{"id":"b90d4cc0-ef70-4665-b6ee-64adeda55c84","arxiv_id":"2606.13270","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalizes localization formula for metric mean dimension to amenable group actions using combinatorial covering lemma and resolves uniformity questions with counterexamples.","lead":"The paper extends Tsukamoto's localization formula for metric mean dimension from R^k and Z^k actions to countable discrete amenable groups by characterizing it via asymptotic entropy of pointwise ε-stable sets. It also constructs counterexamples showing that supremum and limit superior cannot be interchanged in the formula.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the technical step, yet the paper explicitly claims to carry out that replacement and supplies counterexamples for the uniformity issues. Absent a detectable gap in the argument as described, the UNVERDICTED status (stemming from abstract-only review) does not shift.","tokens_in":1696,"tokens_out":278,"duration_ms":16841,"concrete_test":"Specialize the proof of the equivalent definitions (presumably §3) to the case G=ℤ with the standard Følner sequence; verify that the resulting expressions recover Tsukamoto's original localization formula exactly, with no extra terms introduced by the combinatorial covering lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 2.3) asserts that global metric mean dimension equals the asymptotic entropy of pointwise ε-stable sets for countable discrete amenable group actions. This rests on introducing equivalent definitions via topological/packing/Bowen entropies and replacing tiling arguments by Lindenstrauss's combinatorial covering lemma. The abstract states that the lemma suffices for the general amenable case and that explicit counterexamples resolve the three uniformity questions from Yang-Chen-Zhou. No internal inconsistency, hidden assumption on Følner sequences, or mismatch with the metric structure is visible in the stated construction.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends Tsukamoto's localization formula for metric mean dimension from ℤ^k- and ℝ^k-actions to actions of countable discrete amenable groups. The central result (Theorem 2.3) asserts that the global metric mean dimension equals the asymptotic entropy of pointwise ε-stable sets. Equivalent definitions of the invariant are introduced via topological entropy, packing topological entropy, and Bowen's dimensional entropy. Tiling arguments are replaced by Lindenstrauss's combinatorial covering lemma to handle general amenable groups. Counterexamples (Theorem 2.6 and Proposition 5.8) are constructed to resolve the three uniformity questions from Yang-Chen-Zhou, showing that the supremum and limsup in the localization formula cannot be interchanged in general.","tokens_in":1787,"tokens_out":465,"duration_ms":22427,"significance":"If the claims hold, the work generalizes an important invariant to a substantially larger class of group actions, supplies multiple equivalent characterizations that may aid concrete computations, and settles open uniformity questions via explicit counterexamples. The adaptation of the combinatorial covering lemma, if verified to apply without extra conditions, constitutes a useful technical contribution for amenable-group dynamics.","major_comments":[{"comment":"Theorem 2.3: The assertion that Lindenstrauss's combinatorial covering lemma suffices to establish equivalence among the topological, packing, and Bowen formulations (and thereby the localization formula) for arbitrary countable discrete amenable groups is load-bearing; the proof must explicitly confirm that the lemma applies directly to the relevant Følner sequences without additional restrictions on the group or the action.","section":"Theorem 2.3"}],"minor_comments":[{"comment":"The definitions of pointwise ε-stable sets and the three entropy variants should be stated in full before their first use in the statements of the main theorems.","section":null},{"comment":"Notation for the amenable group, the metric, and the Følner sequences should be introduced uniformly at the beginning of Section 2 to avoid later ambiguity.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the positive evaluation of the manuscript. We address the single major comment below.","responses":[{"response":"We agree that an explicit verification of the lemma's hypotheses is necessary for the argument to be fully rigorous. In the revised version we will add a short paragraph immediately after the invocation of Lindenstrauss's combinatorial covering lemma in the proof of Theorem 2.3. This paragraph will record that the Følner sequences employed are chosen to satisfy the exact conditions stated in Lindenstrauss's lemma (in particular, the vanishing of the boundary-to-volume ratio) and that the argument imposes no further restrictions on either the amenable group or the continuous action.","revision_made":"yes","referee_comment":"[Theorem 2.3] Theorem 2.3: The assertion that Lindenstrauss's combinatorial covering lemma suffices to establish equivalence among the topological, packing, and Bowen formulations (and thereby the localization formula) for arbitrary countable discrete amenable groups is load-bearing; the proof must explicitly confirm that the lemma applies directly to the relevant Følner sequences without additional restrictions on the group or the action."}],"tokens_in":1338,"tokens_out":235,"duration_ms":10901,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper extends Tsukamoto's localization formula for metric mean dimension to actions of countable discrete amenable groups. It also constructs counterexamples that resolve the three uniformity questions from Yang, Chen, and Zhou.\n\nThe new work shows that the global metric mean dimension is given by the asymptotic entropy of pointwise ε-stable sets. To get there, they introduce equivalent definitions using topological entropy, packing topological entropy, and Bowen's dimensional entropy. The technical move is to use Lindenstrauss's combinatorial covering lemma instead of tiling arguments, which lets them handle general amenable groups rather than just Z^k or R^k.\n\nIf the proofs go through, this is a solid step forward. The counterexamples are explicit and directly show that the supremum and limit superior cannot be swapped in the localization formula, highlighting that convergence can be non-uniform.\n\nThe equivalent definitions are a plus because they give different ways to compute the invariant in practice. The paper builds cleanly on prior results by Tsukamoto and Lindenstrauss.\n\nA possible soft spot is whether Lindenstrauss's lemma really covers all the cases needed for the equivalences without additional assumptions on the group or the metric. The abstract claims it does, but the details matter for how general the result is.\n\nThis is aimed at people working on topological dynamics and mean dimension for group actions. Anyone studying entropy invariants or localization phenomena in dynamics will get something from the localization result and the uniformity counterexamples.\n\nIt should go to peer review. The claims address open questions and the approach looks reasonable on the surface.","headline":"Extends the localization formula to general amenable groups via the covering lemma and settles the uniformity questions with explicit counterexamples.","tokens_in":2254,"tokens_out":383,"would_cite":true,"duration_ms":20686,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The metric mean dimension of an amenable group action equals the asymptotic entropy of its pointwise ε-stable sets.","keywords":["metric mean dimension","amenable group actions","localization formula","pointwise ε-stable sets","topological entropy","packing entropy","non-uniformity","combinatorial covering lemma"],"falsifier":"An amenable group action where the asymptotic entropy of pointwise ε-stable sets differs from the global metric mean dimension computed by other means would falsify the characterization.","tokens_in":2597,"feed_emoji":"📐","tokens_out":642,"duration_ms":17047,"temperature":0.7,"pith_summary":"This paper extends Tsukamoto's localization formula for metric mean dimension from Z^k- and R^k-actions to actions of general countable discrete amenable groups. It establishes that the global metric mean dimension is recovered exactly from the asymptotic entropy of pointwise ε-stable sets. Equivalent definitions are given in terms of topological entropy, packing topological entropy, and Bowen's dimensional entropy. The proof replaces tiling arguments with Lindenstrauss's combinatorial covering lemma to accommodate the structure of arbitrary amenable groups. Counterexamples demonstrate that the supremum and limit superior in the localization formula cannot be interchanged, confirming non-uniform convergence.","feed_headline":"Metric mean dimension equals asymptotic entropy of stable sets","feed_subtitle":"The result extends localization from Z^k actions to general amenable groups and shows sup and limsup cannot be swapped.","key_machinery":"The localization formula that equates global metric mean dimension to the asymptotic entropy of pointwise ε-stable sets, which works by replacing tiling with Lindenstrauss's combinatorial covering lemma.","core_discovery":"The global metric mean dimension equals the asymptotic entropy of pointwise ε-stable sets for actions of countable discrete amenable groups. This is proved by introducing equivalent definitions via topological entropy, packing topological entropy, and Bowen's dimensional entropy, using Lindenstrauss's combinatorial covering lemma in place of tiling arguments, and constructing counterexamples that show the supremum and limit superior cannot be interchanged in the localization formula.","pith_inferences":["The same covering-lemma technique may allow localization formulas for other entropy-like invariants on amenable groups.","Non-uniformity implies that pointwise dynamical features can dominate the value of global invariants even when averaged.","The new definitions could simplify explicit calculations of metric mean dimension on concrete amenable actions such as shifts on groups with complicated Følner sequences."],"forward_implications":["Equivalent definitions via topological entropy, packing topological entropy, and Bowen's dimensional entropy provide flexible tools for computing metric mean dimension.","The supremum and limit superior in the localization formula cannot generally be interchanged.","The convergence appearing in the localization formula is heterogeneous.","The three uniformity questions from Yang, Chen, and Zhou are resolved by explicit counterexamples."],"fun_headline_variants":["Metric mean dimension matches entropy of stable sets for amenable groups","Localization formula extended to countable amenable group actions","Counterexamples demonstrate non-uniformity in localization formula","Multiple entropy definitions for mean dimension in amenable actions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Lindenstrauss's combinatorial covering lemma suffices to replace tiling arguments when defining the equivalent entropy notions for general amenable groups.","fun_headline_variants_meta":{"raw":{"variants":["Metric mean dimension matches entropy of stable sets for amenable groups","Localization formula extended to countable amenable group actions","Counterexamples demonstrate non-uniformity in localization formula","Multiple entropy definitions for mean dimension in amenable actions"]},"model":"grok-4.3","cost_usd":0.005711,"raw_usage":{"total_tokens":2723,"prompt_tokens":662,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":57112000,"prompt_tokens_details":{"text_tokens":662,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2003,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":662,"tokens_out":58,"duration_ms":13788,"temperature":1.0,"reasoning_tokens":2003,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T05:37:27.357626+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An amenable group action where the asymptotic entropy of pointwise ε-stable sets differs from the global metric mean dimension computed by other means would falsify the characterization.","supporting_citations":[],"review_version":1}