{"id":"e9f73f12-09c9-4fe6-b8f0-491595402484","arxiv_id":"2412.00045","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For f-correlated fuzzy numbers, the alpha-cuts of the second number are f applied to the alpha-cuts of the first; completely correlated numbers preserve LR, triangular, and trapezoidal shapes.","lead":"This paper studies f-correlated fuzzy numbers, which model two uncertain quantities linked by a function f. It shows that the alpha-cuts of the second quantity are the image under f of the alpha-cuts of the first, and that linear correlation preserves LR, triangular, and trapezoidal shapes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof asserts [B]_α = f([A]_α) without deriving it from Definition 7, and the abstract's general LR-shape-preservation claim for nonlinear f is never proved; only the linear complete-correlation case is worked out.","rationale":"The reader's weakest-assumption analysis identifies the same two gaps: the unproved equality [B]_α = f([A]_α) and the missing construction showing that B is LR-type for general f. My stress-test confirms that the equality can be derived from Definition 7 only with an explicit support/domain condition, and that the paper's only worked shape-preservation results are for the linear completely correlated case. The concern is load-bearing because the abstract's main claim is precisely the general shape-preservation statement, and no proof of it appears. However, the claim is not visibly false: for monotone f, a constructive definition of L' and R' exists and can likely be supplied in a short revision. Therefore the conditional verdict is appropriate; the paper should be revised to state the domain hypothesis, derive the alpha-cut equality, and construct the new shape functions for general f.","tokens_in":5347,"tokens_out":8040,"duration_ms":76078,"concrete_test":"Take A triangular with vertices (0,0), (1,1), (2,0), so [A]_α = [α, 2−α], and take f(x) = x^2 on X = [0,∞). From Definition 7, derive B(y) = A(√y) and hence [B]_α = [α^2, (2−α)^2]. Then try to represent B as an LR-type fuzzy number, e.g. with q'_− = q'_+ = 1, a' = 1, b' = 3, and check whether shape functions L'(t) = sqrt(1−t) and a corresponding R' reproduce exactly these alpha-cuts. If the construction succeeds, the general claim is true but the paper still owes the explicit shape-function construction; if it fails, the abstract claim is false for nonlinear f. Separately, re-derive [B]_α = f([A]_α) from Definition 7 and the marginal conditions, and record whether supp A ⊆ X is needed as a missing hypothesis in Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised central claim is that f-correlation preserves LR-type shape for a general monotone injective f. The proof of Theorem 1 starts with the equality [B]_α = f([A]_α), but this equality is neither derived nor cited. It can be obtained from Definition 7 only if one adds an explicit marginal condition: the graph-supported joint distribution has marginal φ_B(f(x)) = φ_A(x), which requires supp A ⊆ X and supp B ⊆ f(X). The paper does not state or prove these domain conditions. Even if that equality is granted, Theorem 1 only locates the endpoints of the alpha-cuts of B; it does not show that B is itself LR-type. That would require constructing new shape functions L' and R' from f, L, and R (for increasing f, one needs L'(t) = L(s) where t = [f(q−) − f(q− − as)]/[f(q−) − f(q− − a)], and similarly for R'), and the paper supplies no such construction. Corollary 1 covers only linear f, and Examples 4 and 5 are exactly the complete-correlation/linear case. Thus the abstract overstates what is established: the general nonlinear shape-preservation claim is unproven as written, although it is plausibly repairable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a class of interactive fuzzy numbers called f-correlated fuzzy numbers, where the joint possibility distribution is supported on the graph of a monotone injective function f. The main claimed result (Theorem 1) is that if A is an LR-type fuzzy number and B is f-correlated to A, then the alpha-cut endpoints of B are obtained by applying f to the alpha-cut endpoints of A. The paper also states a corollary for completely correlated (linear) fuzzy numbers and gives examples for triangular and trapezoidal fuzzy numbers. The abstract further claims that f-correlation preserves LR-type shape in general. The central proof is incomplete: the equality [B]_alpha = f([A]_alpha) is assumed rather than derived, and the abstract's general LR-type claim is not proven.","tokens_in":5603,"tokens_out":7648,"duration_ms":64838,"significance":"The notion of f-correlated fuzzy numbers is a natural generalization of completely correlated fuzzy numbers, and the endpoint-transformation formula, if properly proven, would be a useful tool for computations with interactive fuzzy numbers. The paper includes concrete examples for triangular and trapezoidal fuzzy numbers that are easy to follow. However, the advertised shape-preservation theorem for general nonlinear f is not established in the current version; the proof gap is repairable, but as written the main result is not fully supported. The contribution is modest and would require additional work on shape-function construction and domain conditions to be a complete paper.","major_comments":[{"comment":"The proof begins with the assertion that [B]_alpha = f([A]_alpha) for all alpha in [0,1], but this equality is neither derived from Definition 7 nor stated as a separate lemma. The equality requires the marginal compatibility condition phi_B(f(x)) = phi_A(x) for x in the support of A, which follows from Definition 7 only if the support of A is contained in the domain X of f and if one accounts for the fact that phi_B(y) = 0 for y not in f(X). These domain conditions are not stated in Theorem 1. Because the proof of the theorem relies entirely on this equality, the proof is incomplete as written.","section":"Section 2.1, Theorem 1"},{"comment":"The abstract claims that if two fuzzy numbers are f-correlated and one is an LR-type fuzzy number, then the other is also an LR-type fuzzy number, for a general monotone injective f. Theorem 1, however, only locates the endpoints of the alpha-cuts of B as f applied to the endpoints of the alpha-cuts of A. It does not show that B admits an LR-type representation with some shape functions L' and R'. For an increasing f, one would need to construct L' and R' from f, L, and R (e.g., L'(t) = L(s) where t = [f(q-) - f(q- - as)]/[f(q-) - f(q- - a)]), and the paper supplies no such construction. The only cases where B is actually shown to be LR-type are the linear complete-correlation cases in Examples 4 and 5, and those examples cover only triangular and trapezoidal A. Thus the headline claim of the abstract is not established by the results in the paper.","section":"Abstract and Section 2.1"},{"comment":"Corollary 1 states that if A is an LR-type fuzzy number and B is completely correlated to A (linear f), then B is also an LR-type fuzzy number. No proof is given. The examples that follow cover only triangular and trapezoidal A, not general LR-type fuzzy numbers. A proof would need to handle both q > 0 and q < 0; in the decreasing case, the left branch of B uses the function R and the right branch uses L, so the conclusion that B is 'LR-type' requires either swapping the roles of L and R in the definition or redefining the shape functions. The corollary is therefore unsupported as stated.","section":"Corollary 1"}],"minor_comments":[{"comment":"The formula for R((u - q+)/b) is written as (q+ + b - u)/(q+ - u), which is negative on the interval [q+, q+ + b]; it should be (q+ + b - u)/b. Also, Definition 2 says that the closure of the support of A is [q- - a, q+ + beta], where beta should be b.","section":"Example 1"},{"comment":"The proof uses the set comprehension with the strict inequality sup C(x) > alpha, whereas the alpha-level is defined by >= alpha. For fuzzy numbers with upper semi-continuous membership the two sets coincide, but the proof as written is not faithful to the definition.","section":"Lemma 2"},{"comment":"There is a typo in 'remebember' (should be 'remember'). Also, the statement that continuous monotone injective functions map [a,b] onto [f(a), f(b)] requires that [a,b] be contained in the domain of f, which again connects to the missing domain conditions in Theorem 1.","section":"Section 2.1"},{"comment":"References [5] and [6] are duplicates of the same paper by Carlsson, Fuller, and Majlender; one should be removed.","section":"References"},{"comment":"In the hyperbolic case, the condition 0 notin [A]_0 is stated at the end, but it would be clearer to require that the support of A avoids 0 before defining the joint distribution, so that f is defined on the relevant set.","section":"Example 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is short and largely expository; much of the mathematical content is a special case of known results on completely correlated fuzzy numbers. The main interest is the extension to nonlinear f, but the missing derivation of the alpha-cut equality and the lack of a general LR shape-preservation proof make the paper unsuitable for publication in its current form. The authors should also check the reference list for duplicates and correct the typos in Example 1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my take. The headline: the paper's main theorem, as stated, is not proved. The proof of Theorem 1 begins with [B]_alpha = f([A]_alpha) and never derives that equality from Definition 7. The abstract's broader claim—that f-correlation preserves LR-type shape for any monotone injective f—is never actually established. What is proven, correctly, is the linear/complete-correlation case.\n\nCredit where it's due: the definitions are standard and clearly presented, and Examples 4 and 5 correctly work out the triangular and trapezoidal cases under complete correlation. The interval-mapping property for continuous monotone functions is used correctly. The citation of [4] for f-correlation is appropriate; the self-citation is background, not a circular move.\n\nThe soft spots are real. The missing equality is the load-bearing gap. It holds only if f is injective (assumed), the support of A lies in f's domain, and the marginal of the joint distribution for B is the pushforward of A along f. None of those conditions are stated. Even granting that equality, Theorem 1 only locates the endpoints of alpha-cuts; it doesn't show B itself is LR-type. That would require constructing new shape functions L' and R' from f, L, and R, and the paper supplies no such construction. Corollary 1 only handles linear f and has no separate proof. So the abstract overstates the result.\n\nMinor issues: Example 1 contains typos in the trapezoidal membership function (the R branch is malformed), and the prose has a few slips. These don't change the mathematical idea but reduce precision.\n\nWho is this for? Someone working on interactive fuzzy numbers might find the examples clarifying, but the novelty is low: Theorem 1 largely reduces to Definition 7 plus the cited alpha-cut property from [5]. I wouldn't cite it in my own work.\n\nRecommendation: if this crossed my desk, I'd send it to review, but with a clear expectation of major revision. The underlying idea is plausible and the topic is legitimate, so it deserves referee time; but as written it is not acceptable. Ask the authors to prove the missing equality and either prove the general shape-preservation claim or explicitly restrict the theorem to the linear case.","headline":"The main theorem isn't proven as written—it assumes the key equality it needs to derive, and the advertised general LR-shape preservation is only demonstrated in the linear case.","tokens_in":6130,"tokens_out":2639,"would_cite":false,"duration_ms":24466,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E72"],"pacs":[],"model":"deepseek-v4-flash","headline":"An f-correlated pair has every alpha-cut of one fuzzy number equal to the image under f of the other's alpha-cut, so complete correlation preserves triangular and trapezoidal shape.","keywords":["fuzzy numbers","f-correlated fuzzy numbers","interactive fuzzy numbers","LR-type fuzzy numbers","alpha-cuts","shape preservation","completely correlated fuzzy numbers","extension principle"],"falsifier":"Take A a triangular fuzzy number with support $[-1,1]$ and take $f:[0,1]\\to[1,2]$, $f(x)=x+1$, which is monotone injective and continuous; since the paper does not require the support of A to lie in f's domain, for every $\\alpha<1$ the interval $[A]_\\alpha$ contains points outside $[0,1]$, so the expression $f([A]_\\alpha)$ is undefined and the asserted equality $[B]_\\alpha=f([A]_\\alpha)$ cannot hold.","tokens_in":5150,"feed_emoji":"🔗","tokens_out":15920,"duration_ms":127792,"temperature":0.7,"pith_summary":"The paper introduces f-correlated fuzzy numbers, a class of interactive fuzzy numbers whose joint possibility distribution is concentrated on the graph of a monotone injective function f. Its central claim is that f-correlation preserves LR-type shape: if A is an LR-type fuzzy number with injective shape functions and B is f-correlated to A, then every level set, or alpha-cut, of B is the image under f of the corresponding alpha-cut of A, so the endpoints of B's alpha-cuts are f applied to the endpoints of A's. In the linear special case of complete correlation, triangular and trapezoidal fuzzy numbers stay triangular and trapezoidal. The authors point toward applications in biomathematics, where interactive fuzzy numbers appear in models of dynamical systems.","feed_headline":"One fuzzy number's alpha-cuts dictate its partner's shape","feed_subtitle":"If true, f-correlation maps every alpha-cut of A to that of B, making LR shape transfer easy to compute.","key_machinery":"The argument rides on two pieces: the $\\alpha$-cut representation of an LR-type fuzzy number, $[A]_{\\alpha}=[q^{-}-aL^{-1}(\\alpha), q^{+}+bR^{-1}(\\alpha)]$ for injective shape functions L and R, and the interval-mapping property of continuous monotone injective functions, $f([a,b])=[f(a),f(b)]$ for increasing f with the order reversed for decreasing f. Together with the f-correlated joint possibility distribution, which charges only the graph $y=f(x)$, these turn membership-level information into endpoint-level equality: $[B]_\\alpha=f([A]_\\alpha)$.","core_discovery":"The main result is Theorem 1: if A is an LR-type fuzzy number $A=(q^{-}, q^{+}, a, b)_{LR}$ with L and R injective, and B is f-correlated to A, then for every $\\alpha\\in[0,1]$ the $\\alpha$-cut of B is $[B]_{\\alpha}=f([A]_{\\alpha})=f([q^{-}-aL^{-1}(\\alpha), q^{+}+bR^{-1}(\\alpha)])$. Because a continuous monotone injective function maps intervals to intervals, the endpoints of $[B]_\\alpha$ are exactly the images of the endpoints of $[A]_\\alpha$, with order preserved if f is increasing and reversed if f is decreasing. The paper reads this as a shape-preservation result: Corollary 1 states that if B is completely correlated to A, meaning f is affine, then B is also an LR-type fuzzy number whenever A is. Examples 4 and 5 verify that complete correlation sends triangular fuzzy numbers to triangular fuzzy numbers and trapezoidal fuzzy numbers to trapezoidal fuzzy numbers.","pith_inferences":["A natural extension beyond the paper's explicit statements is to construct the shape functions of B explicitly by composing f with the inverse shape functions $L^{-1}$ and $R^{-1}$; doing so would make the abstract's claim that B is also an LR-type fuzzy number fully explicit for general f.","The endpoint-map view suggests a testable consequence: in fuzzy differential equations with f-correlated pairs, replacing the joint distribution by the endpoint map f should reproduce the same alpha-cut evolution, which a simulation could check.","The affine case offers a template for other structured correlation families: for monotone injective parametric functions such as hyperbolic maps of the form $q/x+r$, analogous closed-form alpha-cut formulas should hold whenever the support of A avoids the singularity.","At the application level, f-correlation could propagate uncertainty through a nonlinear response f while preserving the qualitative triangular or trapezoidal shape of a prior, provided the support of A lies inside f's domain."],"forward_implications":["For any f-correlated pair satisfying Theorem 1, computing the alpha-cuts of B reduces to applying f to the endpoint formulas of A, making fuzzy interval arithmetic on such pairs a one-dimensional endpoint calculation.","In the completely correlated affine case $f(x)=qx+r$, alpha-cut endpoints transform affinely, so triangular fuzzy numbers remain triangular and trapezoidal fuzzy numbers remain trapezoidal under correlation.","Shape preservation means that a model starting with an LR-type fuzzy quantity and introducing an f-correlated partner does not need to switch to a different family of membership functions for the partner.","Because the result separates the effect of f from the shape functions L and R, checking whether an f-correlation is admissible is reduced to checking monotonicity and injectivity of f on the support of A.","For increasing f the ordering of lower and upper endpoints of $[B]_\\alpha$ matches that of $[A]_\\alpha$; for decreasing f the endpoints swap, which is all the information needed for interval arithmetic."],"supporting_citations":[{"why":"defines f-correlated fuzzy numbers; Definition 7 and the hyperbolic interactive example are taken from it.","marker":"[4]"},{"why":"supplies the extension principle with joint possibility distributions and Lemma 2, the alpha-cut equality $[f_C(A_1,\\ldots,A_n)]_\\alpha=f([C]_\\alpha)$ that Theorem 1's proof relies on.","marker":"[5]"},{"why":"gives the LR-type fuzzy number representation and the alpha-cut formula $[q^{-}-aL^{-1}(\\alpha), q^{+}+bR^{-1}(\\alpha)]$ used throughout.","marker":"[10]"},{"why":"defines fuzzy numbers via alpha-levels and establishes that alpha-levels determine a fuzzy set uniquely, the framework for working at the level of alpha-cuts.","marker":"[1]"},{"why":"introduces interactive fuzzy numbers, the ambient notion that f-correlation specializes.","marker":"[9]"},{"why":"founds fuzzy set theory and the extension principle that motivates the joint-possibility construction.","marker":"[12]"}],"fun_headline_variants":["f-correlation preserves LR-type fuzzy shape","Alpha-cuts under f: LR shape is inherited","Fuzzy pairs linked by f share LR structure","Theorem: f-correlation transfers LR form","Shape lock: f-correlated fuzzy numbers stay LR"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the unstated premise that every level set, or alpha-cut, of B equals the image under f of the corresponding level set of A; that equality holds only when f is one-to-one, the support of A lies inside f's domain, and B's membership is exactly the membership of A carried along f.","fun_headline_variants_meta":{"raw":{"variants":["f-correlation preserves LR-type fuzzy shape","Alpha-cuts under f: LR shape is inherited","Fuzzy pairs linked by f share LR structure","Theorem: f-correlation transfers LR form","Shape lock: f-correlated fuzzy numbers stay LR"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001023,"raw_usage":{"total_tokens":4273,"prompt_tokens":866,"completion_tokens":3407,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":482,"completion_tokens_details":{"reasoning_tokens":3336}},"tokens_in":482,"tokens_out":3407,"duration_ms":22643,"temperature":1.0,"reasoning_tokens":3336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:07:11.350094+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take A a triangular fuzzy number with support $[-1,1]$ and take $f:[0,1]\\to[1,2]$, $f(x)=x+1$, which is monotone injective and continuous; since the paper does not require the support of A to lie in f's domain, for every $\\alpha<1$ the interval $[A]_\\alpha$ contains points outside $[0,1]$, so the expression $f([A]_\\alpha)$ is undefined and the asserted equality $[B]_\\alpha=f([A]_\\alpha)$ cannot hold.","supporting_citations":[{"cited_title":"M., Prata, R","cited_arxiv_id":null,"evidence_quote":"defines f-correlated fuzzy numbers; Definition 7 and the hyperbolic interactive example are taken from it."},{"cited_title":"”On weighted possibilistic mean and variance of fuzzy numbers.” Fuzzy Sets and Systems , vol","cited_arxiv_id":null,"evidence_quote":"gives the LR-type fuzzy number representation and the alpha-cut formula $[q^{-}-aL^{-1}(\\alpha), q^{+}+bR^{-1}(\\alpha)]$ used throughout."},{"cited_title":"C., Bassanezi, R","cited_arxiv_id":null,"evidence_quote":"defines fuzzy numbers via alpha-levels and establishes that alpha-levels determine a fuzzy set uniquely, the framework for working at the level of alpha-cuts."},{"cited_title":"”On interactive fuzzy numbers.” Fuzzy Sets and Systems , vol","cited_arxiv_id":null,"evidence_quote":"introduces interactive fuzzy numbers, the ambient notion that f-correlation specializes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"founds fuzzy set theory and the extension principle that motivates the joint-possibility construction."}],"review_version":1}