{"id":"45887e66-0339-49b3-8dc4-faf39c162425","arxiv_id":"2501.12952","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces dynamical pair assignments and proves the space of P-realizable systems is Borel if and only if the associated P-rank is bounded.","lead":"A new framework, dynamical pair assignments, puts entropy pairs and regionally proximal pairs under one roof, and proves that the family of P-realizable systems is Borel exactly when a natural ordinal rank is bounded. Generalists may care because it sharpens the boundary between easy and hard-to-describe classes of dynamical systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.7's auxiliary Condition (S) is false: a valid Borel expansion (convex hull on [0,1]) and a two-point well-order x satisfy (x,{0})∈S although {0}∉CE, so the proof of the E-rank coanalyticity—and with it the derivation of Theorem 3.3—collapses as written.","rationale":"The reader's weakest assumption correctly identified the containment ⊆ in Condition (S) of Theorem 5.7 as the most compressed and load-bearing step. My check confirms that the concern is not merely a missing detail: the stated condition is false. For the convex-hull expansion on [0,1], a two-element well-order x and a one-point set A satisfy the definiens of S, but A is not in CE and the order types differ. Therefore the equality claimed in (S) cannot hold, so the analytic relation Q constructed in the proof does not witness the strict inequality part of a coanalytic rank. Since Theorem 5.7 is the mechanism by which the P-rank and Γ-rank are shown to be coanalytic, and Theorem 3.3 depends on that fact via Theorem 5.11 and the boundedness theorem, the main theorem is not established by the manuscript as written. I am not claiming the main theorem is false—only that the submitted proof has a concrete, reproducible false lemma at a central juncture. A repair would require either redefining S (e.g., by adding conditions that force h(m) to track E^α(A) from above as well as below) or a different proof of coanalyticity for expansions. Because the flaw is demonstrated rather than merely suspected, I move the verdict from CONDITIONAL to REJECT: the paper needs substantive revision before its central claim can be certified.","tokens_in":12777,"tokens_out":16432,"duration_ms":179396,"concrete_test":"Check Condition (S) literally against the following instance: X=[0,1], E(A)=conv(A), x∈WF* with D*(x)={0,1} and 0<*_x 1, h(0)={0}, h(1)={0,1}. Verify each conjunct of (S): h(0)=A, h(m)≠X, E(h(0))={0}⊆h(1), and E(h(0))∪E(h(1))={0}∪[0,1]=X. Then verify that A={0} has |A|_E=0 and is not in CE. If these small computations confirm the counterexample, the asserted universal statement (S) is false, and the proof of Theorem 5.7 fails exactly where the reader suspected.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 5.7 is invalid as stated because the auxiliary condition (S) used to build the analytic witness Q is false for a legitimate Borel expansion. Let X=[0,1] and E(A)=conv(A), the convex hull. This is a Borel expansion in the sense of Definition 5.6. Let x∈2^{N×N} encode the linear order 0<*_x 1, so x∈WF* and |x|_*=2. Set A={0} and define h(0)={0}, h(1)={0,1}. Then h(0)=A, h(1)≠X, and for m=1 the required union is E(h(0))=conv({0})={0}⊆h(1). Also ∪_{m∈D*(x)}E(h(m)) = {0} ∪ conv({0,1}) = [0,1] = X. Thus (x,A)∈S. Yet A∉CE because E∞({0})={0}≠X, and |A|_E=0 while |x|_*=2. So the right-hand side of (S), which requires A∈CE and |x|_*=|A|_E, is not satisfied. This contradicts Condition (S) as stated. Since (S) is used to prove Equation (2), and Equation (2) is used to verify that the E-rank is a coanalytic rank, the proof of Theorem 5.7 does not go through. Consequently the derivation of Theorem 3.3 via Theorem 5.11 currently rests on a false intermediate claim; a different proof or a substantially modified condition (S) is required.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13212,"tokens_out":22600,"duration_ms":242676,"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":[{"comment":"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.","section":"5.2, Condition (S)"},{"comment":"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.","section":"3 and 5.2, Proposition 5.9"},{"comment":"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.","section":"2, Proposition 2.8"}],"minor_comments":[{"comment":"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.","section":"4.1, Definition 4.1"},{"comment":"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.","section":"5.2, proof of Theorem 5.7"},{"comment":"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).","section":"2, Proposition 2.8"},{"comment":"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.","section":"3, Definition 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a plausible and interesting central idea, and the examples are well motivated. However, the proof of the central rank theorem has a false auxiliary condition and the Γ operation is not compactness-preserving as defined; these are load-bearing, not cosmetic. I believe the results may be repairable by redefining Γ with a closure step and by substantially reworking the analytic witnesses in Theorem 5.7, but the current version cannot be accepted. I recommend major revision rather than reject because the overall framework appears salvageable and the intended applications are meaningful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: worth reading, not yet sound. The dynamical pair assignment framework (Def. 2.1) is a good unifying abstraction, and the dichotomy in Thm. 3.3 is the right kind of result. But the proof of Thm. 5.7 has a genuine false claim, and the stress-test example is correct.\n\nWhat's new: the definitions of P-full and P-realizable, the Γ and P ranks, and the Borel-iff-bounded-rank statement. The examples (IE-pairs, regionally proximal pairs) are known but cleanly re-derived. The paper is well-written and shows real command of the descriptive set theory. The citation pattern is honest, including the self-citation to the unpublished note [5].\n\nThe problem: Condition (S) in the proof of Thm. 5.7 is false as stated. Take X=[0,1], E(A)=conv(A), and A={0}. Let x encode the two-point well-order 0<1. Define h(0)={0}, h(1)={0,1}. Then the hypotheses of S hold: h(0)=A, both h(m) are not X, E(h(0))={0}⊆h(1), and the union E(h(0))∪E(h(1))={0}∪[0,1]=X. So (x,A)∈S. But A∉CE, since E∞({0})={0}≠X, and |A|E=0 while |x|*=2. So the equality in (S) fails. This directly breaks the containment proof for Equation (2), which is what makes the E-rank a coanalytic rank. There is also a smaller definitional issue: E^λ(A), defined as a union of increasing compacta, need not be compact, so the iterated expansion is not always in K(X); the proof uses such objects.\n\nNet: the abstraction and the main statement are plausible, and maybe a repaired argument exists. But as written, the load-bearing lemma is unsupported. This is a major revision, not a minor typo-fix. I would still send it to a serious referee—there is enough genuine novelty—but the referee should be asked to focus on Section 5.2. I would not cite it until the proof is repaired.","headline":"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.","tokens_in":13693,"tokens_out":4214,"would_cite":false,"duration_ms":42129,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E15","37B05","54H05","37B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["dynamical pair assignments","P-full systems","P-realizable systems","coanalytic ranks","Gamma-rank","entropy pairs","regionally proximal pairs","Borel complexity"],"falsifier":"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.","tokens_in":12596,"feed_emoji":"🔁","tokens_out":10594,"duration_ms":101118,"temperature":0.7,"pith_summary":"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.","feed_headline":"Rank bound decides when dynamical pair classes are Borel","feed_subtitle":"P-full classes are always Borel; P-realizable classes need a bounded ordinal rank.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the definitions and theorems on coanalytic ranks, derivatives, and the boundedness theorem that power the proof of Theorem 3.3.","marker":"[10]"},{"why":"Provides the independence-set characterization and entropy-pair facts used to prove the entropy pair assignment is a dynamical pair assignment in Proposition 4.5.","marker":"[11]"},{"why":"States the IE-pair properties, including closedness, invariance, and the link between non-diagonal IE-pairs and positive entropy, used in the entropy example.","marker":"[12]"},{"why":"Introduced regionally proximal pairs, whose closure, invariance, and factor-map behavior are used to build the regionally proximal pair assignment in Proposition 4.11.","marker":"[14]"},{"why":"Supplies the local entropy theory background and the identification of CPE and UPE with realizability and fullness for entropy pairs in Proposition 4.6.","marker":"[7]"},{"why":"Records the earlier unpublished note in which some of these results first appeared, connecting the present work to the authors' prior development.","marker":"[5]"}],"fun_headline_variants":["Full classes always Borel; realizable need bounded rank","Borel iff bounded rank for P-realizable systems","Ordinal rank bound decides Borelness of dynamical pair classes","Bounded rank makes pair-realizable systems Borel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Full classes always Borel; realizable need bounded rank","Borel iff bounded rank for P-realizable systems","Ordinal rank bound decides Borelness of dynamical pair classes","Bounded rank makes pair-realizable systems Borel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000469,"raw_usage":{"total_tokens":2300,"prompt_tokens":877,"completion_tokens":1423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":1357}},"tokens_in":493,"tokens_out":1423,"duration_ms":11392,"temperature":1.0,"reasoning_tokens":1357,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:34:41.994837+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definitions and theorems on coanalytic ranks, derivatives, and the boundedness theorem that power the proof of Theorem 3.3."},{"cited_title":"Independence in topological and C*- dynamics","cited_arxiv_id":null,"evidence_quote":"Provides the independence-set characterization and entropy-pair facts used to prove the entropy pair assignment is a dynamical pair assignment in Proposition 4.5."},{"cited_title":"Ergodic Theory: Independence and Di- chotomies","cited_arxiv_id":null,"evidence_quote":"States the IE-pair properties, including closedness, invariance, and the link between non-diagonal IE-pairs and positive entropy, used in the entropy example."},{"cited_title":"The equicontinuous structure relatio n for minimal abelian transformation groups","cited_arxiv_id":null,"evidence_quote":"Introduced regionally proximal pairs, whose closure, invariance, and factor-map behavior are used to build the regionally proximal pair assignment in Proposition 4.11."},{"cited_title":"A note on derivatives, expansions and $\\Pi^1_1$-ranks","cited_arxiv_id":"2107.09866","evidence_quote":"Records the earlier unpublished note in which some of these results first appeared, connecting the present work to the authors' prior development."}],"review_version":1}