{"id":"8412f950-84a6-4d06-bd9f-38abb6bb7b5d","arxiv_id":"2507.04186","paper_version":1,"verdict":"UNVERDICTED","confidence":"HIGH","novelty_score":0.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A review of fractional calculus that restates standard definitions and the FALVA variational approach without new results.","lead":"This review explains fractional calculus, where derivatives and integrals of non-integer order are defined, focusing on the Riemann-Liouville and Caputo versions. It argues the topic could be added to a physics degree program, but the text contains several mathematical errors.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Semigroup proof in Eqs. (12)-(19) fails: Eq. (13) has a wrong substitution exponent and Eq. (16) a false Beta identity, so Eq. (18) cannot imply Eq. (19), undermining the review's trustworthiness.","rationale":"I read the paper as a pedagogical review whose central implicit claim is that the Riemann-Liouville and Caputo definitions and the FALVA result are presented accurately enough for classroom use. That claim requires the derivation of Eq. (19) to be valid and the integral kernels in Eqs. (23)-(26) to be correct. The weakest point is exactly the semigroup proof: the substitution error in Eq. (13) and the false Beta identity in Eq. (16) are not mere typesetting slips, because Eq. (18) then contradicts Eq. (19). The check with f=1, alpha=beta=1/2, a=0 makes the contradiction concrete. I credit the paper for citing standard references and for reproducing the known FALVA equation, and the semigroup property itself is true; the problem is that the derivation cannot be reconstructed as written. This matches the reader's weakest_assumption, so I agree with it. Since the reader already returned UNVERDICTED and my concern does not introduce a different outcome, I recommend UNCHANGED.","tokens_in":7602,"tokens_out":7305,"duration_ms":69915,"concrete_test":"Recompute Eq. (13) from Eq. (11) with t = xi + s(x - xi). The exact result is (x - xi)^(alpha+beta-1)B(alpha,beta), so Eq. (13)'s exponent alpha-beta-1 is false. Then compare Eq. (16) with the standard identity B(alpha,beta) = Gamma(alpha)Gamma(beta)/Gamma(alpha+beta). Finally, evaluate the printed Eq. (18) for a=0, f(x)=1, alpha=beta=1/2: the left side of Eq. (19) is x, while Eq. (18) gives (1/(2Gamma(1/2))) integral_0^x xi^(-1)d xi, which diverges. This one-parameter check settles that the proof as printed cannot be correct.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The semigroup property (19) is a standard true result, but the paper's proof is internally inconsistent. In Eq. (12), the substitution t = xi + s(x - xi) has Jacobian dt = (x - xi) ds, and the integrand becomes (x - xi)^(alpha+beta-1)(1-s)^(alpha-1)s^(beta-1), not the printed (x - xi)^(alpha-beta-1) with dt = (1-s)d xi. Eq. (16) states B(alpha,beta) = Gamma(alpha)Gamma(beta)/(Gamma(alpha)+Gamma(beta)); the correct denominator is Gamma(alpha+beta). Because of these two errors, Eq. (18) carries kernel (x - xi)^(alpha-beta-1) and prefactor 1/(Gamma(alpha)+Gamma(beta)), which cannot equal aI^(alpha+beta)f, whose kernel is (x - xi)^(alpha+beta-1) with prefactor 1/Gamma(alpha+beta). The claimed equality only appears if the two errors cancel, which they do not. This is not a disagreement with consensus; it is an internal contradiction of the derivation. The same kind of exponent mistake reappears in Eq. (23), where the kernel is printed as (x-t)^(n-alpha+1) instead of (x-t)^(n-alpha-1). Since the manuscript's value is pedagogical, these errors are load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a pedagogical review of fractional calculus and its applications in physics. It sketches the historical development, presents the Riemann-Liouville fractional integral and derivative and the Caputo derivative, attempts to prove the semigroup property of the RL integral, and introduces the Fractional Actionlike Variational Approach (FALVA). The paper argues that fractional calculus is a useful topic for inclusion in an undergraduate physics curriculum and points to several application areas such as anomalous diffusion, dissipative systems, and cosmology. The central content is expository: no new theorems or experimental results are presented, and the FALVA equation is a re-derivation of the known El-Nabulsi-Torres result.","tokens_in":7833,"tokens_out":4841,"duration_ms":47988,"significance":"If the equations were correct, this could serve as a compact, accessible introduction for physics students. The paper's value, however, rests entirely on the accuracy of the standard formulas it presents. The semigroup property and the FALVA equation are correct results in the literature, but the manuscript's derivations contain objective algebraic errors that invalidate the proofs as written. Because the stated goal is pedagogical implementation, these errors are load-bearing rather than cosmetic. The paper provides no original mathematical contribution and does not develop the cited applications in any detail, so its significance is modest and contingent on a thorough correction of the technical content.","major_comments":[{"comment":"The change of variables t = ξ + s(x − ξ) has Jacobian dt = (x − ξ) ds, not dt = (1 − s) dξ as stated. The integrand becomes (x − ξ)^{α+β−1}(1 − s)^{α−1} s^{β−1}, so Eq. (13) should carry the exponent α+β−1 rather than α−β−1. This error propagates into Eqs. (14), (17), and (18), making the subsequent derivation of the semigroup property invalid.","section":"Section 2.1, Eqs. (12)-(13)"},{"comment":"The Beta function identity is misstated as B(α,β) = Γ(α)Γ(β)/(Γ(α)+Γ(β)). The correct denominator is Γ(α+β). Along with the exponent error in Eq. (13), this yields Eq. (18) with kernel (x−ξ)^{α−β−1} and prefactor 1/(Γ(α)+Γ(β)), which is not the Riemann-Liouville integral of order α+β. The semigroup property (19) is true in the literature, but it is not proven by the text as written.","section":"Section 2.1, Eq. (16)"},{"comment":"The fractional integral kernel in Eq. (23) is written as (x−t)^{n−α+1} in the denominator, which is equivalent to (x−t)^{α−n−1}; the correct exponent is n−α−1 (or, equivalently, denominator exponent α−n+1). The same typo appears in Eqs. (24) and (25)-(26). Because these formulas define the Riemann-Liouville fractional derivative, the errors are not superficial and must be corrected.","section":"Section 2.2, Eqs. (23)-(26)"},{"comment":"The integration by parts in Eq. (39) is incorrect: the right-hand side should be −∫_a^t d/dτ[∂L/∂q̇ (t−τ)^{α−1}] δq dτ, not −∫_t^a, and the boundary term vanishes only after explicitly using δq(a)=δq(t)=0. In addition, Eq. (40) introduces the symbol tI_α^a f(t) without definition and changes the integration lower limit from a to 0. While the final Euler-Lagrange equation (41) is a known result, the derivation as presented is internally inconsistent and needs to be reworked.","section":"Section 3, Eqs. (39)-(40)"}],"minor_comments":[{"comment":"The text contains numerous typographical and grammatical errors, including 'Leibiniz,' 'strogly believe,' and 'revisits the unfolds who followed this questions.' These should be corrected throughout.","section":"Abstract and Introduction"},{"comment":"In property P.4, the notation aI_α^t f(x)g(x)dx is ambiguous; it should be made clear over which variable the integral acts and how the fractional integral on the right is defined.","section":"Section 2.1, Eq. (6)"},{"comment":"The definition of the space AC^n(Ω) is imprecise: the condition 'f^{(n−1)}(x) ∈ AC^n(Ω)' should likely read f^{(n−1)} ∈ AC(Ω) or a similar standard statement.","section":"Section 2.2, AC^n definition"},{"comment":"The relation between Caputo and Riemann-Liouville derivatives appears to have incorrect index shifts and arguments (e.g., the exponent k−n−α and the argument x are suspicious). Please check against a standard reference such as Samko, Kilbas, and Marichev.","section":"Section 2.3, Eq. (34)"},{"comment":"The notation δ(aI_α^t)f(t) is confusing; the variation should be applied to the action functional S_α[q](t), and the integrand should display the correct variable dependencies.","section":"Section 3, Eq. (38)"}],"recommendation":"major_revision","confidential_remarks":"The paper's novelty is low for a research journal; it is essentially a survey with known results. The main concern is that the technical errors in the core equations and derivations will mislead the intended readership of physics students if published without correction. The authors should be asked to verify every displayed formula against a standard textbook and to have the manuscript proofread. The overlap with El-Nabulsi and Torres (2008) in the FALVA section should also be acknowledged more explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is a review, and the review's own proof of the semigroup property is wrong in at least three places. That's the first thing you should know, because the paper's only value is pedagogical. The historical account and the standard definitions are fine—they're taken from the literature, and the FALVA section is a recognizable restatement of El-Nabulsi and Torres—but the derivation in Eqs. (12)–(19) does not work. The exponent in Eq. (13) should be α+β−1, not α−β−1, and the Jacobian is (x−ξ), not (1−s)dξ. Eq. (16) gives the Beta function with a false denominator, Γ(α)+Γ(β), instead of Γ(α+β). With those errors, Eq. (18) cannot equal the right-hand side of Eq. (19). This is an internal contradiction, not a disagreement with consensus. The same kind of exponent mistake appears in Eq. (23), where the kernel of the RL integral is written with n−α+1 instead of n−α−1. For a paper that is explicitly aimed at teaching, these are load-bearing flaws: a student trying to follow the proof will be misled, and the claimed semigroup property is not established by the text. The writing is also rough—grammar, typos, and notation are inconsistent throughout—but that's secondary to the math. What the paper does well is gather the history and the main definitions into one place, and the motivation for teaching fractional calculus in a physics degree is reasonable; the FALVA section, once corrected, could be a useful worked example. But as it stands, the errors are too central and too numerous to overlook. The target audience is an undergraduate or beginning graduate student looking for an entry point. For that reader, the standard textbooks (Samko, Miller and Ross) and the original El-Nabulsi–Torres paper are better and safer options. I would not send this to peer review in its current state. The corrections are mechanical, but the paper needs to be rewritten carefully before it can serve the pedagogical purpose it sets out to fulfill.","headline":"A well-intentioned review of fractional calculus undermined by a broken semigroup proof and other objective math errors; not publishable as is.","tokens_in":8442,"tokens_out":2639,"would_cite":false,"duration_ms":28873,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26A33","49K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's central claim is that Riemann–Liouville fractional integrals compose by adding their orders and that a fractional action yields a dissipative Euler–Lagrange equation, though the proof as printed contains misprinted identities.","keywords":["Riemann-Liouville","Caputo","fractional calculus","fractional integral","fractional derivative","semigroup property","Euler-Lagrange equation","FALVA"],"falsifier":"Substitute $\\alpha=\\beta=1/2$, $f(\\xi)=1$, and $a=0$ into the paper's Eq. (18): the printed formula gives $\\frac{1}{2\\sqrt{\\pi}}\\int_0^x (x-\\xi)^{-1}d\\xi$, which diverges at $\\xi=x$, while the claimed value of $I^1 1$ is $x$.","tokens_in":7321,"feed_emoji":"⚛️","tokens_out":13950,"duration_ms":140088,"temperature":0.7,"pith_summary":"This paper is a survey of fractional calculus written for physics students, moving from L'Hôpital's 1695 question about half-order derivatives through the Riemann–Liouville and Caputo operators to a variational principle called FALVA. The mathematical result it tries to establish is the semigroup law for the Riemann–Liouville fractional integral, $aI^\\alpha_x[aI^\\beta_x f]=aI^{\\alpha+\\beta}_x f$, and the physical result is a fractional Euler–Lagrange equation whose extra term plays the role of a dissipative force. If both held, fractional integral operators would compose by adding orders, and frictional systems could be derived from a fractional action. The proof of the semigroup law as printed leans on a Beta-function identity and an exponent in the change of variables; the text's versions of those two ingredients are not correct as printed, so that claim is asserted but not demonstrated in the derivation shown. A further goal is to assess whether this material fits into an undergraduate physics curriculum.","feed_headline":"Fractional integrals compose by adding orders, a review claims","feed_subtitle":"A physics-oriented review links the law to a fractional variational principle for dissipative systems.","key_machinery":"The engine is the Riemann–Liouville fractional integral $aI^\\alpha_x f(x)=\\frac1{\\Gamma(\\alpha)}\\int_a^x (x-t)^{\\alpha-1}f(t)\\,dt$, together with the claimed semigroup identity $aI^\\alpha_x[aI^\\beta_x f]=aI^{\\alpha+\\beta}_x f$. The proof is meant to reduce the double integral to a single integral via a change of variables and the Beta function identity $B(\\alpha,\\beta)=\\Gamma(\\alpha)\\Gamma(\\beta)/\\Gamma(\\alpha+\\beta)$; this is the step where the text's printed formulas go astray. The variational engine is the FALVA action, which inserts the kernel $(t-\\tau)^{\\alpha-1}/\\Gamma(\\alpha)$ into the action integral and yields the dissipative correction in Eq. (41).","core_discovery":"On its own terms, the paper's contribution is to present the Riemann–Liouville fractional integral as a semigroup of operators: applying $I^\\beta$ and then $I^\\alpha$ is the same as applying $I^{\\alpha+\\beta}$. It then derives, from the fractional action $S_\\alpha[q]=\\frac{1}{\\Gamma(\\alpha)}\\int_a^x L(\\dot q,q,\\tau)(t-\\tau)^{\\alpha-1}d\\tau$, the fractional Euler–Lagrange equation $\\frac{\\partial L}{\\partial q_i}-\\frac{d}{d\\tau}\\frac{\\partial L}{\\partial \\dot q_i}-\\frac{\\alpha-1}{t-\\tau}\\frac{\\partial L}{\\partial \\dot q_i}=0$, reading the last term as a Rayleigh-type dissipative force. It also positions the Caputo derivative as the fractional derivative whose Laplace transform takes integer-order initial conditions, which is the property that makes it attractive in physical models.","pith_inferences":["The semigroup statement itself is the standard theorem, and the textual errors are typographical, not conceptual.","A corrected derivation of Eq. (19) would go through with the standard Beta identity and the factor $(x-\\xi)^{\\alpha+\\beta-1}$ from the change of variables, so the paper's conclusion is likely right even where its printed proof is not.","Applying the FALVA equation to a damped harmonic oscillator with $L=\\frac12 m\\dot q^2-\\frac12 k q^2$ and comparing its $\\alpha\\to1$ limit to the Rayleigh-dissipation solution would be a direct numerical test of the variational claim."],"forward_implications":["If the semigroup law holds, a fractional integral of order $\\alpha+\\beta$ can be computed as two successive lower-order integrals, which justifies treating fractional integration as a one-parameter semigroup of operators.","The FALVA Euler–Lagrange equation adds a term $-\\frac{\\alpha-1}{t-\\tau}\\frac{\\partial L}{\\partial \\dot q_i}$ to the standard equation, offering a variational description of velocity-dependent dissipative forces.","The Caputo derivative's dependence on integer-order initial conditions makes fractional differential equations easier to cast in physical problems, where initial positions and velocities are usually integer-order data.","The review's scope suggests that the Riemann–Liouville and Caputo toolbox is enough to introduce fractional calculus in an undergraduate physics curriculum, with applications in anomalous diffusion, viscoelasticity, and signal processing.","Applications such as the FALVA-based dark matter model and fractional cosmological equations inherit the formalism's behavior, so the claimed semigroup property underpins the numerical and analytical treatment of those models."],"supporting_citations":[{"why":"Sonin's 1869 paper is the origin of the Riemann–Liouville fractional integral that the semigroup claim concerns.","marker":"[2]"},{"why":"Laurent's open-contour version of Cauchy's formula yields the Riemann–Liouville integral definition used in Eq. (7).","marker":"[5]"},{"why":"These two references supply the left and right Riemann–Liouville derivative definitions and the integer-order limit properties used in Section 2.2.","marker":"[6, 7]"},{"why":"Caputo's paper motivates the Caputo derivative and the integer-order initial conditions that make it physically convenient.","marker":"[9]"},{"why":"El-Nabulsi and Torres introduce the fractional actionlike variational approach that produces Eq. (41).","marker":"[10]"},{"why":"Goldstein's Classical Mechanics supplies the Rayleigh dissipation function used to interpret the fractional term in Eq. (41).","marker":"[11]"}],"fun_headline_variants":["Fractional integrals compose as semigroup, review shows","Fractional Euler-Lagrange gets dissipative term","Caputo derivative simplifies physical initial conditions","Fractional action yields Rayleigh-like dissipative term","Review links fractional calculus to dissipative dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Eq. (19) depends on the evaluation of the Beta function, Eq. (16), and on the power of $(x-\\xi)$ that emerges from the change of variables in Eq. (13); as printed, the denominator in Eq. (16) is $\\Gamma(\\alpha)+\\Gamma(\\beta)$ rather than $\\Gamma(\\alpha+\\beta)$, and the exponent is misprinted, so the semigroup law is not established by the derivation given.","fun_headline_variants_meta":{"raw":{"variants":["Fractional integrals compose as semigroup, review shows","Fractional Euler-Lagrange gets dissipative term","Caputo derivative simplifies physical initial conditions","Fractional action yields Rayleigh-like dissipative term","Review links fractional calculus to dissipative dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3215,"prompt_tokens":1018,"completion_tokens":2197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":2127}},"tokens_in":634,"tokens_out":2197,"duration_ms":16682,"temperature":1.0,"reasoning_tokens":2127,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:53:03.039291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute $\\alpha=\\beta=1/2$, $f(\\xi)=1$, and $a=0$ into the paper's Eq. (18): the printed formula gives $\\frac{1}{2\\sqrt{\\pi}}\\int_0^x (x-\\xi)^{-1}d\\xi$, which diverges at $\\xi=x$, while the claimed value of $I^1 1$ is $x$.","supporting_citations":[{"cited_title":"On differentiation with arbitrary index,Moscow Matem","cited_arxiv_id":null,"evidence_quote":"Sonin's 1869 paper is the origin of the Riemann–Liouville fractional integral that the semigroup claim concerns."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Laurent's open-contour version of Cauchy's formula yields the Riemann–Liouville integral definition used in Eq. (7)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Caputo's paper motivates the Caputo derivative and the integer-order initial conditions that make it physically convenient."},{"cited_title":"and Torres D.F.M","cited_arxiv_id":null,"evidence_quote":"El-Nabulsi and Torres introduce the fractional actionlike variational approach that produces Eq. (41)."},{"cited_title":"Classical Mechanics, Addison-Wesley Publishing Company Inc","cited_arxiv_id":null,"evidence_quote":"Goldstein's Classical Mechanics supplies the Rayleigh dissipation function used to interpret the fractional term in Eq. (41)."}],"review_version":1}