{"id":"a5aadcf3-39df-403a-8185-380c2944fc0a","arxiv_id":"2511.18040","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For amenable group actions, relative zero entropy is preserved under passing to induced probability-measure maps, and positive relative entropy makes the induced relative mean dimension infinite.","lead":"This math paper proves that for actions of amenable groups, a factor map has zero relative topological entropy exactly when its induced map on probability measures does, and positive relative entropy in the original map forces infinite relative mean dimension in the induced map. It extends known results for single systems to factor maps, using combinatorial independence sets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's proof uses an exact finite-empirical representation that Lemma 3.1 cannot provide as stated; the forward half of Theorem 1.1(i) is unsupported until a simultaneous approximation argument is supplied.","rationale":"The main theorem's equivalences rest on Theorem 2.4 (finite relative entropy implies zero relative mean dimension) and on the two one-way statements proved in Sections 3 and 4. The reader's weakest assumption was Theorem 2.4, whose proof is omitted. I agree that this is load-bearing and warrants the CONDITIONAL verdict. However, the most concrete defect I find is in Section 3: Lemma 3.1 is printed as an exact equality that is false for non-atomic measures, and (7) asserts an exact simultaneous empirical representation that is not justified. The proof of Prop. 3.4 would still go through if a density/approximation version of Lemma 3.1 is supplied and if the strict inequalities are shown to be stable under approximation; this is plausible but not in the manuscript. Thus the concern is a proof gap rather than a known false theorem. It does not change the reader's CONDITIONAL verdict, but it gives a sharper, checkable target for revision.","tokens_in":13510,"tokens_out":23804,"duration_ms":222019,"concrete_test":"Verify Lemma 3.1 with X=Y=[0,1], π=id, ν=Lebesgue: (ν,ν) belongs to R_eπ but ν is not a finite equal-weight empirical measure, so Lemma 3.1 is false as stated. Then check whether the intended density version can be used in Prop. 3.4: for each σ choose L_k, y_i^k, x_i^{σ,k} with π(x_i^{σ,k})=y_i^k, (1/L_k)Σδ_y_i^k → ν, and (1/L_k)Σδ_x_i^{σ,k} → λσ; prove that the strict inequalities λσ(h^{-1}A_i)>a_i persist for large k. If they do not, the forward direction fails; if they do, a revision of this step is sufficient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 3.4, after fixing an independence set E for (U1,U2) with respect to eπ, the proof obtains ν and λσ satisfying (6), then invokes Lemma 3.1 to find L, yi, xi^σ with π(xi^σ)=yi and the exact equalities ν=(1/L)Σδ_yi, λσ=(1/L)Σδ_xi^σ (equation (7)). Lemma 3.1 as stated (R_eπ = ∪_n R_eπ_n) is false: take X=Y=[0,1], π=id, and ν Lebesgue; (ν,ν)∈R_eπ, but ν is not in any M_n(X), so (ν,ν) is not in any R_eπ_n. The lemma can at most be a density statement (closure). The proof gives no limiting argument, and the inequalities feeding into Ψσ are strict consequences of (6); it is not automatic that they are preserved under weak* approximation. Because this step is the only bridge from the IE-pair (μ1,μ2) to an independence set for (A1,A2) in X, the forward direction htop(π)=0 ⇒ htop(eπ)=0 is incomplete as written. The other unproved ingredient, Theorem 2.4, is also load-bearing for the converses, but the empirical-approximation gap is the more directly defective step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relation between relative topological entropy and relative mean dimension for factor maps of actions of countably infinite amenable groups, and the corresponding induced factor maps on spaces of Borel probability measures. Theorem 1.1 claims: (i) a factor map π has zero relative topological entropy if and only if the induced factor map eπ does; (ii) π has positive relative topological entropy if and only if eπ has infinite relative mean dimension. The proof strategy is to prove the two hard one-way directions — htop(π)=0 ⇒ htop(eπ)=0 in Section 3 and htop(π)>0 ⇒ mdim(eπ)=∞ in Section 4 — and then to use a relative version of the Lindenstrauss–Weiss theorem (Theorem 2.4) to obtain the converses. Section 3 uses relative independence sets and a Sauer–Shelah-type combinatorial lemma; Section 4 builds large simplexes inside fibers of eπ and applies a tailored Lebesgue lemma. The conclusions are strong and the overall architecture is coherent, but two load-bearing ingredients are problematic as written.","tokens_in":13873,"tokens_out":22903,"duration_ms":196365,"significance":"If the gaps are repaired, this is a significant contribution. The paper establishes a complete quantitative dictionary between fiber complexity of a factor map and that of its induced measure factor: zero relative entropy is preserved exactly, and positive relative entropy forces infinite relative mean dimension in the induced system. This extends the Glasner–Weiss and Burguet–Shi results to relative entropy/mean dimension for amenable group actions, and the independence-based method in Section 3 is a novel route. The construction in Section 4, embedding high-dimensional simplexes in fibers of the induced map, is elegant. The paper also makes good use of machine-checkable-style combinatorial lemmas and gives a self-contained proof of the main combinatorial ingredient (Lemma 3.2). However, the current manuscript contains an unproved theorem and a false lemma that are essential to the main theorem, so the central claims are not yet established as written.","major_comments":[{"comment":"Lemma 3.1 as stated is false. If X=Y=[0,1] and π=id, then R_{eπ} is the diagonal of M(X)×M(X); taking ν to be Lebesgue measure gives (ν,ν)∈R_{eπ}, but ν is not in any M_n(X), since M_n(X) consists of uniform empirical measures on n points. Thus R_{eπ} is not the union of the R_{eπ_n}. At most one has a density statement, R_{eπ} = closure(⋃_n R_{eπ_n}). This matters because Proposition 3.4 uses Lemma 3.1 to pass from (6) to the exact representation (7), with a common L and exact equalities ν=(1/L)Σδ_{y_i} and λ_σ=(1/L)Σδ_{x_i^σ}. Without a simultaneous approximation argument, the strict inequalities in (6) and the subsequent counting argument leading to the independence set for (A_1,A_2) are not justified. The forward direction htop(π)=0 ⇒ htop(eπ)=0 is therefore incomplete as written.","section":"Section 3, Lemma 3.1 and Eq. (7)"},{"comment":"Theorem 2.4 states that finite relative topological entropy implies zero relative mean dimension for a factor map of amenable group actions. The proof is omitted with the comment that it is 'almost the same' as [LW00, Theorem 4.2]. However, [LW00, Theorem 4.2] is the absolute case, and the relative version is load-bearing: after the one-way implications in Sections 3 and 4, Theorem 2.4 is exactly what converts them into the if-and-only-if statements of Theorem 1.1. The authors should either give a full proof or cite a reference that contains the relative amenable-group version. As it stands, the converses of both (i) and (ii) rest on an unproved assertion.","section":"Section 2, Theorem 2.4"}],"minor_comments":[{"comment":"Several entries in the bibliography appear not to be cited in the text: [Dow11], [GTW00], [Lia22]. These should be removed or cited.","section":"References"},{"comment":"Typo: 'assuem' should be 'assume'.","section":"Page 11, Claim 1 proof"},{"comment":"The injectivity of Ψ is used implicitly when identifying L_n with Δ_{[2]^H}^{m_n}. It is true because the sets defining x_E are pairwise disjoint for distinct E, but it should be stated explicitly.","section":"Section 4, definition of Ψ"},{"comment":"The sentence 'Take h_{i,j} ∈ {h_ℓ : ℓ∈[M]} such that h_{i,j} ≠ h_{i,j'} for any i∈[m_n] and distinct j,j′∈[H]' could be misread as requiring distinctness across different i. It is clearer to say 'for each fixed i, the elements h_{i,1},...,h_{i,H} are pairwise distinct'.","section":"Section 4, construction of h_{i,j}"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely true and the proof strategy is promising, but the current version has two load-bearing gaps: a false lemma (Lemma 3.1) and an unproved relative Lindenstrauss–Weiss theorem (Theorem 2.4). The false lemma is particularly concerning because it comes from a paper co-authored by the first-named author; the authors should be asked to supply a correct density statement and a full simultaneous approximation argument, not merely a citation. If these repairs are successful, the paper would be a strong contribution to the relative entropy/mean dimension theory of induced systems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: this paper proves a relative version of two known induced-system phenomena, but the proof as written is not complete. The zero-entropy preservation step rests on a lemma (3.1) that is false as stated, and the 'if' directions of the main theorem rely on Theorem 2.4, which is stated without proof. Both look repairable, and the main ideas are likely sound, but a referee will need to force the repairs.\n\nWhat's new: the result itself, for amenable group actions, is a meaningful extension of Glasner-Weiss (zero entropy of induced systems) and Burguet-Shi / Shi-Zhang (positive entropy implies infinite mean dimension of induced systems) from absolute systems to factor maps. The proof machinery—using IE-pairs and a Sauer-Shelah-type lemma to transfer independence from the induced system back to the original—is a real step forward, and Section 4's simplex embedding argument is an elegant way to get the huge fiber complexity.\n\nThe main proof has two soft spots, in proportion to how soft they are. First, the stress-test is right about Lemma 3.1. In this paper, M_n(X) is the set of uniform empirical measures of length n, so R_{eπ} is not the union of the R_{eπ_n}; the union is only the empirical pairs. The proof needs a simultaneous approximation argument (choose L large, approximate all λ_σ and ν by common empirical averages with small error), and the strict inequalities would then be preserved by choosing the slack carefully. As written, it's a gap, though not one that obviously sinks the result. Second, Theorem 2.4 (finite relative entropy implies zero relative mean dimension) is simply asserted, and it's load-bearing for the 'if' directions. If the proof is really 'almost the same' as Lindenstrauss-Weiss, the authors should write it out, because the relative amenable-group setting is not identical.\n\nThe other direction—positive entropy of π forces infinite relative mean dimension of eπ—looks solid modulo the usual checks. Claim 3's use of the Burguet-Shi Lebesgue lemma is the right tool.\n\nMy overall take: this is a serious paper with a likelier-than-not correct theorem, but it is not ready in its current form. The gaps are fixable and the combinatorial core is plausible. I'd send it to a competent referee—not desk reject it. The referee should ask for a corrected treatment of the empirical approximation step and a full proof of Theorem 2.4.","headline":"The result is important and likely correct, but the written proof has two real gaps: a false empirical-representation lemma in Section 3 and an unproved relative Lindenstrauss-Weiss theorem used for the converses.","tokens_in":14309,"tokens_out":4462,"would_cite":false,"duration_ms":42411,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B40","37B99"],"pacs":[],"model":"deepseek-v4-flash","headline":"For amenable group actions, a factor map has zero relative entropy exactly when its induced factor map does, and positive relative entropy exactly when the induced factor has infinite relative mean dimension.","keywords":["relative topological entropy","relative mean dimension","induced factor maps","amenable group actions","combinatorial independence","IE-pairs","probability measure spaces","Følner sequences"],"falsifier":"Construct a factor map π between two G-systems with G a countably infinite amenable group such that h_top(π,G)=0 but mdim(π,G)>0. Such a map would violate the contrapositive of Theorem 2.4 and therefore the converse direction of Theorem 1.1(ii); the paper's one-way implication (positive entropy yields infinite relative mean dimension of the induced factor) would still hold. Conversely, any example with h_top(π,G)>0 and mdim(eπ,G)<∞ would directly refute Theorem 1.1(ii).","tokens_in":13414,"feed_emoji":"📐","tokens_out":10374,"duration_ms":77940,"temperature":0.7,"pith_summary":"This paper proves a dichotomy for factor maps between compact dynamical systems acted on by a countably infinite amenable group. If the original factor map has zero relative topological entropy—meaning its fibers are not exponentially complex—then the induced factor map on the space of Borel probability measures also has zero relative topological entropy. Conversely, if the original factor map has positive relative topological entropy, the induced factor map has infinite relative mean dimension, so its fibers contain arbitrarily high-dimensional structure. Together these statements say that the relative complexity of the original map is completely mirrored in the induced measure system: the zero-versus-positive entropy threshold becomes a zero-versus-infinite dimension threshold. This matters because it converts a counting question into a geometric one and gives a new way to detect positive relative entropy through infinite-dimensional fibers.","feed_headline":"Positive relative entropy forces infinite dimension on induced maps","feed_subtitle":"For amenable group actions, the induced map's geometry reveals whether relative entropy is zero or positive.","key_machinery":"Combinatorial independence sets for factor maps. For a pair of sets A_1,A_2, a subset I of the group is an independence set if every finite assignment of group elements to A_1 or A_2 can be realized by a single point in a common fiber of π. Positive independence density—having such sets of size at least c|H| inside every finite H—is equivalent to positive relative entropy by a known characterization. Section 3 converts independence for measure-open sets in M(X) into independence for point-open sets in X via an averaging lemma that builds, for each binary coloring of a large group subset, a common fiber point. Section 4 turns an independence set of positive density into a continuous affine in","core_discovery":"The central claim is Theorem 1.1: for a countably infinite amenable group G and a factor map π:(X,G)→(Y,G), the induced factor map eπ:(M(X),G)→(M(Y),G) on Borel probability measures satisfies h_top(π,G)=0 if and only if h_top(eπ,G)=0, and h_top(π,G)>0 if and only if mdim(eπ,G)=+∞. Relative topological entropy counts, per group element, how many orbit segments are needed to separate points within a single fiber; relative mean dimension counts, per group element, how many real parameters are needed to describe a fiber. The paper proves the forward directions directly: an independence set witnessing positive relative entropy of eπ is converted through an averaging argument into an independence","pith_inferences":["The omitted proof of the bridge theorem is the delicate point. If a relative version of 'finite entropy implies zero mean dimension' fails for amenable group actions, the iff statements in Theorem 1.1 would reduce to one-way implications; a concrete falsifier would be a factor map with zero relative entropy and positive relative mean dimension.","The proof uses only independence density and a Lebesgue lemma, so the same scheme could likely be adapted to other induced spaces—such as the space of closed subsets or the measure center of a system—to obtain analogous entropy-versus-dimension dichotomies.","The construction in Section 4 is quantitative: the order of refining covers grows like 2^H while the group size grows polynomially in H, suggesting that when positive relative entropy is present, the relative mean dimension of the induced factor is not merely infinite but diverges at a controlled rate tied to the independence density."],"forward_implications":["Zero relative topological entropy of a factor map between amenable group actions is completely determined by the induced factor map on probability measures: one is zero exactly when the other is.","Positive relative topological entropy of the original map manifests at the induced level as infinite relative mean dimension—a geometric form of complexity, not merely infinite entropy.","Combined with the bridge theorem that finite relative entropy forces zero relative mean dimension, the result implies that in this setting a factor map cannot have both finite positive relative entropy and positive relative mean dimension.","In the absolute case where the factor is a single point, the theorem recovers and unifies known results: a system has zero topological entropy iff its induced system does, and positive entropy iff its induced system has infinite mean dimension."],"fun_headline_variants":["Zero entropy iff zero entropy on induced maps","Positive entropy makes induced maps infinite-dimensional","Entropy sign tied to induced dimension for amenable actions","Zero entropy persists through measure-induction","Relative entropy zero iff induced entropy zero"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the unproved Theorem 2.4—that any factor map with finite relative topological entropy has zero relative mean dimension; the paper says the proof is 'almost the same' as the absolute case but omits it, and the cited theorem it is modelled on covers only the absolute, non-relative setting. If that relative statement fails, the 'only if' halves of Theorem 1.1 collapse while the proved one-way implications survive.","fun_headline_variants_meta":{"raw":{"variants":["Zero entropy iff zero entropy on induced maps","Positive entropy makes induced maps infinite-dimensional","Entropy sign tied to induced dimension for amenable actions","Zero entropy persists through measure-induction","Relative entropy zero iff induced entropy zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1066,"prompt_tokens":597,"completion_tokens":469,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":341,"completion_tokens_details":{"reasoning_tokens":403}},"tokens_in":341,"tokens_out":469,"duration_ms":4434,"temperature":1.0,"reasoning_tokens":403,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T20:50:12.406533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a factor map π between two G-systems with G a countably infinite amenable group such that h_top(π,G)=0 but mdim(π,G)>0. Such a map would violate the contrapositive of Theorem 2.4 and therefore the converse direction of Theorem 1.1(ii); the paper's one-way implication (positive entropy yields infinite relative mean dimension of the induced factor) would still hold. Conversely, any example with h_top(π,G)>0 and mdim(eπ,G)<∞ would directly refute Theorem 1.1(ii).","supporting_citations":[],"review_version":1}