{"id":"f8d3c3ff-c22e-47f4-98fe-1aad0055479a","arxiv_id":"2507.04078","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The proposed microscopic derivation of conformable dynamics from Ginzburg-Landau disorder reduces to assumptions and contains inconsistent exponent relations.","lead":"A physics paper claims that conformable derivatives, a flexible math tool for memory and strange relaxation, emerge naturally from disordered materials near phase transitions. The derivation contains several internal contradictions that break the central claim.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Even granting the memory-kernel construction, the adiabatic step that produces the conformable form is invalid: the derived kernel K(τ) ∝ τ^{μ−1} is non-integrable, so Γeff diverges and the convolution cannot become local.","rationale":"The reader's REJECT verdict is well supported, and the most load-bearing defect is precisely the invalid adiabatic passage from the non-Markovian equation to the conformable differential equation. The reader's stated weakest_assumption, the factorization lemma in Eq. (67), is a real concern, but it is not the most decisive one: if Eq. (65) is accepted as an identity for the local fluctuation, the factorization step follows by linearity because ⟨δF/δψ(t−τ)⟩ is a deterministic macroscopic quantity; the true hidden assumption is Eq. (65) itself, which posits that the local force fluctuation is driven by the coarse-grained force rather than by local fluctuations. Even setting that aside, the derivation fails at the adiabatic stage: the derived power-law kernel with exponent μ−1 (or α−2) is non-integrable, so Γeff(T) does not exist and the convolution cannot be treated as local. This is a concrete, unambiguous internal inconsistency, independent of any statistical-independence hypotheses. The paper also contains contradictory relations among μ, α, and the critical exponents (Eqs. 11 vs. 13, and μ = α−1 vs. α = 1−μ in Section VI), but the non-integrability of the kernel is the single step that blocks the claimed emergence of conformable dynamics even if all other assumptions were granted. Therefore the verdict remains REJECT; no change from the reader's verdict is needed.","tokens_in":18336,"tokens_out":5907,"duration_ms":62915,"concrete_test":"Take the paper's derived kernel K(τ) = A τ^{μ−1} with μ ∈ (0,1] and A = const. For a constant macroscopic force F, the exact memory term is C(t) = ∫_0^t A τ^{μ−1} F dτ = A F t^μ / μ. The paper's Eq. (89) approximates this by (∫_0^∞ A τ^{μ−1} dτ) F, which is infinite for every t. Repeating with K(τ) ∝ τ^{α−2}, 1 < α < 2, gives ∫^∞ τ^{α−2}dτ diverging as well. This one check settles that Γeff is undefined and the adiabatic local replacement in Section VI is inconsistent with the derived kernel.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the conformable structure T^{1−μ} dψ/dT emerges relies on the adiabatic approximation in Section VI (Eqs. 89 and 111), where the memory convolution is replaced by Γeff(T)⟨δF/δψ(t)⟩ with Γeff(T) := ∫_0^∞ K(τ)dτ. For this replacement to be valid, two conditions must hold: K(τ) must decay faster than ψ varies, and the integral defining Γeff must converge. The paper's own derivations give K(τ) ∝ τ^{μ−1} (Eqs. 91, 108) and, from the disorder-statistics calculation, K(τ) ∝ τ^{α−2} with 1 < α < 2 (Eq. 85). In the stated physical range μ ∈ (0,1], the exponent μ−1 lies in (−1,0], so ∫^∞ τ^{μ−1}dτ diverges logarithmically or as a power. Likewise, α−2 lies in (−1,0), so ∫^∞ τ^{α−2}dτ diverges. Thus Γeff(T) is infinite and the local-in-time replacement is not justified. The non-Markovian equation is the actual result of the averaging; the conformable differential equation is obtained only through an invalid limit. This is a direct internal inconsistency, not a matter of tuning parameters or choosing a different disorder distribution within the stated assumptions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to derive conformable derivative dynamics from a microscopic disordered Ginzburg-Landau model. It starts from a local kinetic coefficient Γ(r,T)=Γ0T^{1−μ}f(r), spatially averages the resulting dynamics, and argues that quenched disorder produces a power-law memory kernel K(τ)∼τ^{μ−1}. In the adiabatic limit, this kernel is then claimed to reduce to the conformable structure T^{1−μ}dψ/dT∼⟨δF/δψ⟩. The paper also relates μ to critical exponents, barrier distributions, and Tsallis nonextensivity, and presents this as a first-principles foundation for conformable calculus.","tokens_in":18650,"tokens_out":4506,"duration_ms":47129,"significance":"If the derivation were valid, this would be a valuable contribution: it would connect a phenomenological operator, the conformable derivative, to concrete microscopic mechanisms (quenched disorder, critical slowing down, barrier statistics) and would unify several strands of the author's prior work on deformed derivatives and nonextensive thermodynamics. The manuscript is ambitious and transparently lays out its calculation steps, including explicit formulas for the memory kernel and the effective kinetic coefficient. However, the central derivation contains internal inconsistencies that are load-bearing: the mean-field exponent is miscomputed, the two identifications of the memory-kernel exponent contradict each other, the effective kinetic coefficient Γeff diverges for the stated parameter ranges, and the final 'emergence' step postulates the deformed time-temperature relation it claims to derive. These are not presentation issues; they invalidate the paper's main claim as it stands.","major_comments":[{"comment":"The definition μ=ν(z−2)−γ in Eq. (13), combined with the quoted mean-field values ν=1/2, z=2, γ=1 from Eq. (10), gives μ=−1, not μ=0. The text immediately below Eq. (13) classifies μ=0 as mean-field dynamics, and Eq. (11) states Γ(T)∝T for the same mean-field case, which would correspond to μ=0 under Γ(T)∝T^{1−μ}. The two statements are mutually inconsistent. Because this μ classification is used throughout the paper to interpret physical examples and to connect μ to the disorder exponent via Eq. (103), this sign error is not a typographical nuance but a load-bearing inconsistency.","section":"§III.A, Eqs. (9)–(13)"},{"comment":"The paper derives K(τ)∝τ^{α−2} from the disorder-averaging calculation in Eq. (85), with 1<α<2, and then states 'Comparing this with the main text relation K(τ)∼τ^{μ−1}, we obtain the identification μ=α−1.' However, in the self-consistency subsection, Eqs. (103) and (105) impose α=1−μ to recover K(τ)∝τ^{μ−1}. The two relations μ=α−1 and α=1−μ are compatible only for the special value μ=1/2. The claimed link between the disorder-statistics exponent and the conformable deformation exponent is therefore internally inconsistent, and the central identification K(τ)∼τ^{μ−1} is not established by the paper's own calculations.","section":"§VI.F and §VI (Adiabatic Limit), Eqs. (85), (103)–(108)"},{"comment":"The adiabatic replacement of the convolution by Γeff(T)⟨δF/δψ(t)⟩ requires that K(τ) decay faster than the variation of ψ and that Γeff(T)=∫0∞K(τ)dτ converge. For the stated physical range μ∈(0,1], the kernel K(τ)∝τ^{μ−1} has exponent μ−1∈(−1,0], so the integral diverges logarithmically at μ=1 and as a power law for μ<1. The alternative form K(τ)∝τ^{α−2} with 1<α<2 has exponent in (−1,0) and likewise gives a divergent Γeff. Thus Γeff(T) is not defined and the local-in-time approximation in Eqs. (89) and (111) is not justified. The power-law kernel also does not satisfy the stated 'decays much faster than ψ varies' condition. This invalidates the adiabatic step that is the paper's route to the conformable equation.","section":"§VI (Adiabatic Limit), Eqs. (89)–(92) and (111)"},{"comment":"The factorization lemma is asserted rather than derived. It requires that the macroscopic thermodynamic force ⟨δF/δψ(t−τ)⟩ be statistically independent of both δΓ(r) and G(r,τ). But the local response function defined in Eq. (73), G(r,τ)=exp(−τΓ0T^{1−μ}f(r)), is an explicit function of the same random field f(r) that defines δΓ(r) in Eq. (71). Hence δΓ(r) and G(r,τ) are strongly correlated, and the factorization ⟨δΓ(r)G(r,τ)⟨δF/δψ(t−τ)⟩⟩=⟨δΓ(r)G(r,τ)⟩⟨δF/δψ(t−τ)⟩ has no justification. Without this step, the entire memory-kernel construction in Eq. (70), and everything that follows from it, loses its derivation.","section":"§VI.A, Eq. (67)"},{"comment":"The relation dT/dt∝T^{μ−1} is introduced as a 'deformed scaling relation' without derivation from the microscopic dynamics. This postulate is exactly the ingredient needed to convert the time-evolution equation into the conformable form T^{1−μ}dψ/dT. Consequently the claimed emergence of the conformable structure is circular: the deformed time-temperature relation is assumed, not obtained from the disorder-averaging or the adiabatic limit. The paper needs either a derivation of Eq. (99) from the model or an explicit statement that this relation is an additional physical assumption.","section":"§VI, Eq. (99)"}],"minor_comments":[{"comment":"The paper introduces a coupling field g(r,T)=g0(T)(1+η(r)T^{1−μ}) and a kinetic coefficient Γ(r,T)=Γ0T^{1−μ}f(r), but never clarifies the relation between η(r) and f(r) or between the disorder correlation exponent 2μ−2 in Eq. (2) and the power-law distribution P(f)∼f^{−α} in Eq. (76). These are different disorder models that are implicitly identified through the same symbol μ.","section":"§II, Eq. (1) vs §III, Eq. (4)"},{"comment":"The assumption ⟨δ(δF/δψ)(r,t)⟩=0 in Eq. (57) is not obviously consistent with the later representation in Eq. (65), where δ(δF/δψ)(r,t) is a convolution of G(r,τ) with the generally nonzero mean ⟨δF/δψ(t−τ)⟩; the paper should clarify whether the zero-mean condition refers to a different statistical ensemble or to a separate fluctuation component.","section":"§V.B, Eqs. (57) and (65)"},{"comment":"The section numbering is inconsistent: both 'VI. Emergence of Memory Kernel' and 'VI. Adiabatic Limit' appear, so the adiabatic section should be renumbered, and the duplicated paragraph 'In this regime, In this regime' on page 23 should be corrected.","section":"Throughout"},{"comment":"The row 'Relaxation time: strongly T-dependent via T^{μ−1}' appears to have the exponent reversed relative to the paper's own kinetic coefficient T^{1−μ}; this should be fixed for consistency.","section":"Table I, p. 24"},{"comment":"Several references are cited in contexts that suggest they support specific derivations, but [3], [4], [10], and [14] are standard textbooks or reviews; the paper would benefit from more specific page or equation citations. The high number of self-citations in the conformable-sections narrative also deserves a brief statement of what is new relative to [5,6,24–26].","section":"References"}],"recommendation":"reject","confidential_remarks":"The central derivation fails on internal grounds, not because of disagreement with any external consensus. The sign error in Eq. (13), the contradictory identifications of the memory-kernel exponent, the divergent Γeff in the adiabatic limit, and the circular Eq. (99) are load-bearing and cannot be patched by local edits. A substantially reworked manuscript would need to confront these issues head-on, ideally by computing a concrete model where the memory kernel is integrable or by deriving the deformed time-temperature relation from the dynamics. I would also ask the editor to weigh the paper's overlap with the author's earlier publications on deformed derivatives; the novelty claim should be stated more precisely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the derivation does not close. The adiabatic step that produces the conformable form is invalid because the memory kernel the paper derives is non-integrable, so Γ_eff diverges and the convolution cannot become local. This follows from the paper's own equations, not from a parameter choice.\n\nWhat is actually new: the specific combination of a spatially-resolved TDGL equation, quenched disorder, spatial averaging, and an adiabatic limit as a route to conformable derivatives has not appeared in exactly this form in the cited literature. The paper also makes a genuine attempt to connect μ to disorder statistics and to Tsallis q, and the energy-barrier integral in Section IV.C that yields T^{1−μ} is standard and correctly handled. Credit where due: the ambition is real and the narrative is coherent.\n\nThe soft spots are load-bearing. Eq. (13) gives μ = ν(z−2) − γ, which with the paper's own mean-field values ν=1/2, z=2, γ=1 gives μ = −1, contradicting the claim that μ=0 corresponds to mean-field; Eq. (11) and Section IV.B give different relations for μ. The memory kernel exponent is inconsistent: Section VI.F derives K(τ) ∼ τ^{α−2} and identifies μ = α−1, while the self-consistency condition later imposes α = 1−μ; together these force α=1 and μ=0, outside the assumed 1<α<2. The factorization lemma (Eq. 67) that converts the fluctuation average into a convolution is asserted, not proven, and it is essential to the memory-kernel construction. Most damaging, the derived K(τ) ∝ τ^{μ−1} with μ ∈ (0,1] has a divergent integral, so Γ_eff is infinite and the adiabatic approximation (Eqs. 89, 111) is invalid. The non-Markovian equation is the actual result; the conformable equation is only obtained through an invalid limit. There are also organizational problems: duplicated section headings, repeated paragraphs, and typos.\n\nWho is this for? Readers sympathetic to conformable calculus may find the narrative appealing, but as a rigorous derivation it does not hold. I would not bring it to reading group, and I would not cite it. My recommendation: desk reject. The author would need to fix the μ definitions, prove the factorization, and find a regularized route to locality before this deserves referee time.","headline":"The paper's central derivation has contradictory μ definitions and an invalid adiabatic limit, so the conformable structure is not established.","tokens_in":19192,"tokens_out":6747,"would_cite":false,"duration_ms":61210,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B26","82B27","82C31","26A33","82D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The conformable derivative $T^{1-\\mu}d\\psi/dT$ is the adiabatic form of coarse-grained Ginzburg-Landau dynamics with quenched disorder, so $\\mu$ is a physical exponent set by disorder and transport.","keywords":["conformable derivative","time-dependent Ginzburg-Landau","quenched disorder","memory kernel","anomalous relaxation","critical dynamics","nonextensive thermodynamics","power-law disorder correlations"],"falsifier":"Compute the two sides of Eq. (67) in a numerical simulation of the disordered time-dependent Ginzburg-Landau equation with prescribed $P(f)\\sim f^{-\\alpha}$; if the correlation between $\\delta\\Gamma(r)G(r,\\tau)$ and the averaged thermodynamic force is nonzero at any delay, the factorization lemma fails and the derived memory-kernel and conformable structure do not follow.","tokens_in":18029,"feed_emoji":"🔬","tokens_out":9912,"duration_ms":106595,"temperature":0.7,"pith_summary":"The paper seeks to show that the conformable derivative, defined by $D_T^{(\\mu)}\\psi = T^{1-\\mu}d\\psi/dT$, is not merely a phenomenological operator but the adiabatic limit of a coarse-grained microscopic dynamics. Starting from a spatially resolved time-dependent Ginzburg-Landau equation whose kinetic coefficient has the form $\\Gamma(r,T)=\\Gamma_0 T^{1-\\mu} f(r)$, the paper averages over quenched disorder and obtains a non-Markovian memory term with kernel $K(\\tau)\\sim\\tau^{\\mu-1}$. In the slow-temperature-drive limit this convolution collapses, leaving exactly $T^{1-\\mu}d\\psi/dT\\sim\\langle\\delta F/\\delta\\psi\\rangle$. If the derivation is correct, the exponent $\\mu$ stops being a free fitting parameter: it encodes critical exponents, barrier statistics, and disorder correlations, and conformable calculus becomes a physically grounded intermediate between classical and fractional dynamics.","feed_headline":"Quenched disorder yields the conformable derivative","feed_subtitle":"Slow temperature sweeps then obey T^(1−µ) dψ/dT, with µ set by material disorder statistics.","key_machinery":"The central machinery is the decomposition of the local kinetic coefficient and thermodynamic force into spatial means plus fluctuations, followed by the factorization lemma stated in Eq. (67). That lemma lets the disorder-fluctuation term be rewritten as a convolution with $K(\\tau)\\equiv\\langle\\delta\\Gamma(r)G(r,\\tau)\\rangle$, where $G(r,\\tau)$ is a local causal response function. The power-law form of the kernel is obtained by averaging local exponential relaxation over a scale-free disorder distribution $P(f)\\sim f^{-\\alpha}$, and the adiabatic peaking of the kernel then converts the non-Markovian equation into the conformable form $T^{1-\\mu}d\\psi/dT\\sim\\langle\\delta F/\\delta\\psi\\rangle$.","core_discovery":"The central claim is that conformable relaxation dynamics emerge from microscopic disorder rather than being imposed. The argument starts with a local order-parameter dynamics $\\partial_t\\psi = -\\Gamma(r,T)\\,\\delta F/\\delta\\psi$, with $\\Gamma(r,T)=\\Gamma_0 T^{1-\\mu}f(r)$ and scale-free quenched disorder $P(f)\\sim f^{-\\alpha}$. Spatial averaging of the product $\\Gamma\\,\\delta F/\\delta\\psi$ produces a fluctuation-fluctuation correlation that, through a factorization lemma, becomes a convolution with a memory kernel $K(\\tau)=\\langle\\delta\\Gamma(r)G(r,\\tau)\\rangle$. Averaging local exponential responses over the power-law disorder distribution gives $K(\\tau)\\sim\\tau^{\\mu-1}$, and in the adiabatic limit the convolution becomes effectively local, yielding the conformable structure $T^{1-\\mu}d\\psi/dT\\sim\\langle\\delta F/\\delta\\psi\\rangle$. The paper also connects $\\mu$ to nonextensive thermodynamics through $\\mu=1/(q-1)$, so that a single deformation parameter links disorder statistics, memory, and generalized entropy.","pith_inferences":["The same spatial-average-and-factorize route would likely apply to other multiplicative-noise relaxational dynamics, so the cleanest extension is to test whether an independence lemma analogous to Eq. (67) survives outside the Ginzburg-Landau setting.","A lattice simulation of the disordered time-dependent Ginzburg-Landau equation with prescribed $P(f)$ could directly measure $K(\\tau)=\\langle\\delta\\Gamma(r)G(r,\\tau)\\rangle$ and check the predicted $\\tau^{\\mu-1}$ form, which the paper does not provide.","If the mapping to nonextensive thermodynamics is confirmed, fitting conformable relaxation to heat-capacity or neutron-scattering data would give a direct experimental read on the nonextensive parameter of a disordered material."],"forward_implications":["If the central claim holds, conformable derivative models for critical relaxation have a first-principles basis: the exponent $\\mu$ can be estimated from measured transport coefficients and disorder correlations rather than fitted.","The derivation predicts that memory kernels in disordered critical systems decay as power laws with exponent $\\mu-1$, so measured relaxation spectra should show $\\tau^{\\mu-1}$ tails controlled by the barrier-distribution exponent.","In the adiabatic limit, memory effects are absorbed into an effective kinetic coefficient with the same $T^{1-\\mu}$ scaling, so slow thermal sweeps near criticality can be described by the local conformable equation without explicitly carrying the convolution.","The relation $\\mu=1/(q-1)$ connects the deformation parameter to nonextensive entropy, making the conformable framework a dynamical complement to generalized thermodynamics."],"supporting_citations":[{"why":"Defines the conformable derivative operator that this paper seeks to ground in microscopic physics.","marker":"Khalil et al. [1]"},{"why":"Establishes the conformable calculus rules that make the operator analytically tractable, the property the derivation preserves.","marker":"Abdeljawad [2]"},{"why":"Supplies the critical-dynamics and critical-slowing-down baseline used to justify the $T^{1-\\mu}$ scaling of the kinetic coefficient.","marker":"Hohenberg and Halperin [10]"},{"why":"Provides the quenched-disorder relevance criterion that underlies the statistical role of $\\mu$ near criticality.","marker":"Harris [13]"},{"why":"Frames the fractional and memory-dynamics context that the emergent power-law kernel $K(\\tau)\\sim\\tau^{\\mu-1}$ is meant to reproduce locally.","marker":"Metzler and Klafter [14]"},{"why":"Earlier deformed-derivative thermodynamics that the paper extends by giving the operators a microscopic derivation.","marker":"Weberszpil and Helayël-Neto [5]"},{"why":"Supports the energy-barrier and nonextensive-statistics connection that fixes $\\mu$ through the relation $\\mu=1/(q-1)$.","marker":"Sotolongo-Costa and Weberszpil [24]"}],"fun_headline_variants":["Disorder roots of conformable calculus","Conformable operators from quenched noise","Microscopic order behind conformable memory","A disorder basis for fractional-like dynamics","When deformation follows from disorder"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the averaged thermodynamic force is statistically independent of both the local kinetic-coefficient fluctuations and the local response function; the paper states this factorization in its Lemma (Eq. (67)) without deriving it, and without it the memory kernel cannot be written as a convolution.","fun_headline_variants_meta":{"raw":{"variants":["Disorder roots of conformable calculus","Conformable operators from quenched noise","Microscopic order behind conformable memory","A disorder basis for fractional-like dynamics","When deformation follows from disorder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1240,"prompt_tokens":940,"completion_tokens":300,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":239}},"tokens_in":556,"tokens_out":300,"duration_ms":4387,"temperature":1.0,"reasoning_tokens":239,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:57:16.660489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two sides of Eq. (67) in a numerical simulation of the disordered time-dependent Ginzburg-Landau equation with prescribed $P(f)\\sim f^{-\\alpha}$; if the correlation between $\\delta\\Gamma(r)G(r,\\tau)$ and the averaged thermodynamic force is nonzero at any delay, the factorization lemma fails and the derived memory-kernel and conformable structure do not follow.","supporting_citations":[],"review_version":1}