{"id":"f9054ded-d4ec-496f-94dc-2cc3138a409d","arxiv_id":"2606.28681","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes characterizations of topological conditional entropy via dimensional entropy of stable sets and fibres for amenable group actions, with topological proofs generalizing prior work.","lead":"This paper proves three theorems characterizing topological conditional entropy and relative entropy using dimensional entropy of stable sets and fibres for amenable group actions. A smart generalist might read it to see how entropy concepts from dynamics extend to group actions with purely topological proofs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged the extension step as the key assumption but could not inspect the proofs. The full text supplies those proofs without introducing new unverified conditions, so the UNVERDICTED status is retained pending independent verification of the calculations rather than any detected flaw.","tokens_in":1831,"tokens_out":216,"duration_ms":32181,"concrete_test":"Recompute the equality in Theorem 1.1 for the standard Z-action case using the paper's definition of dimensional entropy on stable sets; if the numerical value matches the known topological conditional entropy, the extension step is verified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript supplies explicit definitions and proofs for extending Bowen's dimensional entropy to amenable actions via Følner sequences, then derives the stated characterizations in Theorems 1.1 and 1.3 by purely topological arguments that avoid measure-theoretic hypotheses. No internal inconsistency, hidden assumption, or failure of the extension is visible in the argument structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends Bowen's dimensional entropy to actions of amenable groups using Følner sequences. It proves three main results: Theorem 1.1 characterizes topological conditional entropy via the dimensional entropy of stable sets (answering a question from Dou-Wang-Zhang 2025 and providing the first such result even for Z-actions); Theorem 1.2 gives a dimensional entropy inequality for factor maps relating the entropy of a set, its image, and the topological entropy of fibres; and Theorem 1.3 shows that the relative topological entropy of a factor map equals the dimensional entropy of the fibres, via a purely topological argument generalizing Oprocha-Zhang 2011 and contrasting with measure-theoretic approaches.","tokens_in":1907,"tokens_out":400,"duration_ms":22672,"significance":"If the results hold, they strengthen the topological foundations of entropy theory for amenable group actions by supplying new characterizations and inequalities that avoid measure-theoretic tools, generalize prior work from single maps to group actions, and resolve an open question on conditional entropy. The explicit use of Følner sequences for the extension and the self-contained topological proofs are notable strengths.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction could include a short explicit statement of the precise definition of Bowen's dimensional entropy on stable sets (e.g., via the Følner-sequence limit) to make the extension from the Z-case immediately visible without consulting the cited references.","section":"Abstract / §1"},{"comment":"Theorem 1.2 is described as the dimensional-entropy counterpart of the packing-entropy result in Dou-Zheng-Zhou 2023; a one-sentence comparison of the two inequalities (e.g., noting where the dimensional version replaces packing with Bowen dimension) would clarify the novelty for readers.","section":"§1"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the referee recognizes the novelty of the characterizations and the purely topological proofs.","responses":[],"tokens_in":1357,"tokens_out":53,"duration_ms":13721,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main points are that they characterize topological conditional entropy by the dimensional entropy of stable sets (Theorem 1.1), answering the open question from Dou-Wang-Zhang 2025, and show that relative topological entropy equals the dimensional entropy of the fibres (Theorem 1.3) with a proof that stays topological instead of using measure theory. They also prove a dimensional entropy inequality for factor maps (Theorem 1.2) that extends earlier single-map work.\n\nWhat is new is the extension to amenable groups via Følner sequences, the fact that the stable-set characterization holds already for Z-actions, and the self-contained topological route to the fibre result that had only been obtained measure-theoretically before. The paper generalizes Oprocha-Zhang 2011 cleanly and avoids any circularity with the cited results.\n\nThe arguments appear solid on the outline given: explicit definitions are supplied and the stress-test finds no internal inconsistency or hidden measure assumptions. The only soft spot is that the abstract is terse on the actual derivations, so a referee would still need to verify the details of how the dimensional entropy behaves under the group action and on the fibres. That is normal for this kind of paper and not a load-bearing flaw.\n\nThis is for people working on entropy in topological dynamics for group actions. A reader who already knows the Dou-Wang-Zhang and Oprocha-Zhang papers will get the most out of it. The work is focused and technically grounded enough to deserve a serious referee.","headline":"The paper gives the first topological characterization of conditional entropy via dimensional entropy on stable sets for amenable actions and a purely topological proof of the fibre formula for relative entropy.","tokens_in":2372,"tokens_out":384,"would_cite":true,"duration_ms":17012,"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":"Topological conditional entropy of amenable group actions equals the dimensional entropy of stable sets.","keywords":["dimensional entropy","amenable group actions","stable sets","fibres","topological conditional entropy","relative topological entropy","factor maps"],"falsifier":"An explicit amenable group action together with a concrete stable set for which the numerical value of Bowen's dimensional entropy differs from the value of the topological conditional entropy.","tokens_in":2725,"feed_emoji":"","tokens_out":615,"duration_ms":21868,"temperature":0.7,"pith_summary":"The paper shows that Bowen's dimensional entropy computed on stable sets recovers the topological conditional entropy exactly for actions of amenable groups. It also proves that the relative topological entropy of any factor map equals the dimensional entropy of the fibers over points in the base space. Both results are obtained with purely topological arguments that avoid measures. The characterizations extend single-map results to group actions and supply the first such description of conditional entropy via dimensional entropy even when the acting group is the integers.","feed_headline":"Stable sets determine conditional entropy for amenable actions","feed_subtitle":"Dimensional entropy on stable sets equals topological conditional entropy, and fiber dimensional entropy equals relative entropy of factor m","key_machinery":"Bowen's dimensional entropy defined on stable sets and on fibers of factor maps for amenable group actions, used to equate classical entropy quantities to covering-based quantities on those sets.","core_discovery":"Bowen's dimensional entropy on the stable sets of an amenable group action equals the topological conditional entropy of the action, and the dimensional entropy on the fibers of a factor map equals the relative topological entropy of that map. These identities hold for the full class of amenable groups and are proved without invoking invariant measures or Shannon-McMillan-Breiman theorems.","pith_inferences":["The stable-set formula may simplify explicit calculations of conditional entropy in systems whose stable sets admit simple covers.","The same covering arguments could be tested on actions of non-amenable groups once a suitable definition of dimensional entropy is chosen.","The fiber formula supplies a way to compare relative entropies across different factor maps by examining only the preimage sets."],"forward_implications":["Topological conditional entropy can be recovered by computing dimensional entropy only on stable sets rather than on the whole space.","Relative topological entropy of a factor map equals the supremum of dimensional entropies of its fibers.","A dimensional-entropy inequality holds for any factor map relating the entropy of a set, its image, and the entropies of the fibers.","The fiber characterization of relative entropy admits a purely topological proof that applies to all amenable groups."],"fun_headline_variants":["Dimensional entropy on stable sets equals conditional entropy","Fibre dimensional entropy equals relative topological entropy","Stable set entropy matches topological conditional entropy","Amenable actions tie fibre entropy to relative topological entropy"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The definitions and covering properties of Bowen's dimensional entropy on stable sets and fibers extend directly to amenable group actions so that the stated equalities hold.","fun_headline_variants_meta":{"raw":{"variants":["Dimensional entropy on stable sets equals conditional entropy","Fibre dimensional entropy equals relative topological entropy","Stable set entropy matches topological conditional entropy","Amenable actions tie fibre entropy to relative topological entropy"]},"model":"grok-4.3","cost_usd":0.004271,"raw_usage":{"total_tokens":2191,"prompt_tokens":750,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":42712000,"prompt_tokens_details":{"text_tokens":750,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1386,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":750,"tokens_out":55,"duration_ms":14600,"temperature":1.0,"reasoning_tokens":1386,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T09:01:52.525646+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit amenable group action together with a concrete stable set for which the numerical value of Bowen's dimensional entropy differs from the value of the topological conditional entropy.","supporting_citations":[],"review_version":1}