{"id":"372df295-d2e8-4227-bcc1-fc9635c76d68","arxiv_id":"2506.23313","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Multivariate diam-mean equicontinuity and frequent stability satisfy Auslander-Yorke dichotomies, and m-regular extensions of the maximal equicontinuous factor are characterized by weakly mean-sensitive tuples.","lead":"This paper proves a dichotomy for group actions on compact spaces: a minimal system either keeps nearby points close on average or spreads them apart, in a multivariate sense. It also shows that for virtually nilpotent or locally Bronstein groups, this dichotomy is equivalent to a 'tuple' condition detecting how many preimages a system has over its maximal equicontinuous factor.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2 is not self-contained: it imports the two decisive tuple-to-regularity implications from the unpublished preprint [26, Cor. 5.3], and the paper gives neither a proof nor a precise statement of that result.","rationale":"The reader's weakest-assumption analysis identified exactly the dependence on [26, Cor. 5.3], and that remains the load-bearing concern after independent reading. The self-contained dichotomies in Theorems 1.1, 1.2, and the first parts of Theorem 5.1 appear sound; their minor proof gaps, such as the fixed-index pigeonhole argument and the tempered-subsequence step in Theorem 4.1, are repairble without changing the statements. The decisive question is whether the unpublished companion result [26] really supplies the strong tuple-to-regularity implications used in Theorem 5.2. Since the current manuscript neither proves nor states that corollary precisely, the central equivalence cannot be fully checked from the material provided. This supports the reader's CONDITIONAL verdict, so no adjustment to the verdict is needed.","tokens_in":15316,"tokens_out":17150,"duration_ms":191115,"concrete_test":"Obtain the proof of Corollary 5.3 of [26] and check whether it establishes the exact statements used in Theorem 5.2 under this paper's definition of 'regular' and the Section 5.1 local-Bronstein condition. Concretely: verify that (a) no essential weakly (m+1)-sensitive tuple implies πeq is regular m′-to-one with m′≤m without invoking Theorem 5.2 of this note, and (b) a regular m-to-one MEF has every fiber containing an essential weakly m-sensitive tuple, using minimality and upper semicontinuity to upgrade the almost-everywhere statement to all fibers. If the proof of [26, Cor. 5.3] is unavailable, modify Theorem 5.2 to state the two directions as conditional on that corollary.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline equivalence in Theorem 5.2 depends on Corollary 5.3 of [26] in two places: (i)⇒(iii) imports the assertion that a regular m-to-one MEF has every fiber containing an essential weakly m-sensitive tuple, and (ii)⇒(i) imports the assertion that absence of an essential weakly (m+1)-sensitive tuple forces a regular m′-to-one MEF with m′≤m. Neither implication is proved in the note, and [26] is an unpublished preprint coauthored by one of the present authors. The paper's own presentation makes verification harder: the local Bronstein condition is restated differently in Section 5.1 from Definition 2.14, and the display in Definition 2.14 is garbled. If [26] yields only a measure-theoretic or residual version, or uses a different convention for 'regular', then condition (iii) 'for every y' and the m′≥m step in (ii)⇒(i) could fail. A smaller internal gap occurs at the end of Theorem 4.1, where the existence of a tempered Følner subsequence is used in the wrong direction to pass from tempered sequences to all Følner sequences; this is likely repairble by choosing K with μ(∂K)=0, but as written the proof of (i)⇒(iv) is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multivariate diam-mean equicontinuity, diam-mean sensitivity, frequent stability, and strong spreading for topological dynamical systems acted on by countable discrete amenable groups. It establishes Auslander-Yorke type dichotomies: for transitive systems, either almost diam-mean m-equicontinuity or diam-mean m-sensitivity (Theorem 1.1), and either almost frequent m-stability or strong m-spreading (Theorem 1.2), with the minimal versions following as corollaries. It also proves a characterization of diam-mean m-sensitivity via essential weakly mean-sensitive m-tuples (Theorem 1.3 / Theorem 5.1) and states a local characterization of m-regular extensions of the maximal equicontinuous factor (Theorem 5.2) under a local Bronstein condition, together with a characterization of strong m-spreading via finite families of tuples (Proposition 5.5).","tokens_in":15596,"tokens_out":6837,"duration_ms":62798,"significance":"If all results hold, the paper would give clean multivariate analogues of classical dichotomies and a local tuple-based description of finite-to-one regularity, extending earlier work in [18] to virtually nilpotent groups and to amenable groups under a local Bronstein condition. The proofs of the dichotomies in Propositions 3.1--3.4 are self-contained and appear correct, and Theorem 5.1 provides a useful and largely elementary tuple characterization. However, the central structural result, Theorem 5.2, is not proven in this manuscript: its decisive implications are imported from an unpublished preprint by one of the authors, and the proof of Theorem 4.1 contains a genuine gap. The significance of the paper is therefore conditional on an external result that is not made available here.","major_comments":[{"comment":"The equivalence in Theorem 5.2 is not self-contained. In direction (ii)⇒(i), the implication 'no essential weakly (m+1)-sensitive tuple implies π_eq is regular m′-to-one for some m′ ≤ m' is imported from [26, Corollary 5.3], and in direction (i)⇒(iii), the statement that in a regular m-to-one extension every fiber contains an essential weakly m-sensitive tuple is also imported from the same corollary. Neither the precise statement nor the proof of Corollary 5.3 is given in this note, and [26] is an unpublished preprint coauthored by one of the present authors. These steps are load-bearing for the main characterization, so Theorem 5.2 is incomplete as written. The authors should either reproduce the needed arguments or state the exact result they rely on and make it verifiable.","section":"§5.1, Theorem 5.2"},{"comment":"The proof of Theorem 4.1 establishes the desired bound only for tempered Følner sequences and then concludes 'As any Følner sequence has a tempered Følner subsequence [25], this completes the proof.' This is the wrong direction: passing to a subsequence can only decrease a limsup, so a bound on the limsup along a tempered subsequence does not imply the same bound for the original Følner sequence. In addition, the proof says it will show that any x is a diam-mean m-equicontinuous point, but the target is diam-mean (m+1)-equicontinuity. Thus the implication (i)⇒(iv) lacks a valid proof; a different argument, for instance selecting K with μ(∂K)=0 and using unique ergodicity along all Følner sequences, is needed.","section":"§4, Theorem 4.1"},{"comment":"The definition of 'locally Bronstein' is garbled: the display in Definition 2.14 uses an m-tuple quantifier '∀i ∈ {1,...,m}' and variables x_i, y_j in what is supposed to be a condition on pairs (x,y), so the displayed formula is not meaningful as written. Moreover, Section 5.1 restates the local Bronstein condition differently, requiring minimal points (x,x′) with π_eq(x)=π_eq(x′) rather than almost periodic pairs satisfying xQy. Since Theorem 5.2 assumes this condition, the manuscript must state one consistent, precise definition and show that the two formulations are equivalent if both are used.","section":"§2.4, Definition 2.14 and §5.1"},{"comment":"The proof of the converse direction in Theorem 5.1 contains several notational slips that make it hard to verify: after 'for any k ∈ F there exist x^k_1, ..., x^k_m ∈ U', the symbol k is used both as a group element and as an index, and later the finite-family construction refers to diameters shrinking like 1/l but initially uses 'Diam(A^1_i) ≤ 1'. These are local and presumably repairable, but they should be corrected so that the construction of the weakly mean-sensitive tuple is unambiguous.","section":"§5.1, Theorem 5.1"}],"minor_comments":[{"comment":"The abstract contains typographical errors: 'action of a is a countable discrete infinite group' and 'vial weakly mean sensitive tuples' should be read as 'action of a countable discrete infinite group' and 'via weakly mean sensitive tuples'.","section":"Abstract and Introduction"},{"comment":"In the proof of invariance, two occurrences of 'x /∈ F Sm_ε' should read 'x /∈ DE^m_ε'; this is presumably a copy-and-paste error.","section":"§3.2, Proposition 3.3"},{"comment":"The display after (4.1) uses an equality where an inequality is needed, and the constant '2ε · μ_eq(K)' should be '(2ε/3) · μ_eq(K)' to match the preceding bound Diam_{m+1}(gB(x,η)) < 2ε/3.","section":"§4, Theorem 4.1, equation (4.1)"},{"comment":"In the definition of the regionally proximal m-relation, the expression 'd(g_n(x'_{i,n}), g_n(y'_{j,n}))' should presumably be 'd(g_n(x'_{i,n}), g_n(x'_{j,n}))' since only the x' sequences are defined.","section":"§2.4, Definition 2.10"}],"recommendation":"major_revision","confidential_remarks":"The heavy reliance on [26, Corollary 5.3], an unpublished preprint coauthored by one of the authors, is a concern both for verifiability and for the self-containedness expected of a journal paper. If the authors do not include a proof or a precise public statement of that corollary, the main equivalence Theorem 5.2 is not established by this manuscript. The rest of the paper contains useful and largely correct material, so the result could be worth publishing after the gap is properly addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [name],\n\nThe paper is a short note with two parts of very different weight. The first part, Theorems 1.1 and 1.2, extends the Auslander-Yorke dichotomies for diam-mean m-equicontinuity/sensitivity and frequent m-stability/strong m-spreading from abelian to all countable discrete amenable groups. That is new, and the proofs are correct, self-contained extensions of the standard Gδ-set arguments. I checked the main steps; the invariant-open-set proofs in Section 3 go through, and the dichotomy conclusions follow as stated. The second part, Theorem 5.2, offers a tuple-based characterization of m-regularity under a local Bronstein condition. This is where the note gets fragile.\n\nThe equivalence (ii)⇔(i) and part of (i)⇒(iii) depend on Corollary 5.3 of [26], an unpublished preprint coauthored by one of the present authors. The paper does not state or prove that corollary. That is not a minor omission: the implication 'no essential weakly (m+1)-sensitive tuple implies regular m'-to-one with m'≤m' is the decisive step, and a referee cannot verify it from the note alone. There is also an internal gap in Theorem 4.1: the argument uses a tempered Følner subsequence in the wrong direction to conclude the bound for all Følner sequences. This is likely repairable by choosing K with μ(∂K)=0, but as written the proof is incomplete. In addition, Definition 2.14 (local Bronstein) has a garbled display, and the restatement in Section 5.1 does not visibly agree with it.\n\nOn the positive side, the paper is honest about the dependency—it clearly labels the result as coming from [26]—and the authors are careful to distinguish their notation from [18] (footnotes 1 and 2). The dichotomies themselves are a genuine contribution for amenable group actions, and Theorem 1.3 (weakly m-sensitive tuples vs. diam-mean m-sensitivity) is a useful reformulation, even if the proof follows [23].\n\nThis is a paper for specialists in topological dynamics, particularly those working on mean equicontinuity and finite-to-one extensions beyond action by Z. It deserves a serious referee, but the referee should be told to verify the [26] dependency and to ask for a fix of Theorem 4.1 and a consistent definition of the local Bronstein condition.\n\nRecommendation: send to review, with an expectation of revision.\n\nBest,","headline":"New and correct Auslander-Yorke dichotomies for amenable groups; the tuple characterization of m-regularity is not self-contained and the note needs cleanup.","tokens_in":16151,"tokens_out":5054,"would_cite":true,"duration_ms":48855,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37A05","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem 5.2 equates m-regularity of the maximal equicontinuous factor, weakly mean-sensitive m-tuples, and the diam-mean sensitivity threshold for minimal locally Bronstein systems.","keywords":["topological dynamics","diam-mean equicontinuity","frequent stability","Auslander-Yorke dichotomy","weakly mean-sensitive tuples","maximal equicontinuous factor","amenable group actions","local Bronstein condition"],"falsifier":"Find a minimal abelian (or virtually nilpotent) system whose first diam-mean-sensitive order differs from the almost-everywhere fiber size of its maximal equicontinuous factor; for example, a Toeplitz subshift with known maximal-factor fiber statistics where an essential weakly 3-sensitive tuple appears although the fibers have size 2, or where the fibers have size 3 but the first sensitive order is 2. Either outcome would falsify Theorem 5.2. A more targeted check is to verify whether implication $(ii)\\Rightarrow(i)$ holds when the imported result from reference [26] is replaced by a direct calculation of maximal-factor fibers.","tokens_in":2435,"feed_emoji":"🌀","tokens_out":3288,"duration_ms":134564,"temperature":0.7,"pith_summary":"This paper establishes exact either/or laws for two multivariate notions of stability in topological dynamics. For a minimal action of a countable discrete amenable group, the system is either diam-mean $m$-equicontinuous or diam-mean $m$-sensitive, and either frequently $m$-stable or strongly $m$-spreading; there is no mixed behaviour at any finite order $m$. The paper then shows that when the system satisfies the local Bronstein condition—a mild approximation property automatic for actions of virtually nilpotent groups, including all abelian groups—these thresholds carry structural meaning: the first order at which diam-mean sensitivity appears is exactly the size of almost every fiber of the maximal equicontinuous factor. This equivalence is certified locally by weakly mean-sensitive tuples: an essential $m$-tuple whose product neighbourhoods are visited by every open set with positive density, together with the absence of such an $(m+1)$-tuple, is equivalent to the factor map being regular $m$-to-one.","feed_headline":"At every order m, minimal systems are stable or spreading","feed_subtitle":"The threshold where diam-mean m-sensitivity first appears equals the fiber size of the maximal equicontinuous factor.","key_machinery":"The proof has two workhorses. One is the set of diam-mean $m$-equicontinuity points $DE^m$ and the set of frequent $m$-stability points $FS^m$: for each sensitivity scale $\\varepsilon$ the approximating sets are open, invariant, and $G_\\delta$, and their complements are closed and invariant. Invariance plus a transitive point forces the whole space to lie on one side of the dichotomy (Propositions 3.2 and 3.4). The other workhorse is the local Bronstein condition, which lets the paper invoke an imported structural result from reference [26]: absence of essential weakly $(m+1)$-sensitive tuples forces the maximal equicontinuous factor map to be regular $m'$-to-one with $m'\\le m$, and regular fibers contain essential weakly $m'$-sensitive tuples. Weakly mean-sensitive tuples are the local certificates that connect the two: an essential $m$-tuple has positive upper Banach density of visits to product neighbourhoods of any open set, which is exactly what turns tuple sensitivity into diam-mean $m$-sensitivity (Theorem 5.1).","core_discovery":"The central claim is Theorem 5.2. It says that for a minimal topological dynamical system satisfying the local Bronstein condition, the following are equivalent for each positive integer $m$: the map to the maximal equicontinuous factor is regular $m$-to-one; there exists an essential weakly mean-sensitive $m$-tuple and no essential weakly mean-sensitive $(m+1)$-tuple; every fiber of the maximal equicontinuous factor contains an essential weakly $m$-sensitive tuple and none of size $m+1$; the system is diam-mean $(m+1)$-equicontinuous but not diam-mean $m$-equicontinuous; and the system is not diam-mean $(m+1)$-sensitive but is diam-mean $m$-sensitive. Theorems 1.1 and 1.2 provide the underlying dichotomies for transitive and minimal systems, respectively, for both diam-mean equicontinuity/sensitivity and frequent stability/strong spreading. In the locally Bronstein setting, the equivalence pins down the integer $m$ at which mean sensitivity begins as the regular fiber cardinality of the maximal equicontinuous factor.","pith_inferences":["The same tuple machinery could be pushed further: the existence of weakly mean-sensitive tuples of size exactly $m$ might be decidable for subshifts of finite type by finite-block search, giving a numerical 'sensitivity spectrum' that predicts the maximal-equicontinuous-factor fiber size without constructing the factor.","The local Bronstein assumption in Theorem 5.2 is likely not optimal; the dichotomy theorems need only minimality, so extending the imported result from reference [26] beyond locally Bronstein systems would immediately widen the equivalence to any minimal amenable action.","Because Proposition 5.5 shows strong $m$-spreading is a collective, finite-family phenomenon, quantitative versions of the dichotomy—rates of divergence or densities of spreading times—may be accessible even when single tuples fail to certify spreading."],"forward_implications":["In every minimal locally Bronstein system, the smallest $m$ for which an essential weakly mean-sensitive $m$-tuple exists equals the smallest $m$ for which diam-mean $m$-sensitivity holds, and both equal the almost-everywhere fiber size of the maximal equicontinuous factor.","Diam-mean $(m+1)$-equicontinuity without diam-mean $m$-equicontinuity is equivalent to diam-mean $m$-sensitivity without diam-mean $(m+1)$-sensitivity, so the two notions record the same threshold from opposite sides.","For minimal actions of abelian groups—where the local Bronstein condition is automatic—the equivalence gives a purely dynamical criterion for $m$-regularity: count the first order at which sensitivity appears, without constructing the maximal equicontinuous factor.","The dichotomy theorems imply that almost diam-mean $m$-equicontinuity and diam-mean $m$-sensitivity exhaust the possibilities for transitive systems, and for minimal systems the 'almost' disappears, making the alternative a genuine partition of the space.","The frequent-stability dichotomy runs in parallel: minimal systems are either frequently $m$-stable or strongly $m$-spreading, with the same threshold structure under the local Bronstein condition."],"supporting_citations":[{"why":"Supplies the imported Corollary 5.3, which connects absence of weakly $(m+1)$-sensitive tuples to regular $m'$-to-one maximal-equicontinuous-factor fibers with $m'\\le m$; used in $(ii)\\Rightarrow(i)$.","marker":"[26]"},{"why":"Gives the abelian-case equivalence between $m$-regularity and diam-mean $(m+1)$-equicontinuity, and between almost $m:1$ and frequent $(m+1)$-stability, which this paper extends; also supplies Lemma 2.7 used in preliminaries.","marker":"[18]"},{"why":"Provides the mean-equicontinuity/mean-sensitivity dichotomy and the invariant-$G_\\delta$-set method adapted in Propositions 3.2 and 3.4.","marker":"[22]"},{"why":"Supplies the weakly mean-sensitive tuple framework and the mean-sensitivity argument adapted in Theorem 5.1.","marker":"[23]"},{"why":"Proves that under the local Bronstein condition the multivariate regionally proximal relation reduces to the pairwise relation, used for tuple-to-factor connections.","marker":"[3]"},{"why":"Shows that virtually nilpotent actions have only trivial minimal proximal flows, which via [16] implies the local Bronstein property.","marker":"[17]"},{"why":"Defines incontractibility and links it to disjointness from minimal proximal flows, bridging [17] to the local Bronstein condition.","marker":"[16]"},{"why":"The classical Auslander-Yorke dichotomy that this paper generalizes to multivariate diam-mean and frequent stability notions.","marker":"[4]"},{"why":"Supplies the pointwise ergodic theorem for amenable groups used in Theorem 4.1 to pass from Følner averages to the unique invariant measure on the maximal equicontinuous factor.","marker":"[25]"}],"fun_headline_variants":["M-sensitivity threshold equals maximal factor's fiber size","Minimal systems: diam-mean m-sensitivity begins at fiber size","Fiber size of maximal equicontinuous factor sets m-sensitivity onset","The m where sensitivity starts equals maximal factor fiber count","For minimal systems, m-sensitivity onset is fiber cardinality of maximal factor"],"cache_read_input_tokens":18176,"weakest_assumption_plain":"The argument leans on an imported, as-yet-unpublished result from reference [26]: if a minimal system has no essential weakly mean-sensitive $(m+1)$-tuple, then the map to its maximal equicontinuous factor must be regular $m'$-to-one for some $m'\\le m$, and each generic fiber must contain an essential weakly $m'$-sensitive tuple; should that theorem fail, the $(ii)\\Rightarrow(i)$ direction of Theorem 5.2 collapses.","fun_headline_variants_meta":{"raw":{"variants":["M-sensitivity threshold equals maximal factor's fiber size","Minimal systems: diam-mean m-sensitivity begins at fiber size","Fiber size of maximal equicontinuous factor sets m-sensitivity onset","The m where sensitivity starts equals maximal factor fiber count","For minimal systems, m-sensitivity onset is fiber cardinality of maximal factor"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001309,"raw_usage":{"total_tokens":5391,"prompt_tokens":1058,"completion_tokens":4333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":4243}},"tokens_in":674,"tokens_out":4333,"duration_ms":29550,"temperature":1.0,"reasoning_tokens":4243,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:48:00.361216+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a minimal abelian (or virtually nilpotent) system whose first diam-mean-sensitive order differs from the almost-everywhere fiber size of its maximal equicontinuous factor; for example, a Toeplitz subshift with known maximal-factor fiber statistics where an essential weakly 3-sensitive tuple appears although the fibers have size 2, or where the fibers have size 3 but the first sensitive order is 2. Either outcome would falsify Theorem 5.2. A more targeted check is to verify whether implication $(ii)\\Rightarrow(i)$ holds when the imported result from reference [26] is replaced by a direct calculation of maximal-factor fibers.","supporting_citations":[{"cited_title":"Independence and sensitivity for group actions","cited_arxiv_id":null,"evidence_quote":"Supplies the imported Corollary 5.3, which connects absence of weakly $(m+1)$-sensitive tuples to regular $m'$-to-one maximal-equicontinuous-factor fibers with $m'\\le m$; used in $(ii)\\Rightarrow(i)$."},{"cited_title":"Multivariate frequent stability and diam-mean equicontinuity, 2025","cited_arxiv_id":null,"evidence_quote":"Gives the abelian-case equivalence between $m$-regularity and diam-mean $(m+1)$-equicontinuity, and between almost $m:1$ and frequent $(m+1)$-stability, which this paper extends; also supplies Lemma 2.7 used in preliminaries."},{"cited_title":"Mean equicontinuity and mean sensitivity","cited_arxiv_id":null,"evidence_quote":"Provides the mean-equicontinuity/mean-sensitivity dichotomy and the invariant-$G_\\delta$-set method adapted in Propositions 3.2 and 3.4."},{"cited_title":"Equicontinuity and sensitivity in mean forms","cited_arxiv_id":null,"evidence_quote":"Supplies the weakly mean-sensitive tuple framework and the mean-sensitivity argument adapted in Theorem 5.1."},{"cited_title":"A group theoretic condition in topological dynamics","cited_arxiv_id":null,"evidence_quote":"Proves that under the local Bronstein condition the multivariate regionally proximal relation reduces to the pairwise relation, used for tuple-to-factor connections."},{"cited_title":"Compressibility properties in topological dynamics","cited_arxiv_id":null,"evidence_quote":"Shows that virtually nilpotent actions have only trivial minimal proximal flows, which via [16] implies the local Bronstein property."},{"cited_title":"Bernoulli disjointness","cited_arxiv_id":null,"evidence_quote":"Defines incontractibility and links it to disjointness from minimal proximal flows, bridging [17] to the local Bronstein condition."},{"cited_title":"Interval maps, factors of maps, and chaos","cited_arxiv_id":null,"evidence_quote":"The classical Auslander-Yorke dichotomy that this paper generalizes to multivariate diam-mean and frequent stability notions."},{"cited_title":"Pointwise theorems for amenable groups","cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise ergodic theorem for amenable groups used in Theorem 4.1 to pass from Følner averages to the unique invariant measure on the maximal equicontinuous factor."}],"review_version":1}