{"id":"129db7b0-01b4-46ec-8fbb-8ac08d8d346d","arxiv_id":"1908.02177","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces topological nearly entropy on nearly compact spaces and an R-space property, then claims, with a flawed proof, that entropy coincides with restriction entropy.","lead":"The paper extends topological entropy to nearly compact spaces and claims that a map's entropy equals the entropy of its restriction to any invariant nearly compact subset. A smart generalist might read it to see how entropy is adapted to weaker compactness, but the central proof has an unjustified step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.5(b) replaces a supremum over all K∈H(X,f) with one fixed K; Theorem 4.3 is only pointwise, so the central equality is unproved.","rationale":"The reader's REJECT verdict is correct: the central coincidence theorem is not proved. My stress-test isolates the most load-bearing step as the unjustified collapse of sup_K{EntN(f,U,K)} to the fixed K, followed by an application of Theorem 4.3 that only holds pointwise. This is independent of the regular-open extension issue, which the reader identified as the weakest assumption. I agree with the reader's rationale, which also flags the sup collapse, but I regard the collapse as even more fundamental, since no topological lemma about regular open covers can repair it. Both concerns point to rejection, so no verdict change is needed. The paper's smaller results, such as Theorem 4.3 pointwise equality and Theorem 4.4, may be salvageable, but they do not support the abstract's strong claim of coincidence.","tokens_in":12417,"tokens_out":23619,"duration_ms":266841,"concrete_test":"Re-derive the chain in Theorem 4.5(b) with the suprema kept separate. Compute EntN(f) via Definition 2.3 as sup_{K∈H} sup_U EntN(f,U,K), and compare with the claimed value Entn(f|K0,U|K0) for a fixed K0. Since Theorem 4.3 only proves EntN(f,U,K0)=Entn(f|K0,U|K0), the equality sup_K EntN(f,U,K)=Entn(f|K0,U|K0) must be justified separately. A concrete witness would be any system with two nested invariant nearly compact sets K1⊂K2 and EntN(f,K1)<EntN(f,K2): the proof applied to K1 forces Entn(f|K1)=EntN(f), contradicting monotonicity and Theorem 2.2. If no such system is known, the symbolic check that no stated lemma supplies the missing equality is sufficient to confirm the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central claim is Theorem 4.5(b), EntN(f)=Entn(f|K) for every K∈H(X,f). Its proof first swaps the suprema correctly, but then writes 'By Theorem 4.3, EntN(f,U)=Entn(f|K,U|K)'. Theorem 4.3 is pointwise: for the particular K fixed in the theorem statement, EntN(f,U,K)=Entn(f|K,U|K). It says nothing about sup_K EntN(f,U,K). The proof never shows that the fixed K realizes this supremum or that Entn(f|K) is independent of K; the displayed chain would force exactly that. The same paragraph also needs the unproved assertion that each regular open A⊂K extends to a regular open U_A⊂X with A=U_A∩K, and that K is itself nearly compact so that Definition 4.2 applies to f|K; equation (4.1) depends on this. These are independent gaps: the sup collapse cannot be fixed by any topological extension lemma, and the extension issue cannot be fixed by the monotonicity of Theorem 2.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new notion of topological nearly entropy, Entn, for R-dynamical systems on nearly compact spaces, and aims to relate it to the previously defined EntN for systems with invariant nearly compact subsets. The main results are: (i) Theorem 4.3, which claims an equality between Entn(f|K, U|K) and EntN(f, U, K) for K in H(X,f); (ii) Theorem 4.4, stating that when X itself is nearly compact, Entn(f)=EntN(f); and (iii) Theorem 4.5, which asserts that for Hausdorff R-spaces, EntN(f,K)=Entn(f|K,K) and, more strongly, EntN(f)=Entn(f|K) for every K in H(X,f). The paper also introduces the notion of R-space and proves several lemmas about it, and concludes with product-space inequalities for EntN and Entn.","tokens_in":12617,"tokens_out":9049,"duration_ms":89958,"significance":"If the main results were correct, they would extend Adler-Konheim-McAndrew topological entropy to a framework for nearly compact spaces via R-maps, and the equality in Theorem 4.5(b) would be a striking simplification: every invariant nearly compact subset would carry the full entropy of the system. The concept of R-space is potentially useful and the lemmas in Section 4 (Lemmas 4.1-4.3) are interesting in themselves. However, the central proofs contain serious gaps: Theorem 4.3 is not well-defined as stated, and Theorem 4.5(b) collapses a supremum over all invariant nearly compact subsets to a single fixed subset without justification. These issues affect the principal claims of the paper, so the contribution is not presently reliable.","major_comments":[{"comment":"The quantity Entn(f|K, U|K) is not well-defined under the paper's own definitions. Definition 4.2 applies only when the underlying space is nearly compact and the cover is a regular open cover. In Theorem 4.3, K is only assumed to be nearly compact relative to X, which does not imply that K is nearly compact as a subspace; moreover, U|K = {U∩K : U∈U} need not be a regular open cover of K. For example, take X = R, K = [0,1] (which is nearly compact relative to R) and U = (0,2) (regular open in R). Then U∩K = (0,1], which is not regular open in the subspace K because int_K(cl_K((0,1])) = (0,1) ≠ (0,1]. Hence the proof of Theorem 4.3 applies the function Nn to a cover that may not be a regular open cover, and the claimed equality is unsupported.","section":"§4, Theorem 4.3"},{"comment":"The proof of Theorem 4.5(b) contains a critical fallacy in the manipulation of suprema. After swapping the order of sup_K and sup_U, the proof writes 'By Theorem 4.3, EntN(f,U) = Entn(f|K,U|K)' and concludes that sup_U EntN(f,U) = Entn(f|K). But Theorem 4.3 is pointwise: it states that for a fixed K ∈ H(X,f), EntN(f,U,K) = Entn(f|K,U|K). It says nothing about the value of sup_{K∈H(X,f)} EntN(f,U,K) and does not show that the fixed K realizes this supremum or that Entn(f|K) is independent of K. The displayed chain of equalities would imply that every invariant nearly compact subset carries the full entropy, a much stronger statement that is not established and is generally false in the classical compact setting. This gap is load-bearing: without it, the equality EntN(f)=Entn(f|K) does not follow.","section":"§4, Theorem 4.5(b)"},{"comment":"The derivation of equation (4.1) relies on unproved extension properties. The proof asserts that for every regular open cover U_K of the subspace K, there exists a regular open cover U of X such that A = U_A∩K for each A∈U_K, with X\\K regular open, and then that the supremum over all such special covers U equals the supremum over all regular open covers of X. Neither assertion is proved, and Lemma 4.3 does not imply them. Moreover, the resulting restriction U|K contains the empty set (from X\\K), which is not a regular open set, so U|K is not a regular open cover of K. The equality of the two suprema in (4.1) is therefore unjustified, and Theorem 4.5(a) is unsupported.","section":"§4, Theorem 4.5(a), Eq. (4.1)"},{"comment":"The notation Entn(f|K, K) and Entn(f|K, UK, K) used in Theorem 4.5 is not defined anywhere. Definition 4.2 defines Entn(f, U) and Entn(f) only for a nearly compact space X. The proof then treats Entn as if it takes a space, a cover, and a subset as arguments. This is not merely a cosmetic issue: it obscures the fact that Entn for a restriction f|K requires the subspace K to be a nearly compact dynamical system in its own right, a property that is never established in the paper.","section":"§4, Definition 4.2 and Theorem 4.5"}],"minor_comments":[{"comment":"The title contains typographical errors: 'NEARL Y ENTROPY', 'COMP ACT', 'NEARL Y COMP ACT' should be corrected.","section":"Title page"},{"comment":"The sentence 'the topological nearly entropy of f and it restriction f|K coincides' should read 'its restriction f|K'.","section":"Abstract"},{"comment":"The symbol 'EntN(f,U)' appears in the proof, but EntN is defined with three arguments: EntN(f,U,K). This should be 'EntN(f,U,K)'.","section":"§4, Theorem 4.5(b) proof"},{"comment":"The claim that 'the only invariant compact subset of R and hence nearly compact subset of R is {0}' is misleading, because near compactness is weaker than compactness; the argument should justify why no other invariant nearly compact subset exists, or weaken the statement accordingly.","section":"§5, Example 5.1"},{"comment":"Some references contain typographical errors: [13] 'Intoduction' should be 'Introduction', and [9] 'Topoogical' should be 'Topological'.","section":"References"}],"recommendation":"reject","confidential_remarks":"The manuscript relies heavily on the authors' prior work [9] for definitions and proofs, which makes independent verification difficult. However, my rejection is based on the concrete technical gaps in Theorems 4.3 and 4.5, which are central to the paper's claims and cannot be repaired by minor editing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: the new definitions (Entn, R-space) are real, but the headline claim — EntN(f) = Entn(f|K) for every invariant nearly compact K in a Hausdorff R-space — is not proved. The proof of Theorem 4.5(b) has a load-bearing error: it identifies sup_K EntN(f,U,K) with EntN(f,U,K) for a fixed K. Theorem 4.3 is pointwise; it says nothing about the supremum. Unless the fixed K realizes the supremum or the value is independent of K, which is not shown, the chain collapses. This cannot be repaired by a topological extension lemma.\n\nThe same theorem relies on an unproved extension property: every regular open cover of K must come from a regular open cover of X. The authors note K is regular closed and then assert that each regular open A⊆K extends to a regular open U_A⊆X with A=U_A∩K, and that X\\K is regular open. The latter is fine; the former is not automatic. Regular open sets in subspaces need not extend. This is an independent gap.\n\nThere is also a quiet problem in Theorem 4.3 itself: to talk about Entn(f|K,U|K), K must be a nearly compact space and U|K must be a regular open cover of K. H(X,f) only gives near compactness relative to X, and intersections of regular open sets with a subset are generally not regular open in the subspace. The 'subspace K' claim needs extra hypotheses.\n\nCredit where due: Theorem 4.4 (X nearly compact ⇒ Entn=EntN) is correct, assuming Theorem 4.3. The R-space separation lemmas 4.1–4.3 are fine. Section 5's product subadditivity is a plausible analogue, though Lemma 5.5 implicitly assumes a regular open product-neighborhood property that does not follow from the stated hypotheses. Example 5.1 is correct but trivial.\n\nBottom line: not publishable in current form. The definitions and some small results are salvageable, but the coincidence theorem is the paper's backbone and it is unsupported. I'd desk reject and invite a corrected resubmission. I wouldn't bring it to reading group except as a case study in where sup-of-entropy arguments go wrong.\n\nBest,\n\n[You]","headline":"The paper's new definitions are fine but its central coincidence theorem has a load-bearing proof gap; the sup-of-entropy argument is invalid.","tokens_in":13171,"tokens_out":7828,"would_cite":false,"duration_ms":78558,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54H20","37B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every nearly compact invariant subset carries the full entropy of the system on Hausdorff R-spaces.","keywords":["topological nearly entropy","nearly compact space","R-map","R-dynamical system","R-space","regular open cover","product system"],"falsifier":"The equality can be tested by finding a Hausdorff $R$-space $X$, an $R$-map $f$, and an invariant nearly compact subset $K$ with a regular open cover of $K$ that is not the restriction of any regular open cover of $X$; if such a cover has entropy exceeding that of every restrictable cover, then $\\mathrm{Ent}_n(f|_K)$ can strictly exceed $\\mathrm{Ent}_N(f)$, refuting the theorem.","tokens_in":12170,"feed_emoji":"♾️","tokens_out":10484,"duration_ms":91010,"temperature":0.7,"pith_summary":"The paper introduces a second version of topological entropy, Ent_n, defined for maps on nearly compact spaces, and proves it coincides with the previously defined Ent_N whenever the whole space is nearly compact. Its main result is that on a Hausdorff R-space — a space in which unions of regular open sets are again regular open — the entropy of a map equals the entropy of its restriction to any invariant nearly compact subset. If correct, the full dynamical complexity of such a system is already present on every invariant nearly compact piece, so adding points outside such a subset does not increase the entropy. The paper also establishes subadditivity inequalities for product systems and gives an explicit example with zero entropy.","feed_headline":"Nearly compact subsets carry the whole entropy","feed_subtitle":"On Hausdorff R-spaces, a map's entropy equals the entropy of its restriction to any invariant nearly compact subset.","key_machinery":"The core objects are the two mutually consistent entropy functions $\\mathrm{Ent}_N$ and $\\mathrm{Ent}_n$, built from the logarithm of the minimal cardinality of finite subcovers of iterated regular open joins. The load-bearing device is the cover-restriction equality of Theorem 4.3, which converts entropy of $f$ on an invariant subset $K$ into entropy of the restriction $f|_K$ on the subspace. The $R$-space hypothesis — that unions of regular open sets remain regular open — supplies the extension step that identifies the regular open covers of $K$ with restrictions of regular open covers of $X$, making the suprema interchangeable in the proof of Theorem 4.5.","core_discovery":"The central claim is Theorem 4.5(b): if $(X,f)$ is a topological $R$-dynamical system, $X$ is Hausdorff and an $R$-space, and $K\\in H(X,f)$, then $\\mathrm{Ent}_N(f)=\\mathrm{Ent}_n(f|_K)$. The argument rests on Theorem 4.3, which equates the two entropy notions with respect to a cover: $\\mathrm{Ent}_N(f,\\mathcal U,K)=\\mathrm{Ent}_n(f|_K,\\mathcal U|_K)$. The authors use the $R$-space property to extend every regular open cover of $K$ to a regular open cover of $X$ (adding $X\\setminus K$), which allows them to interchange suprema and conclude that the supremum over invariant subsets equals the entropy of the restriction. The paper also shows $\\mathrm{Ent}_n(f)=\\mathrm{Ent}_N(f)$ for nearly compact $X$ and proves product inequalities for both entropy notions.","pith_inferences":["If the equality holds for every invariant nearly compact subset, it suggests an analogue of the classical fact that entropy is supported on the non-wandering set; a natural test is whether nearly compact invariant sets are the only carriers of entropy in $R$-dynamical systems beyond Hausdorff $R$-spaces.","The $R$-space condition is strong enough that it may force every open set to be regular open; checking this could reveal that the theorem applies more broadly than its proof suggests, or that the condition can be replaced by a cover-extension axiom.","The zero-entropy example on $\\mathbb R$ hints that nearly entropy may be insensitive to expanding maps on noncompact spaces; comparing $\\mathrm{Ent}_N(f)$ and $\\mathrm{Ent}_n(f)$ on a compactification would clarify whether the notion captures genuine complexity or only compactness effects.","The product inequalities are one-sided; testing equality for mixing systems on nearly compact spaces would show whether the subadditivity is strict, mirroring the classical situation for topological entropy."],"forward_implications":["If Theorem 4.5(b) is correct, then on a Hausdorff $R$-space the topological nearly entropy of the whole system equals that of any invariant nearly compact subset, so entropy is fully localised on such pieces.","Theorem 4.4 unifies the two definitions: whenever $X$ itself is nearly compact, $\\mathrm{Ent}_n(f)=\\mathrm{Ent}_N(f)$, so the new definition is an extension rather than a competing notion.","The product inequality $\\mathrm{Ent}_n(f\\times h)\\le \\mathrm{Ent}_n(f)+\\mathrm{Ent}_n(h)$ extends the classical subadditivity of topological entropy to the nearly compact setting, and the analogous $\\mathrm{Ent}_N$ inequality holds on Hausdorff $R$-space products.","The example $f(x)=kx$ on $\\mathbb R$ winds up with $\\mathrm{Ent}_N(f)=0$, illustrating that the definition can detect simple dynamics as zero-entropy.","The product-space results suggest a route toward a generalised entropy for higher-dimensional nearly compact $R$-dynamical systems, preserving the classical subadditivity pattern."],"supporting_citations":[{"why":"Defines the original topological nearly entropy $\\mathrm{Ent}_N$ and its basic properties, including the class $H(X,f)$, which this paper extends.","marker":"[9]"},{"why":"Supplies the original 1965 definition of topological entropy for compact spaces that is being generalised to nearly compact spaces.","marker":"[1]"},{"why":"Provides the definition of nearly compact spaces and the finite-subcover criterion for regular open covers, plus the product of nearly compact spaces used in Lemma 5.4.","marker":"[12]"},{"why":"Introduces the R-map (preimage of regular open sets is regular open), the map class throughout the paper.","marker":"[5]"},{"why":"Gives topological entropy for arbitrary topological spaces, the framework that motivates entropy without separation axioms.","marker":"[10]"},{"why":"Presents topological H-entropy on H-closed spaces, a related generalisation that the present R-dynamical approach builds on.","marker":"[2]"}],"fun_headline_variants":["R-spaces make entropy restriction-invariant","Nearly compact spaces unify entropy notions","Restriction preserves topological nearly entropy","For R-spaces, entropy lives on subsets","Entropy of map equals entropy on invariant subset"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main equality assumes that every regular open cover of an invariant nearly compact subspace $K$ is the restriction of some regular open cover of $X$ with each set of the form $U_A\\cap K$ and with $X\\setminus K$ regular open; this extension property is stated without proof and is needed to equate the suprema over all covers of $X$ and of $K$.","fun_headline_variants_meta":{"raw":{"variants":["R-spaces make entropy restriction-invariant","Nearly compact spaces unify entropy notions","Restriction preserves topological nearly entropy","For R-spaces, entropy lives on subsets","Entropy of map equals entropy on invariant subset"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000601,"raw_usage":{"total_tokens":2769,"prompt_tokens":867,"completion_tokens":1902,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1837}},"tokens_in":483,"tokens_out":1902,"duration_ms":14938,"temperature":1.0,"reasoning_tokens":1837,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:53:12.264243+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The equality can be tested by finding a Hausdorff $R$-space $X$, an $R$-map $f$, and an invariant nearly compact subset $K$ with a regular open cover of $K$ that is not the restriction of any regular open cover of $X$; if such a cover has entropy exceeding that of every restrictable cover, then $\\mathrm{Ent}_n(f|_K)$ can strictly exceed $\\mathrm{Ent}_N(f)$, refuting the theorem.","supporting_citations":[{"cited_title":"Gulamsarwar and Z","cited_arxiv_id":null,"evidence_quote":"Defines the original topological nearly entropy $\\mathrm{Ent}_N$ and its basic properties, including the class $H(X,f)$, which this paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original 1965 definition of topological entropy for compact spaces that is being generalised to nearly compact spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the definition of nearly compact spaces and the finite-subcover criterion for regular open covers, plus the product of nearly compact spaces used in Lemma 5.4."},{"cited_title":"Carnahan, Some properties related to topological spa ce, Ph.D","cited_arxiv_id":null,"evidence_quote":"Introduces the R-map (preimage of regular open sets is regular open), the map class throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives topological entropy for arbitrary topological spaces, the framework that motivates entropy without separation axioms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents topological H-entropy on H-closed spaces, a related generalisation that the present R-dynamical approach builds on."}],"review_version":1}