{"id":"d410dbab-f63c-4a87-94db-9a1527a82005","arxiv_id":"1908.02115","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rough I-convergence and rough I*-convergence are defined in cone metric spaces, and the two are shown equivalent when the ideal satisfies the (AP) condition, though the paper's counterexample for the converse is flawed.","lead":"This paper defines a flexible notion of approximate convergence, called rough ideal convergence, in cone metric spaces where distances are vectors. It also introduces a companion star version and proves they coincide under a standard condition on the ideal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 3.2's counterexample is invalid: the sequence is actually rough I*-convergent for every positive roughness degree to r*/2, so the claimed failure without (AP) and the 'iff' in Corollary 3 are not established.","rationale":"The reader's weakest_assumption identifies the normality restriction in Theorems 3.17-3.18 as the main risk, but that is a stated hypothesis rather than a logical gap. The genuinely load-bearing weakness is Example 3.2: it is the only evidence that rough I-convergence can occur without rough I*-convergence when (AP) fails, and it is demonstrably false. For any positive roughness degree r, the set M = union_{j>=q} D_j with q large enough belongs to F(I) and makes the subsequence satisfy the rough-convergence condition uniformly, so the full sequence is rough I*-convergent to r*/2. The positive implications (Theorems 3.11 and 3.13) appear correct, and Corollary 3's 'iff' is not a consequence of them; in fact, the 'only if' is false for constant sequences. Thus the paper's central advertised 'relationship' is incomplete, but the flaw is local and can be repaired by removing or replacing Example 3.2 and weakening Corollary 3. This is consistent with the reader's CONDITIONAL verdict, so I leave the verdict unchanged.","tokens_in":15264,"tokens_out":15603,"duration_ms":162660,"concrete_test":"Verify the claimed counterexample numerically: fix r with r* = 0.01, so x* = 0.005. Choose q = 67 (so 1/67 <= 0.015 = 1.5 r*) and let M = union_{j>=67} D_j. Check that M is in F(I) and that every m in M satisfies |x_m - 0.005| <= 0.01, hence d(x_m,x*) << r+epsilon for every epsilon >> 0. If this holds—as it does—the assertion that the sequence is not rough I*-convergent to r*/2 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised separation of rough I- from rough I*-convergence without (AP) rests entirely on Example 3.2, and that example is false. With the ideal I and sequence x_n defined there, fix any r in intP and put x* = r*/2, where r* = min(r1,r2). Choose q in N with 1/q <= (3/2)r* and set M = union_{j>=q} D_j. Then M is in F(I) because N\\M = D_1 union ... union D_{q-1} is in I. For every m in M, m belongs to some D_j with j >= q, so x_m = 1/j and |x_m - x*| <= r* (since 1/j <= 1/q <= (3/2)r*). Hence d(x_m,x*) = (|x_m-x*|, |x_m-x*|) <= (r*, r*) <= (r1+epsilon1, r2+epsilon2) for every epsilon_i > 0, i.e. d(x_m,x*) << r+epsilon. Thus the subsequence {x_m}_{m in M} is rough convergent of degree r to x* (the condition holds for all m), so by Definition 3.4 the full sequence is rough I*-convergent of degree r to x*. This directly contradicts the conclusion of Example 3.2 for positive r. Consequently Corollary 3's 'only if' is unsupported; as stated it is also false, since for a constant sequence both limit sets coincide for any ideal without (AP).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the theory of rough I-convergence to cone metric spaces. It defines rough I-convergence and rough I*-convergence of sequences of roughness degree r, studies basic properties such as I-boundedness, limit sets, cluster points, and subsequences, and proves that rough I*-convergence implies rough I-convergence (Theorem 3.11) and that under the (AP) condition on the ideal rough I-convergence implies rough I*-convergence (Theorem 3.13). It then claims, via Example 3.2, that without (AP) the two notions differ, and concludes in Corollary 3 that the rough I-limit set equals the rough I*-limit set if and only if the ideal satisfies (AP). Additional results under a normality assumption on the cone (Theorems 3.17 and 3.18) relate rough I-convergence to I-convergence of the distance sequence in the Banach space.","tokens_in":15626,"tokens_out":12609,"duration_ms":131287,"significance":"If the main claims were correct, the paper would provide a useful extension of known rough convergence and ideal convergence results to cone metric spaces, and the (AP)-based equivalence would clarify the relationship between rough I- and rough I*-convergence in a general setting. The paper also has genuine strengths: Theorems 3.11 and 3.13 appear correct, the use of the standard filter/ideal machinery is appropriate, and several of the auxiliary results (e.g., Theorems 3.2, 3.5, 3.7, 3.9) are routine but correct. However, the central advertised separation of rough I-convergence from rough I*-convergence without (AP) rests on Example 3.2, and that example is false. Moreover, Corollary 3's 'only if' assertion is false as stated. The paper therefore does not establish its main conceptual contribution, and the correctness of the remaining positive results is not enough to compensate for this.","major_comments":[{"comment":"The counterexample is invalid, and in fact the sequence defined there is rough I*-convergent to x* = r*/2 for every r = (r1,r2) with r* = min(r1,r2) > 0. Choose q ∈ N such that 1/q ≤ (3/2)r*, and set M = ⋃_{j≥q} D_j. Since N∖M = D_1 ∪ ⋯ ∪ D_{q-1} belongs to I, we have M ∈ F(I). For every m ∈ D_j ⊆ M with j ≥ q, x_m = 1/j and |x_m − x*| ≤ r*, so d(x_m,x*) = (|x_m−x*|, |x_m−x*|) ≤ (r*, r*) ≤ (r1+ε1, r2+ε2) for every ε = (ε1,ε2) ∈ intP. Thus the subsequence {x_m}_{m∈M} is rough convergent of degree r to x*, meaning the full sequence is rough I*-convergent to x* by Definition 3.4. This directly contradicts the conclusion of Example 3.2. Additionally, the proof changes r after choosing M: the integer p comes from H = N∖M ∈ I, and then r is chosen so that r* = 1/(3(p+1)). That is not a valid negation of rough I*-convergence for a fixed roughness degree r.","section":"Example 3.2, Section 3"},{"comment":"The 'if and only if' statement is false. The 'if' direction follows from Theorems 3.11 and 3.13, but the 'only if' direction is unsupported once Example 3.2 is discarded, and it is in fact false. Consider a constant sequence x_n = x0 in any cone metric space with any admissible ideal I, whether or not I satisfies (AP). For every fixed r with 0 << r, the rough I-limit set and the rough I*-limit set both coincide: a point y is in either set exactly when d(x0,y) << r+ε for every ε ∈ intP. For instance, the set M = N ∈ F(I) witnesses rough I*-convergence, while the defining condition for rough I-convergence is independent of the ideal because the relevant bad set is either empty or all of N. Hence equality of the two limit sets holds for constant sequences for every admissible ideal, including ideals without (AP), contradicting the claimed equivalence.","section":"Corollary 3, Section 3"}],"minor_comments":[{"comment":"The proof of Theorem 3.14 chooses a set L that depends on ε and then asserts rough I*-convergence of the subsequence; to establish I*-convergence one would need a single set M ∈ F(I) working for all ε. The theorem's conclusion is nevertheless correct, because the displayed inclusion directly verifies Definition 3.1 for the subsequence.","section":"Theorem 3.14"},{"comment":"In the proof of Theorem 3.2, 'there exists a y ∈ x' should read 'there exists a y ∈ X'.","section":"Theorem 3.2 proof"},{"comment":"In the proof of Theorem 3.18, the symbol x is used both for the limit point of {x_n} and for an arbitrary vector in intP in the definition of c; this is confusing and should be replaced by a distinct symbol such as u.","section":"Theorem 3.18 proof"},{"comment":"In Definition 3.2, the expression 'M − d(, x_n)' contains a stray comma and should read 'M − d(x_n, y)'.","section":"Definition 3.2"},{"comment":"There are numerous typos and grammatical slips (e.g., 'devolopments', 'remakable', 'fields' in the introduction) that should be corrected in a revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The decision is driven by the invalid Example 3.2 and the false Corollary 3, which are load-bearing for the paper's main claimed contribution. The positive results Theorems 3.11 and 3.13 appear sound and could form the basis of a substantially revised manuscript if the authors can supply a valid example separating the two notions without (AP) and correct the corollary accordingly. I see no reason to question the novelty or scope; the issue is mathematical correctness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper introduces rough I-convergence and rough I*-convergence for sequences in cone metric spaces, then proves that I* implies I, and that with (AP) I implies I*. Those definitions are new in this setting, and the main positive results are routine but correct translations of the normed-space theory. Theorems 3.5, 3.7, 3.8, 3.9, 3.11, 3.13, 3.17 and 3.18 all check out on a careful reading. They are not deep, but they are honest, standard work.\n\nThe serious problem is Example 3.2, which is meant to show that rough I-convergence does not imply rough I*-convergence when the ideal lacks (AP). The example is invalid. With the ideal and sequence as defined there, for any fixed positive roughness degree r, take x* = r*/2 and M = union_{j>=q} D_j where 1/q <= (3/2)r*. Then M lies in F(I), and on M the sequence is within r of x*. So the sequence is actually rough I*-convergent of degree r to x*, contradicting the example. The authors choose the roughness degree after the index p that depends on the supposed M, rather than keeping r fixed. This is not a minor typo: it removes the only evidence for the claimed separation without (AP). Corollary 3's \"only if\" direction is also false as a general statement, since a constant sequence has equal limit sets for any ideal.\n\nThere is a smaller caveat: Theorem 3.18 and related results assume a normal cone with normal constant K, and the proofs genuinely use that assumption. That is fine as an explicit hypothesis, but it should be flagged more clearly.\n\nThe citation pattern is acceptable. The paper leans on the authors' own earlier work [7] and on standard sources [9,12,22], but the new claims are not circular and the self-citations are legitimate background.\n\nWho is this for? Specialists in summability theory and cone metric fixed point work. It is a niche extension, not a major breakthrough, and the main proof idea is borrowed from earlier rough convergence literature. Still, it deserves a serious referee rather than a desk reject, because the definitions are reasonable and the positive theorems are sound. A referee should ask for a corrected example or a restated corollary. I would not cite it in my own work, but it is citable once the example is fixed.","headline":"The AP equivalence theorems are correct, but the one counterexample that is supposed to separate rough I- from rough I*-convergence does not actually work, and Corollary 3 is false as stated.","tokens_in":16145,"tokens_out":3946,"would_cite":false,"duration_ms":40341,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["40A05","40A99"],"pacs":[],"model":"deepseek-v4-flash","headline":"In cone metric spaces, rough I-convergence and rough I*-convergence coincide if and only if the ideal satisfies the (AP) condition, with Example 3.2 showing the condition is needed.","keywords":["cone metric spaces","rough convergence","rough I-convergence","rough I*-convergence","(AP) condition","ideal convergence","statistical convergence"],"falsifier":"Run the construction of Example 3.2 with an ideal that does satisfy (AP), using a cone metric space with a non-normal cone: if a sequence is rough $I$-convergent without being rough $I^*$-convergent, then Theorem 3.13 and Corollary 3 are false. Conversely, computing the rough $I$- and $I^*$-limit sets in Example 3.2's partition ideal and finding them equal would undermine the claimed necessity of (AP).","tokens_in":15061,"feed_emoji":"📐","tokens_out":8652,"duration_ms":80812,"temperature":0.7,"pith_summary":"The paper extends rough convergence to cone metric spaces, where the distance between points is a vector in a real Banach space ordered by a cone rather than a real number. It defines two ideal-based notions: rough I-convergence, which requires the set of indices where the distance from the limit exceeds $r+\\varepsilon$ to belong to an ideal $I$, and rough $I^*$-convergence, which requires a subsequence indexed by a set in the associated filter to be roughly convergent. The paper's central claim is that rough $I^*$-convergence implies rough $I$-convergence in general, and that if the ideal satisfies the (AP) condition the converse holds, so the rough $I$-limit set equals the rough $I^*$-limit set. It also characterizes $I$-bounded sequences by nonempty rough $I$-limit sets and, under a normal-cone assumption, translates rough $I$-convergence into ideal convergence of the distance sequence.","feed_headline":"Rough I and I* limits agree when the ideal has (AP)","feed_subtitle":"Under the (AP) condition, rough I-convergence and rough I*-convergence share the same limit set in cone metric spaces.","key_machinery":"The machinery is the pair consisting of an admissible ideal $I$ of subsets of $\\mathbb{N}$ and its associated filter $F(I)$, together with the (AP) condition on $I$. Rough $I$-convergence is defined directly through membership of the 'bad index' set in $I$, while rough $I^*$-convergence is defined through a filter set $M$ on which the subsequence is roughly convergent. The (AP) condition is the mechanism that lets the proof of Theorem 3.13 pass from a countable family of filter sets $A_i$, each capturing indices where $d(x_n,x^*)$ is within $r+\\ell/i$, to a single filter set $B$ on which the tail lies within $r+\\varepsilon$; Lemma 3.12 supplies exactly this countable-intersection step. In the final theorems, the normality constant $K$ of a normal cone converts cone inequalities into norm inequalities, which lets the paper identify rough $I$-convergence with ideal convergence of $\\{d(x_n,x^*)-r\\}$ and prove the distance-sequence continuity result.","core_discovery":"The central discovery is an equivalence between two notions of rough limit under property (AP). In a cone metric space $(X,d)$, a sequence $x=\\{x_n\\}$ is rough $I$-convergent of degree $r$ to $x^*$ if for every $0\\ll\\varepsilon$ the index set $\\{n: r+\\varepsilon-d(x_n,x^*)\\notin\\operatorname{int}P\\}$ lies in $I$, and rough $I^*$-convergent of degree $r$ if some $M\\in F(I)$ indexes a subsequence that is roughly convergent to $x^*$. Theorem 3.11 shows $I^*$ always implies $I$. Theorem 3.13 shows that when $I$ satisfies (AP), rough $I$-convergence of degree $r$ to $x^*$ implies rough $I^*$-convergence of the same degree, so Corollary 3 identifies the rough $I$-limit set $I-\\operatorname{LIM}^r_x$ with the rough $I^*$-limit set $I^*-\\operatorname{LIM}^r_x$ exactly under (AP). Example 3.2 demonstrates that the (AP) hypothesis is needed: with a partition-based ideal without (AP), a sequence can be rough $I$-convergent to a whole interval of limits while failing to be rough $I^*$-convergent to one of them.","pith_inferences":["Because Example 3.2 is built from the partition ideal that fails (AP), one testable extension is to check whether the same gap between $I$ and $I^*$ limits can be produced with an ideal satisfying (AP) but with a non-normal cone; if so, Theorems 3.17 and 3.18 would not survive without normality.","The translation of rough $I$-convergence into ideal convergence of the distance sequence suggests that known results for ideal convergence of real sequences could be imported directly to prove geometric facts about rough-limit sets in normal-cone metric spaces.","One implicit consequence of Corollary 3 is that the inequality $I^*-\\operatorname{LIM}^r_x \\subseteq I-\\operatorname{LIM}^r_x$ is strict exactly in the absence of (AP), so the size of the gap between the two limit sets could serve as a quantitative measure of how far an ideal is from having (AP)."],"forward_implications":["In every cone metric space, rough $I^*$-convergence of degree $r$ implies rough $I$-convergence of degree $r$, so $I^*-\\operatorname{LIM}^r_x$ is always a subset of $I-\\operatorname{LIM}^r_x$.","When $I$ satisfies (AP), the two limit sets coincide, and the rough $I$-limit set is nonempty if and only if the sequence is $I$-bounded.","Rough $I$-limit sets are monotone in the roughness degree: if $r_1>r$ then $I-\\operatorname{LIM}^r_x \\subset I-\\operatorname{LIM}^{r_1}_x$.","If a sequence is rough $I$-convergent of degree $r$, every subsequence inherits those rough $I$-limits under the same ideal.","In a cone metric space with normal cone of constant $K$, a sequence is rough $I$-convergent of degree $r$ to $x$ exactly when $\\{d(x_n,x)-r\\}$ is $I$-convergent to $0$ (when the shifted distances remain in $P$), and rough $I$-convergence of $x_n$ and $y_n$ at degree $r/(4K+2)$ forces $\\{d(x_n,y_n)\\}$ to be rough $I$-convergent to $d(x,y)$ with degree $\\|r\\|$."],"supporting_citations":[{"why":"Introduces cone metric spaces, the cone ordering, and the convergence definition used throughout the paper.","marker":"[18]"},{"why":"Introduces rough convergence of sequences and the notion of roughness degree $r$ that the paper adapts to cone metric spaces.","marker":"[23]"},{"why":"Defines rough $I$-convergence in normed linear spaces, serving as the template for Definition 3.1.","marker":"[9]"},{"why":"Supplies the definitions of $I$-convergence and $I^*$-convergence in cone metric spaces that the rough versions extend.","marker":"[25]"},{"why":"Introduces ideal convergence and the (AP) condition that is central to Theorems 3.11 through 3.13.","marker":"[12]"},{"why":"Establishes rough convergence in cone metric spaces and provides the lemmas on normal cones used in Theorems 3.17 and 3.18.","marker":"[7]"},{"why":"Provides Lemma 2.2, the Archimedean property used in the proof of Theorem 3.13.","marker":"[15]"},{"why":"Supplies Lemma 3.12, the countable-filter-intersection result that turns (AP) into a single filter set in Theorem 3.13.","marker":"[22]"}],"fun_headline_variants":["Rough I and I* convergence align under (AP)","When ideals satisfy (AP), rough limits coincide","Equivalence of rough I and I* limits under (AP)","Property (AP) unifies rough I and I* convergence","Under (AP) rough I and I* limits match"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equivalence between rough $I$-convergence and rough $I^*$-convergence depends on the ideal satisfying the (AP) condition, and the distance-sequence theorems further assume the cone is normal with a known constant; if either hypothesis fails, the paper's arguments do not establish the stated results.","fun_headline_variants_meta":{"raw":{"variants":["Rough I and I* convergence align under (AP)","When ideals satisfy (AP), rough limits coincide","Equivalence of rough I and I* limits under (AP)","Property (AP) unifies rough I and I* convergence","Under (AP) rough I and I* limits match"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000673,"raw_usage":{"total_tokens":3033,"prompt_tokens":885,"completion_tokens":2148,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2067}},"tokens_in":501,"tokens_out":2148,"duration_ms":14819,"temperature":1.0,"reasoning_tokens":2067,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:54:45.777485+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the construction of Example 3.2 with an ideal that does satisfy (AP), using a cone metric space with a non-normal cone: if a sequence is rough $I$-convergent without being rough $I^*$-convergent, then Theorem 3.13 and Corollary 3 are false. Conversely, computing the rough $I$- and $I^*$-limit sets in Example 3.2's partition ideal and finding them equal would undermine the claimed necessity of (AP).","supporting_citations":[{"cited_title":"Long-Guang and Z","cited_arxiv_id":null,"evidence_quote":"Introduces cone metric spaces, the cone ordering, and the convergence definition used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces rough convergence of sequences and the notion of roughness degree $r$ that the paper adapts to cone metric spaces."},{"cited_title":"D¨ undar and C","cited_arxiv_id":null,"evidence_quote":"Defines rough $I$-convergence in normed linear spaces, serving as the template for Definition 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definitions of $I$-convergence and $I^*$-convergence in cone metric spaces that the rough versions extend."},{"cited_title":"26(2), 2000/2001, pp.- 669-686","cited_arxiv_id":null,"evidence_quote":"Introduces ideal convergence and the (AP) condition that is central to Theorems 3.11 through 3.13."},{"cited_title":"Rough convergence of sequences in a cone metric space","cited_arxiv_id":"1805.10257","evidence_quote":"Establishes rough convergence in cone metric spaces and provides the lemmas on normal cones used in Theorems 3.17 and 3.18."},{"cited_title":"Khani, M","cited_arxiv_id":null,"evidence_quote":"Provides Lemma 2.2, the Archimedean property used in the proof of Theorem 3.13."},{"cited_title":"Nabiev, S","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.12, the countable-filter-intersection result that turns (AP) into a single filter set in Theorem 3.13."}],"review_version":1}