and use uniqueness of measures to obtain the conditional probability kernel identity

\operatorname{Law}_{\lambda_Z}((y_i)_{i\in I}\mid E) =\lambda_Y(\,\cdot\mid P=p). \tag{Column-A}

Taking marginal $b$ and using the one-state case of A Proposition 3.3 gives

\operatorname{Law}_{\lambda_Z}(y_b\mid E)=m_{Y,w}. \tag{Corner-A}

Since $w=\kappa(E_b)$ , reducing the conditioning from $E$ to $E_b$ leaves the right side unchanged. Row $b$ has marginal $m_Z$ , so a single $Z$ -state satisfies

\operatorname{Law}_{m_Z}(y_b\mid e)=m_{Y,\kappa(e)}. \tag{Single-corner}

Thus even after conditioning on the new parameter $e$ , the corner law depends only on the old parameter $\kappa(e)$ . This comes from the specified stability, not from a general principle that added conditioning leaves independence unchanged.

3. Type- $b$ stability directly determines the column function's conditional mean

Source: A p.18, equation (6.5).

For the same extension $\theta:Z\to Y$ , now use type $b$ . The old data are $(P,y_b)$ and the new data are $(E,z_b)$ . Taking the old test to be $q$ , (Stable-A) gives

\mathbb E_{\lambda_Z}[q((y_i)_{i\in I})\mid E,z_b] =g(p,y_b). \tag{Direct-A}

The old complete state $y_b$ corresponds to the new whole-row state $z_b$ . This is exactly the purpose of the single-state conditioning field $\mathcal D_b(Y)=\sigma(P,y_b)$ . The argument requires neither joint independence of the other $n-1$ rows nor a general-system pointwise theorem with $n-1$ factors.

Multiply (Direct-A) by $q(z_b)$ and condition on $E$ . Then

\mathbb E_{\lambda_Z}[q((y_i)_i)q(z_b)\mid E] =\mathbb E_{\lambda_Z}[g(p,y_b)q(z_b)\mid E].

The one-position case of A Proposition 3.3 for $Z$ says that, given $E$ , the state $z_b$ has law $m_{Z,E_b}$ . Within this fiber, at position $b$ condition further on the state $y_b$ . By (u-A), the conditional mean of $q$ is $u(E_b,y_b)$ . Combining this with (Single-corner) gives A (6.4):

K(E)=\mathbb E_{\lambda_Z}[q((y_i)_i)q(z_b)\mid E] =\int_Yg(p,y)u(E_b,y)\,\mathrm{d}m_{Y,w}(y). \tag{Pair-A}

The integral is a real inner product of two potentially different vectors. Its sign is not known in advance.

3.1 Why parameter-dependent tests are legitimate

On the fixed one-state space $Z$ , choose the regular conditional probability kernel $m_{Z,e}$ . For a countable generating algebra of $Y$ and its sets $B$ , the definition of $u$ and (Single-corner) imply

\int q(z)\mathbf1_B(\theta z)\,\mathrm{d}m_{Z,e}(z) =\int_Bu(e,y)\,\mathrm{d}m_{Y,\kappa(e)}(y)

simultaneously for almost every $e$ . Both sides are bounded finite signed measures on $Y$ , so the identity holds for every bounded Borel test in $y$ . The kernel identity can therefore be tested with $y\mapsto g(p,y)$ , even though $p$ is itself part of the given $E$ .

The marginal of $E_b$ is $(\omega_Z)_*m_Z$ , so the exceptional old-kernel set of $e$ remains null in the square array. Equation (Direct-A) is likewise an identity of function classes for the actual array measure. Multiplication by a bounded row function and further conditioning are legitimate. No common exceptional set over uncountably many parameters is introduced.

4. Conditional independence of the two parameter arrays over the corner parameter

Source: A Lemma 6.2, pp.18–19.

The conclusion is

p\text{ and }E_b\text{ are conditionally independent given }w\text{ }. \tag{PI-A}

4.1 Expressing the integral by the deterministic-length parameter law

Take bounded real $F_0$ on $W_Z$ and bounded real $H_i$ on $W_Y$ . Under the one-state parameter law $\nu_Z=(\omega_Z)_*m_Z$ , choose $\overline F_0$ satisfying

\overline F_0(\kappa(e)) =\mathbb E_{\nu_Z}[F_0(e)\mid\kappa(e)].

All variables are functions of the new parameter array $E$ . Apply (Parameter-n) from Part I to $Z$ to express

\mathbb E_{\lambda_Z}\left[F_0(E_b)\prod_{i\in I}H_i(p_i)\right]

as the limit, as $L\to\infty$ , of the deterministic-length integrals

\frac1L\int_0^L\int_Z F_0(\omega_Z(S_Z^{bt}z)) \prod_{i\in I}H_i(\kappa(\omega_Z(S_Z^{it}z))) \,\mathrm{d}m_Z(z)\,\mathrm{d}t. \tag{Fixed-L}

4.2 A measure-preserving substitution at each fixed time

Fix $t$ and set $z'=S_Z^{bt}z$ . The spatial integral in (Fixed-L) becomes

\int_ZF_0(\omega_Z(z')) \prod_{i\in I}H_i(\kappa(\omega_Z(S_Z^{(i-b)t}z'))) \,\mathrm{d}m_Z(z').

At this fixed $t$ , every factor in the product is $\kappa\circ\omega_Z$ -measurable. Indeed,

\kappa\circ\omega_Z=\omega_Y\circ\theta, \quad\theta\circ S_Z^s=S_Y^s\circ\theta

holds modulo null sets at each fixed time, and every flow time preserves the old factor $W_Y$ . Reading the old parameter at time $(i-b)t$ is therefore a measurable function of the time-zero old parameter. The invertible flow defines negative times.

Replace $F_0(\omega_Z(z'))$ by its conditional expectation onto that old parameter, namely $\overline F_0(\kappa(\omega_Z(z')))$ . After reversing the substitution, for every deterministic $L>0$ the value of (Fixed-L) is exactly unchanged when $F_0$ is replaced by $\overline F_0\circ\kappa$ .

The substitution changes neither $t$ nor $[0,L]$ . There is no endpoint error and no displacement of a point-dependent selector $N_a$ : the formula is already a deterministic-length parameter average. All integrands on the original flow are jointly measurable and bounded. Fubini permits fixed-time identities modulo null sets inside the time integral, without a factor model pointwise equivariant simultaneously at all real times.

4.3 Conditional independence from the test identities

Let $L\to\infty$ to obtain

\mathbb E_{\lambda_Z}\left[F_0(E_b)\prod_iH_i(p_i)\right] =\mathbb E_{\lambda_Z}\left[\overline F_0(w)\prod_iH_i(p_i)\right].

Parameter rectangle tests on $p$ generate its $\sigma$ -algebra. Hence

\mathbb E[F_0(E_b)\mid p]=\overline F_0(w).

The marginal of $E_b$ is $\nu_Z$ , so the right side also equals $\mathbb E[F_0(E_b)\mid w]$ . Since $w=p_b$ is a function of $p$ , this is (PI-A). Choosing versions using countable generating tests gives a kernel identity for the conditional joint law.

5. Comparing $g$ and $u$ in the same old fiber

Source: A §6.3, pp.19–20, equations (6.6)–(6.7).

5.1 Two parameter measures

Let $\alpha_w$ be the law of the column parameter array $p$ conditional on its parameter at position $b$ , denoted $w$ . Let $\beta_w$ be the law of $E_b$ conditional on $w$ . Equation (PI-A) gives

\operatorname{Law}(p,E_b\mid w)=\alpha_w\otimes\beta_w. \tag{Parameter-product}

Row $b$ and column $b$ both have the old law $\lambda_Y$ , and in both full old parameter arrays, coordinate $b$ is $w$ . Thus

R_*\beta_w=\alpha_w \quad\text{for almost every }w. \tag{Pushforward}

Equal row and column marginals give only (Pushforward). Writing their joint parameter law as a product also requires the conditional independence just proved.

5.2 Strongly measurable random vectors in the fibers

For almost every fixed $w$ , consider the real Hilbert space

\mathcal H_w=L^2(m_{Y,w};\mathbb R).

Rational simple functions from a countable generating algebra of the standard Borel space $Y$ are dense in each such $L^2$ space. Hence $\mathcal H_w$ is separable.

Fix globally bounded Borel representatives $g,u$ . Use increasing finite partitions from the algebra to average them conditionally in the variable $y$ . Each approximation lies in a finite-dimensional subspace. Its coefficients are integrals against the fixed fiber measure, measurable in $p$ or $e$ ; set coefficients on null atoms to zero. These approximations converge to the original functions in $\mathcal H_w$ . Thus

p\longmapsto g(p,\cdot), \quad e\longmapsto u(e,\cdot)

are strongly measurable random vectors. Their norms are bounded by $\|q\|_\infty$ , so they are square-integrable.

5.3 From the scalar tower property to Hilbert-space-valued conditional expectation

In one $Z$ -state, denote the corner state by $y$ . By (Single-corner),

\operatorname{Law}(e,y\mid w) =\beta_w(\mathrm{d}e)m_{Y,w}(\mathrm{d}y). \tag{State-product}

On the other hand, $m_Z=\lambda_Y$ and $P=R(e)$ make the scalar tower property read

\mathbb E[u(e,y)\mid R(e),y]=g(R(e),y).

Equation (State-product) says that, after further conditioning on $R(e)$ , the law of $e$ has no additional dependence on $y$ . This conditional product law is essential.

To place the identities on one $w$ -full-measure set, let $s_j$ range over the rational simple functions above and $B$ over a countable Borel generating algebra of the old parameter space. For almost every $w$ ,

\int_{R(e)\in B} \langle u(e,\cdot),s_j\rangle_{\mathcal H_w} \,\mathrm{d}\beta_w(e) = \int_B\langle g(p',\cdot),s_j\rangle_{\mathcal H_w} \,\mathrm{d}\alpha_w(p'). \tag{Tests-A}

First take a common full-measure set for the countably many $(B,j)$ . Fix such a $w$ . Uniqueness of measures extends $B$ to all Borel sets, and density of $s_j$ identifies the vector. Together with (Pushforward), these are exactly the test identities for

\mathbb E_{\beta_w}[u(e,\cdot)\mid R(e)] =g(R(e),\cdot) \quad\text{in }\mathcal H_w. \tag{Hilbert-tower}

6. A nonzero random vector cannot be almost surely orthogonal to an independent copy

Source: A p.20, the argument following (6.8).

Let $V$ be a square-integrable random vector in a separable real Hilbert space and $V'$ an independent copy with the same law. If $\langle V,V'\rangle=0$ almost surely, then $V=0$ almost surely.

Suppose $V$ is not almost surely zero. Choose $\varepsilon>0$ so that the event $\|V\|\ge\varepsilon$ has positive probability. Separability provides a countable cover by balls of radius $\varepsilon/4$ . One ball meets this event with positive probability, so independence gives positive probability that both copies lie in that intersection. On this event,

\|V\|,\|V'\|\ge\varepsilon, \quad\|V-V'\|\le\varepsilon/2,

and therefore

\langle V,V'\rangle =\frac{\|V\|^2+\|V'\|^2-\|V-V'\|^2}{2} \ge\frac{7\varepsilon^2}{8}>0,

a contradiction.

7. The row–column correlation cannot vanish almost everywhere

Suppose $K(E)$ in (Pair-A) vanished almost everywhere. By (Parameter-product), for almost every $w$ ,

\langle g(p,\cdot),u(e,\cdot)\rangle_{\mathcal H_w}=0 \quad\text{for }\alpha_w\otimes\beta_w\text{-almost every }(p,e).

Fix almost every $p$ . In the $e$ variable, condition on $R(e)=p'$ . Boundedness permits interchange of conditional expectation with the continuous linear functional defined by the inner product. Equation (Hilbert-tower) and $R_*\beta_w=\alpha_w$ then yield

\langle g(p,\cdot),g(p',\cdot)\rangle_{\mathcal H_w}=0 \quad\text{for }\alpha_w\otimes\alpha_w\text{-almost every }(p,p').

Under parameter law $\alpha_w$ , apply the lemma in §6 to the random vector $g(p,\cdot)$ . It must vanish for almost every $w$ . The old selected law has the single-state law $m_{Y,P_b}$ conditional on the full parameters. Consequently,

\begin{aligned} \|g(P,y_b)\|_{L^2(\lambda_Y)}^2 &=\int_{W_Y}\int \|g(p,\cdot)\|_{\mathcal H_w}^2 \,\mathrm{d}\alpha_w(p)\,\mathrm{d}\nu_Y(w)\\ &=0. \end{aligned}

This contradicts the nonvanishing in (g-A). Thus $K$ is not the zero function.

The independent copies here are two $g$ vectors in a conditional parameter product space. No independence of the actual whole row and column has been asserted.

8. One more auxiliary position on the same minimal support

Source: A §6.4, p.20, equation (6.9).

Take the bounded real parameter function $\Theta(E)=\operatorname{sgn}K(E)$ . Then

\mathbb E_{\lambda_Z}[\Theta(E)q((y_i)_i)q(z_b)] =\mathbb E_{\lambda_Z}|K(E)|>0. \tag{New-correlation}

This product still uses only the positions $J$ . The column function contributes one factor at each $J\setminus\{b\}$ , and the row function $q(z_b)$ occupies position $b$ . Check each type against (Corr-A) from Part II.

8.1 Old auxiliary positions remain auxiliary

If $i\in A$ , the column factor is $h_i(d_Y(y_{ib}))$ . The new state at position $i$ is $z_i$ , in which the full auxiliary list at old position $b$ was retained. The function therefore depends only on new auxiliary variables, ignoring other listed coordinates if necessary, and remains an auxiliary-position test.

The coordinates it uses have their old auxiliary product marginal under the new spatial product law, so its mean remains zero.

8.2 The remaining original-variable positions retain an admissible proper subset

If $i\in J\setminus(A\cup\{b\})$ , the old list is

L_i=(x_0,d_{Y,H_i}), \quad H_i\subsetneq\{1,2,3\}.

The new column function uses $x_0(y_{ib})$ , exactly the new system's original variable, together with the auxiliary variables at old position $b$ formerly belonging to $H_i$ . Their group labels were retained. Hence the function depends on the new original variable and the new groups $H_i$ , ignoring their additional coordinates. Since $H_i$ remains proper, this is an admissible original-variable list.

The product marginal of the used coordinates agrees with the old list law, so the mean remains zero. This is where A needs its broader list definition than B.

8.3 Position $b$ becomes a new auxiliary position

The function $q(z_b)$ uses the newly added lists $Q_j$ , $j\in J\setminus\{b\}$ , hence only auxiliary variables. Under the full new auxiliary product law, those lists are independent and each $h_j$ is centered. Thus

\int q\,\mathrm{d}(\text{new auxiliary product measure}) =\prod_{j\in J\setminus\{b\}} \int h_j\,\mathrm{d}\kappa_j=0.

The set $J\setminus\{b\}$ is nonempty because $k\ge4$ . Thus $q(z_b)$ is also an admissible centered auxiliary test.

Equation (New-correlation) is therefore a nonzero correlation with the same support $k$ and auxiliary-position count $r+1$ , contradicting maximality of $r$ at minimal support.

9. The conclusion of this conditional proof

Source: A p.21, the completion of Theorem 1.1.

A Theorem 1.1 claims that, if $(X,\mathcal F,\mu,T)$ is an invertible, bimeasurable, probability-preserving mixing system, then for each fixed integer $n\ge2$ and fixed bounded measurable tuple $f_1,\ldots,f_n$ ,

\frac1N\sum_{l=1}^{N}\prod_{i=1}^{n}f_i(T^{il}x) \longrightarrow\prod_{i=1}^{n}\int_Xf_i\,\mathrm{d}\mu \quad\text{for almost every }x,

as $N$ runs through all positive integers. The exceptional null set may depend on the system, $n$ , and the fixed tuple. No simultaneous assertion over every bounded function is required.

The contradiction developed in Unit 5 is as follows. Pointwise failure leaves a nonempty support after centering. Selected lengths and a phase weight retain a strictly positive correlation. The conditional law for at most three states forces minimal support $k\ge4$ . Minimal support and maximal auxiliary-position count permit regrouping into three groups. Single-state stability and fiber nonvanishing then produce $r+1$ , a contradiction.

The deduction is complete conditional on the stated smooth triple, relative structure, and higher-order mixing inputs. The original space need not be standard: use the fixed tuple's countable orbit-reading factor and pull back the Borel full-measure convergence set. The supplement gives the full standard-model argument and addresses completion.

For any fixed collection of pairwise distinct positive integer slopes, insert the constant $1$ at missing positions to obtain a direct consequence. This does not extend the arbitrary-finite-order statement to general slopes of mixed signs.

Guide to the original text

A p.17, equations (6.1)–(6.3): $q,g,u$ , the two parameter arrays, and the square-array coordinates.
A p.18, Lemma 6.1, equations (6.4)–(6.5): type-zero and single-state stability, the corner law, and the row–column pairing.
A pp.18–19, Lemma 6.2: changing the time origin on the original flow to obtain conditional parameter independence.
A pp.19–20, equations (6.6)–(6.8): the fiber parameter product, Hilbert-space-valued tower property, and independent-copy lemma.
A pp.20–21, equation (6.9) and completion of the main theorem: increasing auxiliary positions at fixed support and full-length convergence for a fixed tuple.

Unit 5 develops this conditional proof through A §§2–6. The projection count in Appendix A corresponds to the proof of B §4 in Unit 4. The full higher-order mixing proof in R, all external analytic proofs needed by C, and the original full proofs of the relative structure theory remain outside this coverage.