{"id":"0b1c71ed-4c21-42d3-8e41-e4dbafae2333","arxiv_id":"2412.00046","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The central theorem that correlated and standard sums coincide is invalid for decreasing correlation functions, and the proof contains a false decomposition of extrema.","lead":"The paper claims that for f-correlated fuzzy numbers, the correlated sum equals the standard sum and the correlated product is contained in the standard product. The sum claim is false for decreasing functions f, because the proof treats x and f(x) as if they could vary independently.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's correlated-sum equality fails for decreasing f; the proof's Eq. (20) to Eq. (21) separates x and f(x) in the diagonal set {x+f(x)}, and the paper's own f(x)=-x case contradicts the theorem.","rationale":"The reader's weakest assumption identifies the exact invalid step in Theorem 1's proof: decomposing the infimum and supremum of {x+f(x)} by treating x and f(x) as varying independently. The reader's verdict of REJECT is supported both by this internal proof flaw and by the contradiction between Theorem 1 and Theorem 2/Eq. (10) for f(x)=-x. No independent verification, machine-checked proof, or parameter-free derivation is present to offset the error. The correlated product containment appears sound because the diagonal set is a subset of the product set, so that portion could be salvaged, but the headline claim on sums is unsound as stated. My stress-test pass therefore does not change the reader's rejection. I also note Example 1 incorrectly states [A·id A]_0 = [0,2]; for supp A=(-2,1), the closure of {x²} is [0,4]. This strengthens the impression of unchecked computational slips, though the decisive issue remains the invalid proof of Theorem 1.","tokens_in":5423,"tokens_out":2605,"duration_ms":24409,"concrete_test":"Take a triangular fuzzy number A with α-levels [L(α), U(α)] = [α, 2-α] and f(x) = -x. Lemma 2 gives the correlated sum α-level as closure{x + f(x) : x ∈ [α, 2-α]} = {0}, hence [0,0] for every α. The paper's Eq. (10) with q=-1, r=0 gives the standard sum α-level as [L(α)+qU(α)+r, U(α)+qL(α)+r] = [2α-2, 2-2α]. At α=1/2 this is [-1,1], not [0,0]. This direct computation, using only equations stated in the paper, falsifies Theorem 1 for decreasing f.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the correlated and standard sums of f-correlated fuzzy numbers coincide. The proof of Theorem 1 reduces the correlated sum α-level to [inf{x+f(x)}, sup{x+f(x)}] over {x: φ_A(x)>α} (Eqs. 19-20), then rewrites it as [inf x + inf f(x), sup x + sup f(x)] (Eq. 21), i.e., as [A]_α + f([A]_α). This decomposition is only valid if x and f(x) can be minimized and maximized independently, which is false on the diagonal set {x+f(x)} unless compensating monotonicity conditions hold. For decreasing f it generally fails. The paper itself supplies the decisive counterexample: with f(x)=-x, Lemma 2 gives the correlated sum α-level as {x-x} = [0,0] (used in Theorem 2), while Eq. (10) with q=-1, r=0 gives the standard sum as [L(α)-U(α), U(α)-L(α)]. These coincide only in trivial cases, so Theorem 1 is internally contradicted by the paper's own formulas. The product-containment claim, obtained by including the diagonal in the product set, is not affected by this flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a class of interactive fuzzy numbers, called f-correlated fuzzy numbers, where a fuzzy number B is related to a fuzzy number A through a continuous monotone injective function f. The authors derive formulas for the alpha-levels of the correlated sum and product, A +_f B and A *f B, using the diagonal set {x, f(x)}. The main claimed result (Theorem 1) is that for any continuous monotone injective f, the correlated sum of f-correlated fuzzy numbers equals the standard (non-interactive) sum, while the correlated product is a subset of the standard product. The paper also presents special cases (linear and hyperbolic f), a counterexample showing strict containment for the product, and a theorem on additive and multiplicative inverses for correlated operations.","tokens_in":5630,"tokens_out":2631,"duration_ms":23377,"significance":"If valid, the claim that correlated sums reduce to ordinary interval arithmetic on alpha-levels would be a convenient and powerful simplification for practitioners of interactive fuzzy arithmetic. The product-containment result is plausible and correctly argued via the inclusion of the diagonal set in the Cartesian product. However, the central sum-coincidence claim is false for decreasing f, as shown by the paper's own equations. Consequently, the main theorem does not hold in the stated generality, and the paper cannot serve as a reliable reference for computing correlated sums. The concrete example distinguishing [A *id A] from [A * A] is a useful observation, but it does not rescue the paper's principal claim.","major_comments":[{"comment":"The proof of Theorem 1 invalidly replaces the infimum and supremum of the diagonal set {x + f(x) : φ_A(x) > α} with the separated expressions inf x + inf f(x) and sup x + sup f(x). This assumes that x and f(x) can be extremized independently, which is false for decreasing f. The paper itself provides the counterexample: with f(x) = -x, Eq. (15) (or Lemma 2) gives [A +_f B]_α = {x - x} = [0,0], whereas the standard sum computed in Eq. (10) with q = -1, r = 0 is [L(α) - U(α), U(α) - L(α)], a non-degenerate interval for any non-crisp A. Thus Theorem 1 is contradicted by the paper's own formulas and the claimed coincidence of correlated and standard sums is false for decreasing f.","section":"§2.3, Eq. (20)–(21)"},{"comment":"The displayed formula in Eq. (21) also contains a typo: the rightmost term is written as 'sup xf(x)' rather than 'sup f(x)'. More substantively, the passage from Eq. (20) to Eq. (21) relies on the assertion that the infimum and supremum preserve addition, which is only an inequality (inf(x+y) ≥ inf x + inf y and sup(x+y) ≤ sup x + sup y) in general; equality requires additional monotonicity or independence assumptions that are not stated and do not hold for decreasing f.","section":"§2.3, Eq. (21)"},{"comment":"Lemma 2 defines the alpha-level sets of the correlated sum and product using the condition φ_A(x) > α, whereas the paper's introductory definition of alpha-levels (for α > 0) uses φ_A(x) ≥ α, with φ_A(x) > 0 only for α = 0. This inconsistency is not harmless: for α = 1, the strict inequality excludes the core and can change the resulting interval when the supremum is attained only at membership exactly one. The proofs of Theorems 1 and 2 inherit this issue, and the paper should clarify whether the intended definition is strict or non-strict.","section":"§2, Lemma 2"},{"comment":"The formulas for f(x) = q/x + r assume x ≠ 0 and do not state the domain of f relative to the support of A. If the support of A contains 0, the expressions q/L(α) and q/U(α) are undefined or infinite. This is a gap in the derivation of Eqs. (12)–(14) and (17)–(18), and it affects the claimed generality of the results for the hyperbolic case.","section":"§2.1.2 and §2.2, hyperbolic case"}],"minor_comments":[{"comment":"The abstract contains a typo ('fuz zy' should be 'fuzzy') and overstates the result by saying 'We proved that their correlated and standard sum coincide' when, as noted above, this is false for decreasing f.","section":"Abstract"},{"comment":"References [8] and [9] are the same work by Carlsson, Fuller, and Majlender, 'Additions of completely correlated fuzzy numbers,' with identical bibliographic details; one duplicate should be removed and citations renumbered.","section":"References"},{"comment":"The product-containment proof is essentially correct, but the notation is imprecise: the set {x_1 f(x_2) : φ_A(x_1), φ_A(x_2) > α} is not a Cartesian product of two independent alpha-level sets unless the same membership condition is interpreted consistently. This is a minor presentational issue, as the intended reasoning is clear.","section":"§2.3, Eq. (24)–(25)"},{"comment":"The paper would benefit from stating explicitly where the monotonicity of f is used and from identifying the exact condition under which the sum equality holds (e.g., f increasing). Currently the text gives the impression that the result holds for all continuous monotone injective f, which is misleading.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central theorem of the paper is false for a natural and explicitly considered case (decreasing linear f), and the error is a load-bearing invalid inference in the proof. The paper's own Example/TTheorem 2, which uses f(x) = -x, directly contradicts Theorem 1. This is not a matter of presentation or a minor gap: the main claim cannot be repaired within the stated scope. The product-containment result and the explicit example showing strict containment are salvageable pieces, but they do not justify acceptance as is. I would suggest the authors reconsider whether they intend to restrict the main theorem to increasing f and verify the relevant lemmas under a consistent definition of alpha-levels."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the main theorem is false. Theorem 1 claims the f-correlated sum equals the standard sum for continuous monotone f. That equality fails for decreasing f. The proof's step from Eq. (20) to Eq. (21) replaces inf/sup of x+f(x) over the diagonal by inf x + inf f(x) and sup x + sup f(x). That treats x and f(x) as independently variable, which is only valid when the monotonicities align. For decreasing f it is generally wrong. The paper itself contains the counterexample: for f(x) = -x, Lemma 2 gives [A +_f B]_α = {0}, while Eq. (10) gives [L(α)-U(α), U(α)-L(α)]. These coincide only when A is crisp. So the central claim is internally contradicted by the authors' own formulas.\n\nWhat is actually good: the product containment, [A ·_f B]_α ⊂ [A]_α · f([A]_α), is correct and follows immediately from the diagonal being a subset of the product set. The explicit formulas for linear and hyperbolic f are routine but correct, and Theorem 2 (existence of additive and multiplicative inverses via f(x)=-x and f(x)=1/x) is a nice observation that also holds.\n\nNow the soft spots. The invalid decomposition is load-bearing; without it, Theorem 1 collapses. There is also a smaller numerical slip in Example 1: for a fuzzy number with support (-2,1), [A · id_A]_0 is {x^2 : x∈[-2,1]} = [0,4], not [0,2] as written. That indicates the example wasn't checked. The citation pattern is unobjectionable; Lemmas 1-2 come from the authors' own prior work (refs [6,7]), which is normal in a research program.\n\nBottom line: this isn't a paper you can build on. The correct fragments (product containment, inverse theorem, the formulas) could form a short note if the sum claim is removed or restricted to increasing f with a corrected proof. But as it stands, the paper is internally inconsistent, not just incomplete.\n\nRecommendation: I would not send this to a referee; desk reject is appropriate. If the authors resubmit with the comparison claim fixed or dropped, it becomes a different, much smaller paper that could merit review.","headline":"The paper's central sum-comparison theorem is false for decreasing f; the proof separates x and f(x) in {x+f(x)}, and the paper's own f(x)=-x case contradicts it.","tokens_in":6227,"tokens_out":4829,"would_cite":false,"duration_ms":41359,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E72","26E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For f-correlated fuzzy numbers, the paper claims the correlated sum equals the ordinary sum level by level, and the correlated product is contained in the ordinary product.","keywords":["fuzzy numbers","f-correlated fuzzy numbers","interactive fuzzy numbers","arithmetical operations","alpha-levels","extension principle","interval arithmetic","correlated inverses"],"falsifier":"Let $A$ be any fuzzy number with support $[1,2]$ and take $f(x)=-x$; at $\\alpha=0$ the correlated sum is $\\{0\\}$ while the standard sum is $[1,2]+[-2,-1]=[-1,1]$, so this calculation contradicts the equality asserted in Theorem 1.","tokens_in":5152,"feed_emoji":"➕","tokens_out":12439,"duration_ms":107465,"temperature":0.7,"pith_summary":"This paper is about a special class of interactive fuzzy numbers: pairs $(A,B)$ in which $B$ is obtained from $A$ by a continuous monotone injective function $f$, so the joint uncertainty of the pair is concentrated on the curve $y=f(x)$. The authors derive direct formulas for the correlated sum and product of such pairs using only real operations, interval arithmetic, and the function $f$. Their main theorem says that the correlated sum coincides with the standard fuzzy sum for every alpha-level, and that every alpha-level of the correlated product is contained in the corresponding level of the standard product. If that is right, dependent fuzzy arithmetic can be performed without constructing a full joint distribution. The paper closes by exhibiting additive and multiplicative inverses for the correlated operations.","feed_headline":"Correlated fuzzy sums match ordinary sums; products stay inside","feed_subtitle":"If right, arithmetic on linked fuzzy numbers reduces to plain interval calculations on alpha-levels.","key_machinery":"The central object is the $f$-correlated fuzzy number: two fuzzy numbers whose joint possibility distribution $\\phi_C(x,y)=\\phi_A(x)\\chi_{\\{y=f(x)\\}}$ is concentrated on the graph of a continuous monotone injective $f$, so that knowing $A$ determines $B$. The carrying identities are Lemma 1, $[B]_\\alpha=f([A]_\\alpha)$, and Lemma 2, $[A+_f B]_\\alpha=\\overline{\\{x+f(x):\\phi_A(x)>\\alpha\\}}$ and $[A\\cdot_f B]_\\alpha=\\overline{\\{xf(x):\\phi_A(x)>\\alpha\\}}$. These identities reduce correlated operations to one-dimensional sets, and Theorem 1 compares them with standard interval arithmetic by splitting infimums and supremums over sums and by embedding the diagonal set $\\{xf(x)\\}$ into the product set $\\{x_1f(x_2)\\}$.","core_discovery":"The paper's central claim is Theorem 1: for $f$-correlated fuzzy numbers $A$ and $B$ with $B=f(A)$, for every $\\alpha\\in[0,1]$, $[A+_f B]_\\alpha=[A]_\\alpha+f([A]_\\alpha)$, which the authors identify with the standard sum $[A+B]_\\alpha$; and $[A\\cdot_f B]_\\alpha\\subseteq[A]_\\alpha f([A]_\\alpha)=[A\\cdot B]_\\alpha$. The correlated product level is derived as the closure of $\\{xf(x):\\phi_A(x)>\\alpha\\}$, and the proof of containment uses the fact that this diagonal set lies inside the product set $\\{x_1 f(x_2):\\phi_A(x_1),\\phi_A(x_2)>\\alpha\\}$. The paper also gives explicit formulas for linear and hyperbolic $f$, and it proves that choosing $f(x)=-x$ gives an additive inverse while $g(x)=1/x$ gives a multiplicative inverse.","pith_inferences":["Editorial inference: the equality proof works by separating the infimum and supremum of $x+f(x)$; framing the theorem with an explicit monotonicity hypothesis on $f$ would make the scope precise, since the separation step behaves differently for increasing and decreasing functions.","Editorial inference: the product containment suggests correlated multiplication is the diagonal restriction of ordinary fuzzy multiplication, which connects directly to constrained interval arithmetic where dependency between variables is enforced at the interval level.","Editorial inference: the inverses from Theorem 2 make it natural to define correlated subtraction via $f(x)=-x$ and correlated division via $g(x)=1/x$, opening a route to fuzzy differential equations in which the derivative and the state are interactively related.","Editorial inference: for increasing monotone dependence, the sum coincidence implies that adding two such fuzzy numbers can be done level-wise with ordinary intervals, which would simplify simulation codes for models with functional dependence between uncertain parameters."],"forward_implications":["For $f(x)=qx+r$ with $q>0$, the correlated sum level is $(q+1)[A]_\\alpha+r$, so it can be produced by scaling and shifting the original level.","For $f(x)=qx+r$, the correlated product level is $q[A\\cdot_{\\mathrm{id}} A]_\\alpha+r[A]_\\alpha$, so only the product of $A$ with itself and ordinary level arithmetic are needed.","For $f(x)=q/x+r$, the correlated product is $q+r[A]_\\alpha$, a translated and scaled copy of $A$, and the correlated sum is $[A]_\\alpha+q\\overline{\\{1/x:\\phi_A(x)>\\alpha\\}}+r$.","Choosing $f(x)=-x$ gives $[A+_f B]_\\alpha=\\{0\\}$ and choosing $g(x)=1/x$ gives $[A\\cdot_g C]_\\alpha=\\{1\\}$, so correlated operations have additive and multiplicative inverses.","The inclusion $[A\\cdot_f B]_\\alpha\\subseteq[A\\cdot B]_\\alpha$ implies correlated multiplication never produces wider alpha-levels than the standard product."],"supporting_citations":[{"why":"Defines the extension principle and interval arithmetic through which standard fuzzy operations are obtained.","marker":"[1]"},{"why":"Supplies the real-analysis fact that infimum and supremum are additive, used in the proof of Theorem 1.","marker":"[5]"},{"why":"Introduces f-correlated fuzzy numbers and the alpha-level formulas for their operations.","marker":"[6]"},{"why":"Provides the f-correlated framework and the lemmas on alpha-levels that the paper takes as its starting point.","marker":"[7]"},{"why":"Gives the interactive multiplication setting against which the correlated product is compared.","marker":"[11]"},{"why":"Defines interactive fuzzy numbers, the general class that f-correlated fuzzy numbers specialize.","marker":"[12]"}],"fun_headline_variants":["Correlated fuzzy sums equal standard; products contained","Fuzzy sums match for f-linked numbers; products subset","f-correlated fuzzy arithmetic: sums equal, products inside","Linked fuzzy sums coincide; products remain within standard","Interactive fuzzy numbers: sums equal, products nested"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorem rests on assuming that the smallest and largest values of $x+f(x)$ over a level set can be obtained by adding the smallest and largest values of $x$ and $f(x)$ separately, an independence that fails when $f$ is decreasing.","fun_headline_variants_meta":{"raw":{"variants":["Correlated fuzzy sums equal standard; products contained","Fuzzy sums match for f-linked numbers; products subset","f-correlated fuzzy arithmetic: sums equal, products inside","Linked fuzzy sums coincide; products remain within standard","Interactive fuzzy numbers: sums equal, products nested"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1296,"prompt_tokens":850,"completion_tokens":446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":466,"tokens_out":446,"duration_ms":4420,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:06:05.586888+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Let $A$ be any fuzzy number with support $[1,2]$ and take $f(x)=-x$; at $\\alpha=0$ the correlated sum is $\\{0\\}$ while the standard sum is $[1,2]+[-2,-1]=[-1,1]$, so this calculation contradicts the equality asserted in Theorem 1.","supporting_citations":[{"cited_title":"C., Bassanezi, R","cited_arxiv_id":null,"evidence_quote":"Defines the extension principle and interval arithmetic through which standard fuzzy operations are obtained."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the real-analysis fact that infimum and supremum are additive, used in the proof of Theorem 1."},{"cited_title":"M., Prata, R","cited_arxiv_id":null,"evidence_quote":"Introduces f-correlated fuzzy numbers and the alpha-level formulas for their operations."},{"cited_title":"M., Prata, R","cited_arxiv_id":null,"evidence_quote":"Provides the f-correlated framework and the lemmas on alpha-levels that the paper takes as its starting point."},{"cited_title":"”On multiplication of intera ctive fuzzy numbers.” 2013 IEEE 11th International Symposium on Intelligent Systems and Informatics (SISY) , pp","cited_arxiv_id":null,"evidence_quote":"Gives the interactive multiplication setting against which the correlated product is compared."},{"cited_title":"”On interactive fuzzy numbe rs.” Fuzzy Sets and Systems , vol","cited_arxiv_id":null,"evidence_quote":"Defines interactive fuzzy numbers, the general class that f-correlated fuzzy numbers specialize."}],"review_version":1}