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REVIEW 3 major objections 4 minor 15 references

Dynamical pair assignments

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that for any dynamical pair assignment, the class of P-realizable systems is Borel exactly when a natural transfinite rank is bounded.

desk verdict Novel framework, broken central lemma: Condition (S) in Theorem 5.7 is false, so the proof of Theorem 3.3 does not go through as written. read the letter →

arxiv 2501.12952 v1 pith:NSJF27YT submitted 2025-01-22 math.DS math.LO

Udayan B. Darji , Felipe García-Ramos This is my paper · ORCID
classification math.DSmath.LO MSC 03E1537B0554H0537B40
keywords dynamicalpairassignmentsP-fullsystemsP-realizablecoanalyticranksGamma-rankentropypairsregionallyproximalBorelcomplexity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces dynamical pair assignments, a single template for families of pairs of points in topological dynamical systems: to each compact metrizable space X the assignment P gives a Borel map P_X sending every continuous map T to a compact set of pairs, with the properties that the set is T×T-invariant and that factor maps push it forward. Entropy pairs and regionally proximal pairs are shown to fit this template. For any such assignment, the paper proves that the class of P-full systems, where P_X(T)=$X^{2}$, is always a Borel set; the more delicate case is P-realizability, where the assigned pairs generate the whole space as the smallest closed invariant equivalence relation. The central result characterizes exactly when this class is Borel: it is Borel if and only if the associated P-rank, a transfinite closure rank, is bounded on X. A sympathetic reader should care because the theorem turns a descriptive-complexity question about large families of dynamical systems into a concrete rank computation, and it applies uniformly to several known pair notions.

What carries the argument

Three objects carry the argument. A dynamical pair assignment is a family of Borel maps $P_X : C(X,X) \to K(X^2)$ satisfying $T\times T$-invariance and factor-map push-forward; the paper's two featured examples are entropy pairs and regionally proximal pairs. The $\Gamma$-rank is the transfinite length of the closure operation $\Gamma(E)=E^+\cup\Delta_X$ applied to $P_X(T)$. A coanalytic rank is an ordinal-valued function on a coanalytic set whose sublevel sets $\{x : \phi(x)\leq\phi(y)\}$ are analytic and coanalytic uniformly in $y$; the boundedness theorem for such ranks says that a coanalytic set carrying one is Borel if and only if the rank is bounded. The expansion construction of Section 5.2 is what connects these: it packages $\Gamma$ as a Borel expansion on $K(X\times X)$ and proves that the exhaustive set $C_E$ is coanalytic with a coanalytic rank, which is exactly the fact that makes the P-rank coanalytic on R(P_X).

What would settle it

Scrutinize the proof of Condition (S) in Theorem 5.7 and test it on a concrete Borel expansion, for instance the $\Gamma$-expansion on a Cantor set: choose a well-order $x$ and a set $A$ whose E-rank equals $|x|_*$, then check directly whether the defining condition for S holds. A single pair $(x,A)$ for which the containment fails, or for which the asserted order-preserving function $f$ is not order-preserving, would break the coanalytic-rank conclusion and with it the boundedness argument behind Theorem 3.3.

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Extended reading notes

Core claim

The central claim is Theorem 3.3: for a dynamical pair assignment P and a compact metrizable space X, the set R(P_X) = {T ∈ C(X,X) : (X,T) is P-realizable} is a Borel subset of C(X,X) if and only if the P-rank is bounded on X. The P-rank of T is the $\Gamma$-rank of the compact pair set $P_X(T)$, obtained by iterating the operation $E \mapsto E^+ \cup \Delta_X$, which adds the diagonal and closes under finite chains, until the set stabilizes; the rank is the first countable ordinal at which stabilization occurs. The proof establishes that this rank is a coanalytic rank on R(P_X): its initial segments are uniformly definable by analytic and coanalytic relations. The boundedness theorem for coanalytic ranks then yields the dichotomy, because a coanalytic set carrying a coanalytic rank is Borel exactly when the rank is bounded. Thus the descriptive nature of an entire family of dynamical systems is governed by whether a transfinite closure process always terminates by a fixed countable stage.

Load-bearing premise

The load-bearing premise is Theorem 5.7, the general claim that every Borel expansion on compact sets yields a coanalytic rank on its exhaustive sets; the proof's most compressed step is the containment in Condition (S), and if that containment fails, the P-rank need not be coanalytic and Theorem 3.3 loses its foundation.

Editorial extensions

If this is right

  • For every dynamical pair assignment, the class of P-full systems $F(P_X)$ is Borel without any extra hypothesis, since it is the preimage of the single closed set $X^2$ under a Borel map.
  • For entropy pairs, the result identifies CPE systems as the P-realizable class and UPE systems as the P-full class; the corollary is that CPE systems on X form a Borel set exactly when the entropy-pair rank is bounded, and the paper cites constructions with arbitrarily high rank that make the class non-Borel on Cantor spaces.
  • For regionally proximal pairs, Q-realizability is Borel if and only if the Q-rank is bounded; for minimal systems, where the proximal-pair relation is already an equivalence relation, Q-full and Q-realizable coincide, so the rank criterion applies directly there.
  • The theorem gives a uniform strategy for any new pair notion satisfying the three axioms: to decide whether its realizability class is Borel, compute a single ordinal rank rather than analyzing the class case by case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same dichotomy should extend to any pair notion that fits the three axioms, such as proximal pairs, asymptotic pairs, transitivity pairs, or mean-dimension pairs; the authors list these as candidates, and if they qualify, their realizability classes will be Borel exactly when the corresponding rank is bounded.
  • The P-rank may serve as a quantitative measure of how far a realizability class is from being Borel: a bounded rank gives Borelness, while an unbounded coanalytic rank suggests the class is properly coanalytic and can be stratified by the rank's ordinal values. This is an extrapolation beyond the paper's theorem, not a claim the paper proves.
  • One testable extension is to compute the rank bound for concrete spaces: for the entropy-pair assignment on the Cantor set, exhibiting systems whose E-rank is arbitrarily high already shows CPE is not Borel, and the same computational route could decide Borelness for other assignments once their rank is understood.
  • If a candidate pair notion is found whose full class is not Borel, then by Proposition 2.6 it cannot satisfy all three axioms; this gives a quick consistency check when trying to extend the framework to new pair families.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces the notion of a dynamical pair assignment P, a family of Borel maps P_X assigning to each continuous self-map T of a compact metrizable space X a closed, T×T-invariant subset P_X(T) of X^2 that behaves naturally under factor maps. It defines P-full and P-realizable systems, generalizing UPE, CPE, weak mixing, and related notions. The main results are that the set F(P_X) of P-full systems is always Borel, while the set R(P_X) of P-realizable systems is Borel if and only if an associated P-rank is bounded. The P-rank is introduced via a transfinite closure process Γ, and the proof is built on a general theorem (Theorem 5.7) asserting that every Borel expansion on K(X) induces a coanalytic rank on its exhaustive set. The paper applies the framework to IE-pairs and regionally proximal pairs.

Significance. If the main dichotomy is established, it would provide a useful unified descriptive-set-theoretic framework for local entropy theory and other pair-based dynamical notions. The examples connecting the abstract results to IE-pairs and regionally proximal pairs are natural and potentially valuable. The paper also makes a credible attempt to adapt Kechris' derivative/rank machinery to expansions, which could be of independent interest. However, the current proof of the central rank theorem contains a false intermediate condition and a serious compactness problem in the definition of Γ, so the significance is conditional on a substantial revision.

major comments (3)
  1. [5.2, Condition (S)] Condition (S) as stated is false, and the counterexample in the stress-test note is correct. Let X=[0,1], E(A)=conv(A), x∈2^{N×N} encode the linear order 0<*_x 1, and A={0}. With h(0)={0} and h(1)={0,1}, one has h(0)=A, h(m)≠X for m∈D*(x), E(h(0))=conv({0})={0}⊆h(1), and ∪_{m∈D*(x)}E(h(m))={0}∪[0,1]=X. Thus (x,A)∈S. But A∉CE because E∞({0})={0}≠X, and |x|_*=2 while |A|_E=0. This contradicts the asserted equality {A∈K(X):(x,A)∈S}={A∈CE:|x|_*=|A|_E}. Since Condition (S) is used to prove Equation (2), the proof that the E-rank is a coanalytic rank on CE is invalid as written. The condition needs to be reformulated, for example by encoding the transfinite orbit E^α(A) inside the witness h.
  2. [3 and 5.2, Proposition 5.9] The map Γ(E)=E^+∪Δ_X does not in general take K(X×X) into itself. For a concrete example, take X=2^N with the shift T and let E be the graph of T together with its reverse and the diagonal. Then E is compact, symmetric, and T×T-invariant, but E^+ is the shift-orbit equivalence relation. The sequence x_n=0^n1^∞, y_n=1^∞ satisfies (x_n,y_n)∈E^+ and converges to (0^∞,1^∞), yet (0^∞,1^∞)∉E^+ because 0^∞ is fixed by T. Hence Γ(E) is not closed. This invalidates the claim that Γ is a Borel expansion on K(X×X): Lemma 5.8 only applies when the countable union is compact, and Proposition 3.1's closed-chain argument does not apply to the iterates. It also casts doubt on Equation (4), which identifies R(PX) with P_X^{-1}(CΓ). The natural repair is to define Γ(E)=closure(E^+∪Δ_X) and prove the corresponding compactness, Borelness, and rank properties for this map; the same compactness issue affects the map ∪_n on K(X)^N used later in the proof of analyticity of S.
  3. [2, Proposition 2.8] The equality "Π_1(A) is precisely the set of all T which do not belong to R(PX)" is not established. Lemma 2.7 proves one direction: if a nontrivial factor satisfies P_{X1}(T1)⊆Δ_{X1}, then T is not P-realizable. For the converse, the proof must produce, from a non-realizable T, a nontrivial factor with P_{X1}(T1)=Δ_{X1} (or at least with P_{X1}(T1)⊆Δ_{X1}). The natural quotient by the smallest closed invariant equivalence relation containing P_X(T) does not obviously have this property, because Definition 2.1(2) is one-way and the quotient may itself be P-realizable. The present argument, which appeals to Lemma 2.7 for the reverse implication, is therefore incomplete.
minor comments (4)
  1. [4.1, Definition 4.1] The phrase "uniform positive entropy (CPE)" in the second sentence appears to be a typo; uniform positive entropy is usually abbreviated UPE, and CPE is defined in the preceding sentence.
  2. [5.2, proof of Theorem 5.7] The sentence "Let ∆ be the diagonal of ∆ = {(A,A): A∈K(X)\setminus{X}}" contains a duplicated symbol; it should define the diagonal of K(X)\setminus{X} or of K(X)^2.
  3. [2, Proposition 2.8] The notation P(S) is ambiguous because P is a family of maps indexed by compact metrizable spaces; it should read P_{φ(T)}(S).
  4. [3, Definition 3.2] The phrase "the P rank is bounded on X" is terse: the quantifier is over T∈C(X,X), not over points of X. Consider rephrasing as "bounded on C(X,X)" for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.3 rests on independent coanalytic-rank machinery; the self-citation to [5] is not load-bearing.

full rationale

The paper's central claim, Theorem 3.3, is not obtained from its own conclusion. The P rank is an independently defined ordinal invariant of P_X(T) under the Γ-closure process, and R(P_X) is defined separately as the set where Γ∞(P_X(T)) = X². The proof that R(P_X) is Borel iff the P rank is bounded proceeds through Kechris' boundedness theorem for coanalytic ranks (Theorem 5.12) and a separate coanalyticity proof for the P rank (Theorem 5.11). That coanalyticity proof, in turn, rests on Theorem 5.7, which constructs analytic witnesses R and S rather than importing the target theorem. No fitted parameter is renamed as a prediction, no definition reduces to the claimed theorem, and no uniqueness result by the authors is used to force a choice. The only self-citation is the introduction's disclosure that 'Some of the results of this paper appeared in an unpublished note by the same authors [5]'; that note is not cited in the proofs of Theorem 5.7, 5.11, or 3.3, so it is not load-bearing. The proof of Theorem 5.7 is compressed, especially the verification of Condition (S), where the authors write that 'The proof of containment ⊆ also uses an equivalent order preserving function f' without supplying the full argument; the skeptical counterexample suggests this may be a correctness gap in an intermediate claim, but that is not circularity. Even if Theorem 5.7 required repair, the derivation would still not reduce to its own inputs. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The central theorem assumes the standard descriptive set theory toolkit of coanalytic ranks (Kechris), the Polish subspace topologies on C(X,X) and K(X), and the correctness of the Borel expansion construction in Theorem 5.7. No free parameters are fitted; the only ad hoc element is the axiomatic definition of a dynamical pair assignment itself, which is the paper's own postulation.

assumptions (6)
  • standard math Kechris boundedness theorem for coanalytic ranks (Theorem 5.12)
    Used as the black box connecting boundedness of a coanalytic rank to Borelness of the underlying set in the proof of Theorem 3.3.
  • standard math For compact metrizable X, C(X,X) with uniform topology is Polish and K(X) with Vietoris topology is compact metrizable
    The entire descriptive setting depends on these Polish space facts, used throughout Sections 2 and 5.
  • standard math In a compact metrizable space, any strictly increasing chain of closed sets is countable
    Justifies that the Γ rank and E rank stabilize at a countable ordinal (Proposition 3.1 and the paragraph after Definition 5.6).
  • standard math Every coanalytic set admits a coanalytic rank (Kechris, [10])
    Invoked to justify the existence of coanalytic ranks in the general framework, though the paper constructs the specific ranks directly.
  • domain assumption Known characterizations of IE-pairs (Theorem 4.3, cited from [12] and [11])
    Used to prove the entropy pair assignment is a dynamical pair assignment; if these citations were wrong, the example would fail, but the abstract theory would survive.
  • ad hoc to paper The defining axioms of a dynamical pair assignment (Definition 2.1) are postulated for arbitrary P
    The framework assumes such assignments exist and are Borel; the existence is checked for entropy pairs and regional proximal pairs, but the general framework is a new postulate.
invented entities (3)
  • Dynamical pair assignment P
    purpose: Abstraction of all pair-selection rules (entropy pairs, regionally proximal pairs) into a Borel map PX:C(X,X)->K(X^2) with T×T-invariance and factor-map properties.
    The paper's main object of study; a mathematical definition with no predictive handle outside the paper.
  • Γ rank and P rank
    purpose: Ordinal-valued ranks measuring the transfinite closure needed to reach the P-realizable relation; used as the boundary for Borelness of R(PX).
    New ordinal invariants introduced for the proof; they are defined internally and carry no external falsifiable content.
  • Expansion operator (Definition 5.6)
    purpose: Complement of a derivative on K(X), used to define coanalytic ranks for the Γ closure process.
    A formal device adapted from Kechris' derivative framework; it is a mathematical construction without independent empirical evidence.

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Pith. "Pith review of Dynamical pair assignments." pith.science (2026). https://pith.science/paper/NSJF27YT

@misc{pith2026250112952,
  author       = {Pith},
  title        = {Pith review of: Dynamical pair assignments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSJF27YT}},
  note         = {Machine review of arXiv:2501.12952}
}
abstract

Relations between points in the phase space are central to the study of topological dynamical systems. Since many of these relations share common properties, it is natural to study them within a unified framework. To this end, we introduce the concept of \textit{dynamical pair assignments} $\mathcal{P}$. We then introduce the notions of a dynamical system being $\mathcal{P}$-full and $\mathcal{P}$-realizable, which generalize several existing concepts in the field like CPE, weak mixing and UPE. Our results establish that the space of $\mathcal{P}$-full systems is always a Borel set, while the space of $\mathcal{P}$-realizable systems is Borel if and only if an associated natural rank is bounded.

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Reference graph

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