{"id":"37262cda-37ae-4694-905c-5ab291dc8d52","arxiv_id":"2506.04257","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A fractional Maxwell model is applied to a non-planar strike-slip fault, giving displacement and strain solutions, but the stress evolution after fault movement is derived incorrectly.","lead":"This study builds a mathematical model of aseismic deformation around a bent, three-segment strike-slip fault in a viscoelastic Earth with fractional-order memory. It derives and plots displacement, stress, and strain before and after fault creep, but the post-creep stress formula contains an algebraic error.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Post-fault stress inverse Laplace transform in Eq. (40) is incorrect; the spurious t1 term invalidates the stress accumulation/release results.","rationale":"I read the paper as proposing an analytical extension of planar strike-slip fault models to a three-segment non-planar fault in a fractional Maxwell half-space. The Green's function / correspondence-principle setup is plausible, and the pre-fault solution (19) is at least dimensionally consistent with the stated boundary conditions. The load-bearing problem is the inversion of Eq. (24): the paper's Eq. (40) does not follow from the Laplace-domain expression. This is not a matter of a parameter choice or an alternative convention; the known Mittag-Leffler transform pair is unambiguous, and the α=1 limit exposes the extra t1 term. Because the post-fault stress formulas feed directly into the final solution (42) and all stress figures, the main geophysical conclusion—that stress accumulates before time T and is released after fault movement—is not established by this manuscript. The displacement and strain results are separable from this error, so a revised version that corrects the inversion and redoes the numerical analysis might be salvageable, but as it stands the central claim is unsupported. I agree with the reader's identification of this same weakness.","tokens_in":15689,"tokens_out":8795,"duration_ms":83064,"concrete_test":"Compute the inverse Laplace transform of s^{α-2}/(s^α+μ/η) numerically for α=0.5 and the §4 values μ=3.5×10^10 N/m^2, η=5×10^19 Pa·s using a Gaver-Stehfest or Talbot contour routine, at t1 = 1, 10, 100 years. Compare against the paper's time factor t1 + (η/μ)(1 - E_α(-μt1^α/η)) and the Mittag-Leffler factor t1 E_{α,2}(-μt1^α/η). The numerical inversion should match the latter, not the former, settling that Eq. (40) is incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (24) gives (τ̄12)_2 = μV/(2π) · s^{α-2}/(s^α + μ/η) [Φ2+Ψ2+χ2]. The standard Laplace pair L{t1 E_{α,2}(-a t1^α)} = s^{α-2}/(s^α + a) shows the correct inverse is (μV/(2π)) t1 E_{α,2}(-μt1^α/η)[Φ2+Ψ2+χ2]. Equation (40) instead contains t1 + (η/μ)(1 - E_α(-μt1^α/η)), which is not this inverse; for α=1 it reduces to t1 + (η/μ)(1-e^{-μt1/η}) rather than the correct (η/μ)(1-e^{-μt1/η}). Thus (40) has an extra linear-in-t1 term, making the post-fault stress grow without bound and generating the stress-release curves in Figs. 4–6. This error is inherited by the complete solution (42) for τ12 and τ13, and the paper's central claim about stress accumulation and release is therefore unsupported. Displacement (39) and strain (41) do not rely on this inversion and are not affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a two-dimensional model of an infinitely long, surface-breaking, non-planar strike-slip fault with three planar segments, embedded in a fractional Maxwell viscoelastic half-space representing the lithosphere-asthenosphere system. The authors use the Laplace transform, the correspondence principle, and Green's function techniques to obtain expressions for displacement, stress, and strain before and after fault movement. The pre-fault solution accumulates stress until a critical time T, after which the fault creeps with constant velocity V. Analytical results are presented in Eqs. (19), (39)-(42) and are illustrated graphically for various creep velocities and fractional orders α. The central claim is that the model captures the influence of fault geometry and fractional rheology on displacement, stress accumulation, and stress release.","tokens_in":16054,"tokens_out":7673,"duration_ms":75948,"significance":"If the derivations were correct, the paper would offer a useful extension of earlier non-planar fault models (Mondal and Debsarma 2023) to fractional Maxwell rheology, providing closed-form field expressions that could serve as benchmarks for numerical geophysical simulations. The use of standard Green's functions and the correspondence principle is appropriate, and the explicit handling of three tilted fault segments is a worthwhile generalization. However, a central mathematical error in the post-fault stress inversion invalidates the paper's primary new results, and additional inconsistencies in the numerical treatment of the critical time undermine the graphical comparisons. As submitted, the paper does not establish its main conclusions.","major_comments":[{"comment":"The inverse Laplace transform used for the post-fault stress is incorrect. From Eqs. (24) and (38), the transformed post-fault stress is (τ̄12)_2 = (μV/2π) s^{α-2}/(s^α + μ/η) [Φ2+Ψ2+χ2]. The standard Laplace pair L{t E_{α,2}(-a t)} = s^{α-2}/(s^α + a) shows that the correct inverse is (μV/2π) t1 E_{α,2}(-μ t1^α/η) [Φ2+Ψ2+χ2]. For α=1 this reduces to (ηV/2π)(1 - e^{-μ t1/η}) [Φ2+Ψ2+χ2]. The paper's expression contains an extra factor t1 in the bracket, yielding t1 + (η/μ)(1 - E_α(-μ t1^α/η)). This spurious linear term causes the post-fault stress to grow without bound rather than relax to a finite value, and it directly shapes the stress-release curves in Figs. 4-6. Equation (42) inherits the error for both τ12 and τ13, so the paper's central claim that the model describes stress accumulation and release after fault movement is unsupported.","section":"Eq. (40) and Eq. (42)"},{"comment":"The critical time T is computed only for α=0.5, where T=114.01 years, but Figures 4b and 5b plot time series for α=0.1, 0.4, 0.7, and 1.0 with apparent onset of stress release at the same T. Since the pre-fault stress in Eq. (19) depends explicitly on α through t^α terms, the time at which τ12 reaches the critical stress τc must differ for each α. If T is not recomputed per α, the comparison is physically inconsistent because the initial condition for the post-fault phase is wrong; if T is recomputed, the values should be stated. The authors' remark that 'This critical time can be different for different values of critical stress and for different values of α' does not resolve how the figures were generated.","section":"Section 4, after Eq. (43)"},{"comment":"Equation (14) for the Laplace transform of the τ13 constitutive relation has ∂/∂y2 on the right-hand side, but by symmetry with Eq. (13) it should be ∂/∂y3. As written, the expression contradicts the stress-equilibrium equation (15) and the subsequent derivation that the coefficient B in the trial solution is zero. The final τ13 result may be correct, but the typo obscures the derivation and should be corrected.","section":"Eq. (14)"}],"minor_comments":[{"comment":"The statement 'Omitting the first and higher-order derivatives of u1, as their magnitudes are negligible for 0 < α ≤ 1' is vague; please specify which approximation is being made and why the omitted terms are negligible.","section":"Section 3, heading"},{"comment":"The solution u1 in Eq. (19) has dimensions that depend on fractional powers of time unless time is implicitly dimensionless. The figures label the vertical axis 'Displacement (meter/year)' though the derived quantity appears to be a displacement (meters) or a rate; please clarify the units and the relationship between the plotted quantity and u1.","section":"Eq. (19) and Fig. 2"},{"comment":"The reference list jumps from item 23 to item 25; reference 24 appears to be missing. Please renumber the bibliography.","section":"References"},{"comment":"The comparison with a planar fault uses only segment AB of the model; this choice should be stated explicitly in the text, as the length and orientation of the planar counterpart affect the comparison.","section":"Fig. 3"},{"comment":"The notation U(t1) is used both for the trial solution in Eq. (12) and for the dislocation amplitude in Eq. (20); please use distinct symbols to avoid confusion.","section":"Section 2.5, Eq. (20)"}],"recommendation":"reject","confidential_remarks":"The post-fault stress inversion error is elementary and central; the authors should re-derive Eq. (40) using, for example, the known Laplace transform of the Mittag-Leffler function. The additional critical-time inconsistency and the typo in Eq. (14) suggest that the manuscript would need substantial revision and re-plotting before it could be reconsidered. Given that the main quantitative conclusions are invalid as they stand, I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The right call here is reject, and the stress-test note finds the real reason. The paper is a genuine extension — non-planar three-segment fault geometry (from Mondal–Debsarma) transplanted into a fractional Maxwell medium (following Mahato et al.) — and the displacement/strain part of the machinery is put together competently. The coordinate rotations, the Green's function integrals over the three segments, and the comparison against the planar case in Fig. 3 are all reasonable. Credit where due: the paper is clearly structured and the citations to prior work are honest.\n\nThe soft spot is load-bearing. The post-fault stress formula (40) is not the inverse Laplace transform of the expression in (24). The transformed stress is μV/(2π) · s^(α−2)/(s^α + μ/η) times the geometric factors, and its inverse is (μV/(2π)) t1 E_{α,2}(−μ t1^α/η)[Φ2+Ψ2+χ2], not the paper's t1 + (η/μ)(1 − E_α(...)) form. For α=1 the paper's expression adds a linear-in-time term, so the stress grows without bound instead of relaxing, and it fails to satisfy the constitutive equation (4). Since the stress-release curves in Figs. 4–6 and the complete solution (42) all inherit this term, the paper's central claim about stress accumulation and release after fault movement is unsupported.\n\nThe displacement (39) and strain (41) parts do not rely on this inversion and are probably fine. Also, the early step where the authors 'omit first and higher-order derivatives of u1' to get the governing equation is hand-waved and needs justification. And a paper that leans this heavily on plotted results should ship the MATLAB code.\n\nBottom line: this is a fixable but significant error, not a fake result. With the inverse Laplace transform corrected and the figures redrawn, the paper would be a modest but legitimate contribution for readers working on viscoelastic fault deformation with memory. For now, the published version should not stand.\n\nRecommendation: send it to peer review rather than desk-reject — the geometry/rheology combination is novel enough, and the error is specific and catchable — but the decision letter should be reject as written, with the path to revision stated.","headline":"Novel geometry-rheology combination, but the post-fault stress formula is wrong and the stress-release figures rest on that error.","tokens_in":16550,"tokens_out":5346,"would_cite":false,"duration_ms":53982,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74D05","86A15","26A33"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives analytical expressions for displacement, stress, and strain around a three-segment non-planar strike-slip fault in a fractional Maxwell viscoelastic half-space, linking fault-bend geometry to surface deformation and…","keywords":["non-planar fault","strike-slip fault","fractional Maxwell model","viscoelastic half-space","Mittag-Leffler function","Green's function technique","stress accumulation","Laplace transform"],"falsifier":"Directly verify equation (24) by evaluating the inverse Laplace transform of $\\mu s^{\\alpha}/(s^{\\alpha}+\\mu/\\eta) \\cdot V/(2\\pi s^2) \\Phi_2$. Standard Mittag-Leffler identities give $\\mu V/(2\\pi)\\, t_1 E_{\\alpha,2}(-\\mu t_1^{\\alpha}/\\eta)\\,\\Phi_2$, and for $\\alpha=1$ this equals $(\\eta V/(2\\pi))(1-e^{-\\mu t_1/\\eta})\\Phi_2$, with no linear $t_1$ term; the paper's formula (40) contains an extra $V t_1$ term. Checking whether the two agree at $\\alpha=1$ settles the claim.","tokens_in":15514,"feed_emoji":"🌍","tokens_out":10404,"duration_ms":93089,"temperature":0.7,"pith_summary":"The paper aims to establish that a three-segment, non-planar strike-slip fault in a fractional Maxwell viscoelastic half-space has an analytically tractable deformation field, with stress building quasi-statically until a critical time $T$ and then relaxing once fault creep begins. It derives explicit formulas for displacement $u_1$, shear stresses $\\tau_{12}$ and $\\tau_{13}$, and shear strains $e_{12}$ and $e_{13}$ by combining Laplace transforms, fractional calculus, the correspondence principle, and Green's functions for dislocations on the fault surface. The claimed result is that fault-bend geometry, creep velocity, and the fractional order $\\alpha$ all leave measurable signatures in surface deformation and in the rate of stress accumulation and release. If the solutions hold, geodetic observations of creep rate and strain near a bent fault could be mapped back to fault geometry and rheology.","feed_headline":"Non-planar fault stress solved analytically","feed_subtitle":"New model links fault-bend geometry, creep velocity, and fractional memory to surface deformation and stress release.","key_machinery":"The machinery is a boundary-value problem for a quasi-static strike-slip dislocation. The constitutive law is the fractional Maxwell relation $(1/\\eta)\\tau + (1/\\mu) D^{\\alpha}_t \\tau = D^{\\alpha}_t(\\partial u_1/\\partial y_2)$, with an analogous equation for $\\tau_{13}$, using the Caputo fractional derivative of order $0<\\alpha\\le 1$. The fault is a surface of displacement discontinuity split into three planar segments with local coordinate rotations; Green's functions $G_{12}$ and $G_{13}$ give the displacement response to a point dislocation, and the correspondence principle converts the viscoelastic problem into an elastic one in the Laplace domain. The inverse Laplace transform of the resulting $s$-domain stress is what produces the time-dependent Mittag-Leffler factors in the final solution, and therefore carries the rheological content of the paper.","core_discovery":"The central claim is that the complete inter-seismic and post-fault solution is $u_1 = (u_1)_1 + (u_1)_2$, $\\tau_{12} = (\\tau_{12})_1 + (\\tau_{12})_2$, $\\tau_{13} = (\\tau_{13})_1 + (\\tau_{13})_2$, $e_{12} = (e_{12})_1 + (e_{12})_2$, $e_{13} = (e_{13})_1 + (e_{13})_2$, where the pre-fault part relaxes through the Mittag-Leffler function $E_{\\alpha}(-\\mu t^{\\alpha}/\\eta)$ while tectonic loading grows linearly, and the post-fault part is obtained from a dislocation $U(t_1)=V t_1$ on the fault starting at $t_1=t-T$. The paper asserts, in particular, that the post-fault shear stress has the form $(\\tau_{12})_2 = \\mu V H(t_1)/(2\\pi) [ t_1 + (\\eta/\\mu)(1 - E_{\\alpha}(-\\mu t_1^{\\alpha}/\\eta)) ] (\\Phi_2 + \\Psi_2 + \\chi_2)$. The geometry enters through the integrals $\\Phi_1, \\Psi_1, \\chi_1$ built from the three planar segments inclined at $\\theta_1, \\theta_2, \\theta_3$, and their derivatives $\\Phi_2, \\Psi_2, \\chi_2, \\Phi_3, \\Psi_3, \\chi_3$. The claimed effect is that higher creep velocity produces stronger and wider stress and strain peaks near the fault, lower $\\alpha$ produces slower stress relaxation, and non-planarity creates asymmetric surface displacement compared with a planar fault.","pith_inferences":["Beyond the paper, the same additive Green's-function decomposition could be applied to bent faults with more than three segments, so the method generalizes to arbitrary piecewise-planar fault traces.","The predicted surface-displacement asymmetry could be tested with GPS or InSAR profiles across a known stepped or bent strike-slip fault, where the model's peak location and scale would be compared directly.","If the fractional Maxwell rheology is replaced by a more complex two-stage fractional rheology, the correspondence-principle structure should remain, but the Mittag-Leffler relaxation factors would change into combinations of two relaxation functions."],"forward_implications":["Before fault movement, the stress $\\tau_{12}$ approaches the linearly growing tectonic value $\\tau_{\\infty}(0)(1+kt)$, with the transient term $((\\tau_{12})_0-\\tau_{\\infty}(0))E_{\\alpha}(-\\mu t^{\\alpha}/\\eta)$ decaying according to the fractional order; lower $\\alpha$ delays the approach and shifts the critical time $T$.","After fault creep begins at $T$, the displacement grows linearly as $V t_1 H(t_1)$ times the geometry factors, so surface creep rate is predicted to be proportional to $V$ and concentrated near the fault trace.","The post-fault stress release is larger for larger creep velocity $V$ and for $\\alpha$ closer to 1, giving a testable ordering of relaxation rates across fractional orders.","Non-planar faulting produces an asymmetric surface-displacement profile and sharper peaks near the fault; comparing such profiles to planar-fault solutions isolates the geometric contribution."],"supporting_citations":[{"why":"Supplies the quasi-static equilibrium equation and the strike-slip assumption that only $u_1$, $\\tau_{12}$, $\\tau_{13}$, $e_{12}$, $e_{13}$ are nonzero.","marker":"Mukhopadhyay et al. (1980)"},{"why":"Gives the fractional Maxwell constitutive relations used throughout the model.","marker":"Debnath (2003)"},{"why":"Defines the Caputo fractional derivative used for the memory kernel.","marker":"Caputo (1969)"},{"why":"Provides the Green's function technique for a dislocation surface that yields the integral representation of the post-fault displacement.","marker":"Maruyama (1966)"},{"why":"Supplies the correspondence principle that lets the viscoelastic problem be solved in the Laplace domain.","marker":"Rybiki (1968)"},{"why":"Provides the three-segment non-planar fault geometry, parameter values, and the integer-order comparison baseline.","marker":"Mondal and Debsarma (2023)"},{"why":"Supplies parameter ranges and the planar-fault fractional model against which displacement and strain orders are compared.","marker":"Mahato et al. (2022)"}],"fun_headline_variants":["Fractional Maxwell memory sharpens fault-bend stress","Non-planar fault creep: fractional memory in action","Fault-bend geometry meets fractional viscoelasticity","Creep velocity and memory shape non-planar fault stress","Fractional model links fault bends to stress peaks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the inverse Laplace transform that converts the post-fault stress from the Laplace domain into the time-domain formula (40); if that transform is incorrect, the stress-release curves in Figures 4–6 and the final solution (42) do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Maxwell memory sharpens fault-bend stress","Non-planar fault creep: fractional memory in action","Fault-bend geometry meets fractional viscoelasticity","Creep velocity and memory shape non-planar fault stress","Fractional model links fault bends to stress peaks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000575,"raw_usage":{"total_tokens":2792,"prompt_tokens":1103,"completion_tokens":1689,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":1612}},"tokens_in":719,"tokens_out":1689,"duration_ms":12726,"temperature":1.0,"reasoning_tokens":1612,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:39:55.718585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly verify equation (24) by evaluating the inverse Laplace transform of $\\mu s^{\\alpha}/(s^{\\alpha}+\\mu/\\eta) \\cdot V/(2\\pi s^2) \\Phi_2$. Standard Mittag-Leffler identities give $\\mu V/(2\\pi)\\, t_1 E_{\\alpha,2}(-\\mu t_1^{\\alpha}/\\eta)\\,\\Phi_2$, and for $\\alpha=1$ this equals $(\\eta V/(2\\pi))(1-e^{-\\mu t_1/\\eta})\\Phi_2$, with no linear $t_1$ term; the paper's formula (40) contains an extra $V t_1$ term. Checking whether the two agree at $\\alpha=1$ settles the claim.","supporting_citations":[{"cited_title":"Bulletin Society of Earthquake Technology 17(1):1–10","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-static equilibrium equation and the strike-slip assumption that only $u_1$, $\\tau_{12}$, $\\tau_{13}$, $e_{12}$, $e_{13}$ are nonzero."},{"cited_title":"International Journal of Mathematics and Mathema tical Sciences 2003(54):3413–3442","cited_arxiv_id":null,"evidence_quote":"Gives the fractional Maxwell constitutive relations used throughout the model."}],"review_version":1}