The multiple-mixing input in A and B: a guide to R

When adding auxiliary variables, A and B need correlations of fixed bounded functions at widely separated times to converge to the product of their integrals. They attribute this input to Theorem 1.1 of R. This supplement follows that dependency through the main proof in R, continuing the relevant close-reading units for A and B.

The full title of R is Rokhlin’s multiple-mixing problem for one transformation. References below to a source theorem mean the statement in that preprint; the chapters reconstruct its argument in detail. This supplement does not assess its reception, historical priority, or publication status.

1. The precise statement used by A and B

R Theorem 1.1 (p.2) asserts the following. If $T$ is invertible, bimeasurable, probability-preserving, and mixing in the ordinary sense, then for each fixed $k\ge3$, fixed measurable sets $A_1,\ldots,A_k$, and positive integer gaps $n_1,\ldots,n_{k-1}$, as the smallest gap tends to infinity,

The original probability space need not be standard. The order is the number of sets in the correlation.

A Theorem 2.1 (p.3) and B equation (2.1) (p.4) use the fixed bounded-function version: for every fixed finite $s\ge2$ and fixed $g_1,\ldots,g_s\in L^\infty(\mu)$, at integer times satisfying $\min_{i\ne j}|t_i-t_j|\to\infty$,

These quantifiers provide no mixing rate uniform over varying input functions. Every later assertion of uniformity must identify whether it comes from a fixed finite family of functions, finitely many fixed offset patterns, or a specified norm estimate.

This input enters the extension-closure proof of A Proposition 4.3 (pp.11–14) and its counterpart, B Proposition 5.3 (pp.15–17). It controls both integrals involving several positions in one existing output and near orthogonality in inner products between two time configurations. Although B has only four positions, combining two configurations can involve eight time points.

Terminology

• Slot: an index in the finite set $I$. We also use the word position.
• Parent slot: a position whose complete state is included among the added parameters. Given those parameters, the state is known. In a square array, the corresponding term is parent row.
• Active space: the closed linear span of conditional expectations onto the current coordinate of functions of the specified other coordinates. Using a row or column gives a row-active or column-active space.
• Exact line: the line generated by a modulus-one function $\chi$, namely $\mathbb C\chi$, where $\chi$ can be multiplied by modulus-one functions in the other positions of the same row or column to give a constant phase.
• Parameter tuple: the ordered parameter list $p=(p_1,\ldots,p_k)$.
• Conditional fibre: after fixing the local parameter $u$, the probability space $(Y,\eta_u)$, with $H_u=L^2(\eta_u)$.

These explanations describe the source objects without changing their notation.

2. Reading order: the input and output of each chapter

Chapter 1: start with actual time configurations

From failure configurations to a zero-entropy witness and ordered arrays

Source: Proposition 2.1, pp.4–6, and §3.1, pp.6–7.

The input is a hypothesized minimal failure order $k$. The first output is a standard finite-alphabet process $X$ with zero entropy rate and mixing of every order below $k$. Alongside it come fixed centered one-site functions $f_i$, a positive number $\delta$, separated time configurations $d_i(b)$, and ordered array laws $\rho_n$ at every level.

The chapter develops the past-tail replacement and finite-entropy calculation, then proves the existence of array laws, compatibility on arbitrary coordinate faces, invariance under common time shifts, and joint independence of every proper family of last-axis planes. The Bernoulli joining $(U,V,UV)$ explains why the reduction must use actual time configurations.

Chapter 2: obtaining conditional locality

Saturated array extensions and conditional locality

Source: Definition 3.1, Proposition 3.2, Lemmas 3.3–3.4, and Proposition 3.5, pp.7–13.

The input consists of all laws $\rho_n$ from Chapter 1, beginning with the constant-parameter object. The output is a saturated array object preserving the original $X$ factor and witness correlation, together with product laws for proper row and column subfamilies and exact conditional kernels in the square.

The chapter develops inverse-limit closure of the admissible class, a countable schedule of energy tests, and induction on smaller parent sets. Equation (8) first gives the local dependence of the conditional kernel, which then permits additional conditioning. A finite square example shows why column independence alone does not give locality after parents are added.

Chapter 3: operator analysis at fixed parameters

Operator squares and completeness of exact lines

Source: Definition 4.1 and Lemmas 4.2–4.7, pp.14–21.

The input is a grouped three-by-three square at fixed typical parameters, with the conditional kernels from Chapter 2. The conclusion is that, at each rectangular cell, row-active and column-active projections commute, and their intersection is spanned by exact lines satisfying both row and column multiplicative relations.

The chapter develops the two rectangle identities, projection intertwinings, a fixed positive part, cancellation through dense ranges, and a commuting unitary group. It then explains how a positive trace-class average supplies a common discrete eigenbasis. The reverse inclusion in the completeness statement is proved separately, without assuming active spaces are function algebras.

Chapter 4: using fibrewise conclusions across parameters

Measurable labels and finite moving blocks

Source: Lemma 3.6, pp.13–14; Proposition 4.8, pp.22–23; and §5, pp.23–33.

This chapter returns to the earlier parameter geometry, unnecessary for Chapter 3 but needed now: two parameter tuples meeting diagonally are conditionally independent and identically distributed given the local parameter. Restoring all coordinates from the grouped result gives a full decomposition of one active space and measurable local label groups.

The outputs also include finite blocks whose cardinality, for each fixed block index, is the same in almost every fibre, their time permutations, and a fixed finite block $B$ whose measurable absolute coefficient sum $s_r(u)$ is uniformly bounded and has, for every $r$, the same integral $c>0$.

The chapter develops measurable spectral enumeration, descent to local parameters, counting-measure lifts, the construction of finite blocks, and transfer of the witness. Identifications with an abstract group may be nonmeasurable. Finite sums that are integrated must instead be expressed intrinsically and measurably. Permutation of whole blocks is not strengthened to a constant time rule for individual labels.

Chapter 5: ruling out the finite-block witness

Entropy, single-orbit algebra, and cancellation

Source: Proposition 6.1 and Lemmas 6.3–6.5, pp.33–37; and §7, pp.37–38.

The inputs are zero entropy and ordinary mixing of the original $X$, multiplication of label functions up to constant phases, and the finite blocks from Chapter 4. The conclusion is Cesàro cancellation of absolute correlations along every fixed abstract label orbit, and hence convergence to zero of finite-block averages in each conditional fibre.

The chapter develops an exponential estimate for deterministic words, a union bound for high-probability words, algebraic choices on one orbit, polynomial identities with torsion, and Hilbert-space differencing. Bounded convergence is applied only at the end to measurable, uniformly bounded finite-block sums, contradicting their fixed positive integral.

3. Notation and scope

The same letter may have different local roles. The table distinguishes them rather than identifying the objects.

The source also uses $P,A,B$ for three row groups, a local convention inside a square unrelated to the finite block $B$. In different sections, $N$ may denote an averaging length or $G^p-\operatorname{id}$; the relevant formula specifies its meaning.

4. Source statements and expanded arguments

Named and numbered propositions, lemmas, and theorems, and equations carrying source numbers, refer to the corresponding statements in the fixed R manuscript. The additional conditional-expectation calculations, parameter choices, probability-kernel decompositions, spectral-recursion details, and orders of limits are explanatory derivations around those statements.

The boundary examples have distinct roles. A Bernoulli joining prevents replacing actual time configurations by arbitrary joinings. A finite square with a common hidden bit shows why saturation cannot be omitted. A bilateral shift shows that commuting unitaries alone need not have a discrete common eigenbasis. A torsion example explains the extra operation required in the nilpotent argument. None is presented as a numerical verification of the full infinite theorem.

5. Fixed version and reading scope

The fixed R manuscript

• Repository commit: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
• R PDF: 40 pages, 524156 bytes
• SHA-256:45000cec6bcec95daede6af976803b7fc2ea995c5953cc2448ec48218db2b86d
• Date printed on the title page and in the listing: September 23, 2026
• PDF creation metadata: October 6, 2026, 09:37:03 UTC
• Fixed repository commit time: October 8, 2026, 05:20:00 UTC

These dates mean different things; the printed date is not treated as an established first-publication date. The reading covers §§2–7 used in the main proof. Section 8's consequences for noninvertible measure-preserving maps and continuous flows, and parts of the historical attribution, are outside its conclusions.

6. Quantifiers to retain when returning to A and B

The passage from sets to the fixed bounded-function statement in A and B uses finite simple-function approximation. First scale $|g_i|$ to $\le1$ and choose $|h_i|\le1$ with $\|g_i-h_i\|_1\le\varepsilon$. For every time tuple,

The products of means have the same error bound. Simple functions expand into a finite sum, and there are only finitely many time orderings. Subtract the smallest time and use the source set statement. The $\limsup$ of the difference is therefore at most $2s\varepsilon$; letting $\varepsilon\downarrow0$ gives the bounded-function statement.

This gives no rate uniform over arbitrary varying families. The subsequent A/B arguments involving finite offset patterns, bad-graph degree bounds, and selected lengths must still use this input in their own parameter order.